mathematical physics, analysis and geometry - volume 11

Math Phys Anal Geom (2008) 11:1–9DOI 10.1007/s11040-008-9037-8

On the Flag Curvature of Invariant Randers Metrics

Hamid Reza Salimi Moghaddam

Received: 10 December 2007 / Accepted: 6 February 2008 /Published online: 21 March 2008© Springer Science + Business Media B.V. 2008

Abstract In the present paper, the flag curvature of invariant Randers metricson homogeneous spaces and Lie groups is studied. We first give an explicitformula for the flag curvature of invariant Randers metrics arising frominvariant Riemannian metrics on homogeneous spaces and, in special case, Liegroups. We then study Randers metrics of constant positive flag curvature andcomplete underlying Riemannian metric on Lie groups. Finally we give someproperties of those Lie groups which admit a left invariant non-RiemannianRanders metric of Berwald type arising from a left invariant Riemannianmetric and a left invariant vector field.

Keywords Invariant metric · Flag curvature · Randers space ·Homogeneous space · Lie group

Mathematics Subject Classifications (2000) 22E60 · 53C60 · 53C30

1 Introduction

The geometry of invariant structures on homogeneous spaces is one of theinteresting subjects in differential geometry. Invariant metrics are of theseinvariant structures. K. Nomizu studied many interesting properties of in-variant Riemannian metrics and the existence and properties of invariantaffine connections on reductive homogeneous spaces (see [14, 16]). Also some

H. R. S. Moghaddam (B)Department of Mathematics, Shahrood University of Technology, Shahrood, Irane-mail: [email protected]

2 H.R.S. Moghaddam

curvature properties of invariant Riemannian metrics on Lie groups hasstudied by J. Milnor [15]. So it is important to study invariant Finsler metricswhich are a generalization of invariant Riemannian metrics.

S. Deng and Z. Hou studied invariant Finsler metrics on reductive homo-geneous spaces and gave an algebraic description of these metrics [12, 13].Also, in [10, 11], we have studied the existence of invariant Finsler metricson quotient groups and the flag curvature of invariant Randers metrics onnaturally reductive homogeneous spaces. In this paper we study the flag cur-vature of invariant Randers metrics on homogeneous spaces and Lie groups.Flag curvature, which is a generalization of the concept of sectional curvaturein Riemannian geometry, is one of the fundamental quantities which associatewith a Finsler space. In general, the computation of the flag curvature of Finslermetrics is very difficult, therefore it is important to find an explicit and applica-ble formula for the flag curvature. One of important Finsler metrics which havefound many applications in physics are Randers metrics (see [2, 3]). In thisarticle, by using Püttmann’s formula [17], we give an explicit formula for theflag curvature of invariant Randers metrics arising from invariant Riemannianmetrics on homogeneous spaces and Lie groups. Then the Randers metrics ofconstant positive flag curvature and complete underlying Riemannian metricon Lie groups are studied. Finally we give some properties of those Lie groupswhich admit a left invariant non-Riemannian Randers metric of Berwald typearising from a left invariant Riemannian metric and a left invariant vector field.

2 Flag Curvature of Invariant Randers Metrics on Homogeneous Spaces

The aim of this section is to give an explicit formula for the flag curvature ofinvariant Randers metrics of Berwald type, arising from invariant Riemannianmetrics, on homogeneous spaces. For this purpose we need the Püttmann’sformula for the curvature tensor of invariant Riemannian metrics on homoge-neous spaces (see [17]).

Let G be a compact Lie group, H a closed subgroup, and g0 a bi-invariantRiemannian metric on G. Assume that g and h are the Lie algebras of G and Hrespectively. The tangent space of the homogeneous space G/H is given by theorthogonal compliment m of h in g with respect to g0. Each invariant metric gon G/H is determined by its restriction to m. The arising AdH-invariant innerproduct from g on m can extend to an AdH-invariant inner product on g bytaking g0 for the components in h. In this way the invariant metric g on G/Hdetermines a unique left invariant metric on G that we also denote by g. Thevalues of g0 and g at the identity are inner products on g which we denote as< ., . >0 and < ., . >. The inner product < ., . > determines a positive definiteendomorphism φ of g such that < X, Y >=< φX, Y >0 for all X, Y ∈ g.

Now we give the following lemma which was proved by T. Püttmann(see [17]).

On the flag curvature of invariant Randers metrics 3

Lemma 1 The curvature tensor of the invariant metric < ., . > on the compacthomogeneous space G/H is given by

<R(X, Y)Z , W >=1/2(< B−(X, Y), [Z , W] >0 +< [X, Y], B−(Z , W) >0

) ++ 1/4

(< [X, W], [Y, Z ]m >−< [X, Z ], [Y, W]m > −− 2 < [X, Y], [Z , W]m >

) ++ (

< B+(X, W), φ−1 B+(Y, Z ) >0 −− < B+(X, Z ), φ−1 B+(Y, W) >0

), (1)

where the symmetric resp. skew symmetric bilinear maps B+ and B− aredefined by

B+(X, Y) = 1/2([X, φY] + [Y, φX]),

B−(X, Y) = 1/2([φX, Y] + [X, φY]),

and [., .]m is the projection of [., .] to m.

Let X be an invariant vector field on the homogeneous space G/H such that

‖ X ‖=√

g(X, X) < 1. A case happen when G/H is reductive with g = m ⊕ h

and X is the corresponding left invariant vector field to a vector X ∈ m suchthat < X, X >< 1 and Ad(h)X = X for all h ∈ H (see [13] and [10]). By usingX we can construct an invariant Randers metric on the homogeneous spaceG/H in the following way:

F(xH, Y) = √g(xH)(Y, Y) + g(xH)

(Xx, Y

) ∀Y ∈ TxH(G/H). (2)

Now we give an explicit formula for the flag curvature of these invariantRanders metrics.

Theorem 1 Let G be a compact Lie group, H a closed subgroup, g0 a bi-invariant metric on G, and g and h the Lie algebras of G and H respectively.Also let g be any invariant Riemannian metric on the homogeneous spaceG/H such that < Y, Z >=< φY, Z >0 for all Y, Z ∈ g. Assume that X is aninvariant vector field on G/H which is parallel with respect to g and g(X, X) <

1 and XH = X. Suppose that F is the Randers metric arising from g and X,and (P, Y) is a flag in TH(G/H) such that {Y, U} is an orthonormal basis of Pwith respect to < ., . >. Then the flag curvature of the flag (P, Y) in TH(G/H)

is given by

K(P, Y) = A(1+ < X, Y >)2(1− < X, Y >)

, (3)

4 H.R.S. Moghaddam

where A = α. < X, U > +γ (1+ < X, Y), and for A we have:

α = 1/4(< [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φX] >0

) ++ 3/4 < [Y, U], [Y, X]m >+1/2 < [U, φX] + [X, φU], φ−1

([Y, φY])>0 −− 1/4 < [U, φY] + [Y, φU], φ−1

([Y, φX] + [X, φY]) >0, (4)

and

γ = 1/2 < [φU, Y] + [U, φY], [Y, X] >0 ++ 3/4 < [Y, U], [Y, U]m > + < [U, φU], φ−1([Y, φY]) >0 −− 1/4 < [U, φY] + [Y, φU], φ−1

([Y, φU] + [U, φY]) >0 . (5)

Proof X is parallel with respect to g, therefore F is of Berwald type and theChern connection of F and the Riemannian connection of g coincide (see [6],page 305.), so we have RF(U, V)W = Rg(U, V)W, where RF and Rg are thecurvature tensors of F and g, respectively. Let R := Rg = RF be the curvaturetensor of F (or g). Also for the flag curvature we have [18]:

K(P, Y) = gY(R(U, Y)Y, U)

gY(Y, Y).gY(U, U) − g2Y(Y, U)

, (6)

where gY(U, V) = 1/2 ∂2

∂s∂t (F2(Y + sU + tV))|s=t=0.By a direct computation for F we get

gY(U, V) = g(U, V) + g(X, U).g(X, V) − g(X, Y).g(Y, V).g(Y, U)

g(Y, Y)3/2×

× 1√

g(Y, Y)

{g(X, U).g(Y, V) + g(X, Y).g(U, V) +

+ g(X, V).g(Y, U)}. (7)

Since {Y, U} is an orthonormal basis of P with respect to < ., . >, by using theformula (7) we have:

gY(Y, Y).gY(U, U) − gY(Y, U) = (1+ < X, Y >)2(1− < X, Y >). (8)

Also we have:

gY(R(U, Y)Y, U) = < R(U, Y)Y, U > + < X, R(U, Y)Y > . < X, U > ++ < X, Y > . < R(U, Y)Y, U > ++ < X, U > . < Y, R(U, Y)Y >, (9)

now let α=< X, R(U, Y)Y >, θ =<Y, R(U, Y)Y > and γ =< R(U, Y)Y, U >.


By using Püttmann’s formula (see Lemma 1) and some computations wehave:

α = 1/4(< [φU, Y] + [U, φY], [Y, X] >0 + < [U, Y], [φY, X] + [Y, φX] >0

) ++ 3/4 < [Y, U], [Y, X]m > +1/2 < [U, φX] + [X, φU], φ−1([Y, φY]) >0 −− 1/4 < [U, φY] + [Y, φU], φ−1([Y, φX] + [X, φY]) >0, (10)

θ = 0, (11)

and

γ = 1/2 < [φU, Y] + [U, φY], [Y, U] >0 +3/4 < [Y, U], [Y, U]m > ++ < [U, φU], φ−1([Y, φY]) >0 −−1/4 < [U, φY] + [Y, φU], φ−1([Y, φU] + [U, φY]) >0 . (12)

Substituting (7), (8), (9), (10), (11) and (12) in the (6) completes the proof. ��

Remark In the previous theorem, If we let H = {e} and m = g then we canobtain a formula for the flag curvature of the left invariant Randers metricsof Berwald types arising from a left invariant Riemannian metric g and a leftinvariant vector field X on Lie group G.

If the invariant Randers metric arises from a bi-invariant Riemannian metricon a Lie group then we can obtain a simpler formula for the flag curvature, wegive this formula in the following theorem.

Theorem 2 Suppose that g0 is a bi-invariant Riemannian metric on a Lie groupG and X is a left invariant vector field on G such that g0(X, X) < 1 and X isparallel with respect to g0. Then we can define a left invariant Randers metric Fas follows:

F(x, Y) = √g0(x)(Y, Y) + g0(x)

(Xx, Y

).

Assume that (P, Y) is a flag in TeG such that {Y, U} is an orthonormal basis ofP with respect to < ., . >0. Then the flag curvature of the flag (P, Y) in TeG isgiven by

K(P, Y) = < [Y, [U,Y]], X>0 .< X,U >0 +< [Y, [U,Y]], U >0 (1+< X, Y >0)

4(1+ < X, Y >0)2(1− < X, Y >0).

Proof Since X is parallel with respect to g0 the curvature tensors of g0 andF coincide. On the other hand for g0 we have R(X, Y)Z = 1/4[Z , [X, Y]],therefore by substituting R in (6) and using (7) the proof is completed. ��

6 H.R.S. Moghaddam

3 Invariant Randers Metrics on Lie Groups

In this section we study the left invariant Randers metrics on Lie groups and,in some special cases, find some results about the dimension of Lie groupswhich can admit invariant Randers metrics. These conclusions are obtainedby using Yasuda–Shimada theorem. The Yasuda–Shimada theorem is oneof important theorems which characterize the Randers spaces. In the year2001, Shen’s examples of Randers manifolds with constant flag curvaturemotivated Bao and Robles to determine necessary and sufficient conditions fora Randers manifold to have constant flag curvature. Shen’s examples showedthat the original version of Yasuda–Shimada theorem (1977) is wrong. ThenBao and Robles corrected the Yasuda–Shimada theorem (1977) and gave thecorrect version of this theorem, Yasuda–Shimada theorem (2001) (see [5]; fora comprehensive history of Yasuda–Shimada theorem see [4]).

Suppose that M is an n-dimensional manifold endowed with a Riemannianmetric g = (gij(x)) and a nowhere zero 1-form b = (bi(x)) such that ‖b‖ =bi(x)b j(x)gij(x) < 1. We can define a Randers metric on M as follows

F(x, Y) =√

gij(x)YiY j + bi(x)Yi. (13)

Next, we consider the 1-form β = bi(b j|i − bi| j)dxi, where the covariant deriv-ative is taken with respect to Levi–Civita connection to M. Now we give theYasuda–Shimada theorem from [4].

Theorem 3 (Yasuda–Shimada; see [4]) Let F be a strongly convex non-Riemannian Randers metric on a smooth manifold M of dimension n � 2. Letgij be the underlying Riemannian metric and bi the drift 1-form. Then:

(+) F satisfies β = 0 and has constant positive flag curvature K if and only if:

– b is a non-parallel Killing field of g with constant length;– the Riemann curvature tensor of g is given by

Rhijk = K(1 − ‖b‖2

) (ghkgij − ghjgik

) ++ K

(gijb hb k − gikb hb j

) −− K

(ghjb ib k − ghkb ib j

) −− bi| jb h|k + bi|kb h| j + 2b h|ib j|k

(0) F satisfies β = 0 and has zero flag curvature ⇔ it is locally Minkowskian.(–) F satisfies β = 0 and has constant negative flag curvature if and only if:

– b is a closed 1-form;– bi|k = 1/2σ(gik − bib k), with σ 2 = −16K;– g has constant negative sectional curvature 4K, that is,

Rhijk = 4K(gijghk − gikghj).


Since any Randers manifold of dimension n = 1 is a Riemannian manifoldfrom now on we consider n > 1.

An immediate conclusion of Yasuda–Shimada theorem is the followingcorollary.

Corollary 1 There is no non-Riemannian Randers metric of Berwald type withβ = 0 and constant positive flag curvature.

Now by using the results of [8] we obtain the following conclusions.

Theorem 4 Let Fn = (M, F, gij, bi) be an n-dimensional parallelizable Randersmanifold of constant positive flag curvature with β = 0 on M and completeRiemannian metric g = (gij). Then the dimension of M must be 3 or 7.

Proof By using Theorem 2.2 of [8] M is diffeomorphic with a sphere ofdimension n = 2k + 1. But a sphere Sm is parallelizable if and only if m = 1, 3or 7 (see [1]). Therefore n = 3 or 7. ��

A family of Randers metrics of constant positive flag curvature on Liegroup S3 was studied by D. Bao and Z. Shen (see [7]). They produced, foreach K > 1, an explicit example of a compact boundaryless (non-Riemannian)Randers spaces that has constant positive flag curvature K, and which is notprojectively flat, on Lie group S3. In the following we give some results aboutthe dimension of Lie groups which can admit Randers metrics of constantpositive flag curvature. These results show that the dimension 3 is important.

Corollary 2 There is no Randers Lie group of constant positive flag curvaturewith β = 0, complete Riemannian metric g = (gij) and n = 3.

Proof Any Lie group is parallelizable, so by attention to Theorem 4 and thecondition n = 3, n must be 7. Since G is diffeomorphic to S7 and S7 can notadmit any Lie group structure, hence the proof is completed. ��

Similar to the [15] for the sectional curvature of the left invariant Rie-mannian metrics on Lie groups, we compute the flag curvature of the leftinvariant Randers metrics on Lie groups in the following theorem.

Theorem 5 Let G be a compact Lie group with Lie algebra g, g0 a bi-invariantRiemannian metric on G, and g any left invariant Riemannian metric on G suchthat < X, Y >=< φX, Y >0 for a positive definite endomorphism φ: g −→ g.Assume that X ∈ g is a vector such that < X, X >< 1 and F is the Randersmetric arising from X and g as follows:

F(x, Y) = √g(x)(Y, Y) + g(x)

(Xx, Y

),

where X is the left invariant vector field corresponding to X, and we haveassumed X is parallel with respect to g. Let {e1, · · · , en} ⊂ g be a g-orthonormal

8 H.R.S. Moghaddam

basis for g. Then the flag curvature of F for the flag P = span{ei, e j}(i = j) at thepoint (e, ei), where e is the unit element of G, is given by the following formula:

K(P = span{ei, e j}, ei) = X j. < R(e j, ei)ei, X > +(1 + Xi). < R(e j, ei)ei, e j >

(1 + Xi)2(1 − Xi),

where X = Xkek,

< R(e j, ei)ei, X > = − 1/4(< [φe j, ei], [ei, X] >0 + < [e j, φei], [ei, X] >0

+ < [e j, ei], [φei, X] >0 + < [e j, ei], [ei, φX] >0) +

+ 3/4 < [e j, ei], [ei, X] > −− 1/2 < [e j, φX] + [X, φe j], φ−1

([ei, φei])

>0 ++ 1/4 < [e j, φei] + [ei, φe j], φ−1

([ei, φX] + [X, φei])

>0

and

< R(e j, ei)ei, e j > = − 1/2(< [φe j, ei], [ei, e j] >0 + < [e j, φei], [ei, e j] >0

) ++ 3/4 < [e j, ei], [ei, e j] >−< [e j, φe j], φ−1

([ei, φei])

>0 ++ 1/4 < [e j, φei] + [ei, φe j], φ−1

([ei, φe j] + [e j, φei])

>0 .

Proof By using Theorem 1, the proof is clear. ��

Now we give some properties of those Lie groups which admit a leftinvariant non-Riemannian Randers metric of Berwald type arising from a leftinvariant Riemannian metric and a left invariant vector field.

Theorem 6 There is no left invariant non-Riemannian Randers metric ofBerwald type arising from a left invariant Riemannian metric and a left invariantvector field on connected Lie groups with a perfect Lie algebra, that is, a Liealgebra g for which the equation [g, g] = g holds.

Proof If a left invariant vector field X is parallel with respect to a left invariantRiemannian metric g then, by using Lemma 4.3 of [9], g(X, [g, g]) = 0. Since g

is perfect therefore X must be zero. ��

Corollary 3 There is not any left invariant non-Riemannian Randers metric ofBerwald type arising from a left invariant Riemannian metric and a left invariantvector field on semisimple connected Lie groups.

Corollary 4 If a Lie group G admits a left invariant non-Riemannian Randersmetric of Berwald type F arising from a left invariant Riemannian metric g anda left invariant vector field X then for sectional curvature of the Riemannianmetric g we have

K(X, u) � 0

for all u, where equality holds if and only if u is orthogonal to the image [X, g].


Proof Since F is of Berwald type, X is parallel with respect to g. By usingLemma 4.3 of [9], ad(X) is skew-adjoint, therefore by Lemma 1.2 of [15] wehave K(X, u) � 0. ��

References

1. Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72(2),20–104 (1960)

2. Antonelli, P.L., Ingarden, R.S., Matsumoto, M.: The Theory of Sprays and Finsler Spaces withApplications in Physics and Biology. Kluwer, Dordrecht (1993)

3. Asanov, B.S.: Finsler Geometry, Relativity and Gauge Theories. Kluwer, Dordrecht (1985)4. Bao, D.: Randers space forms. Period. Math. Hungar. 48(1), 3–15 (2004)5. Bao, D., Robles, C.: On randers spaces of constant flag curvature. Rep. Math. Phys. 51(1), 9–42

(2003)6. Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann–Finsler Geometry. Springer-

Verlag, Berlin (2000)7. Bao, D., Shen, Z.: Finsler metrics of constant positive curvature on lie group S3. J. Lond. Math.

Soc. 66(2), 453–467 (2002)8. Bejancu, A., Farran H.R.: Randers manifolds of positive constant curvature. Internat. J. Math.

Math. Sci. 2003(18), 1155–1165 (2003)9. Brown, N., Finck, R., Spencer, M., Tapp K., Wu, Z.; Invariant metrics with nonnegative

curvature on compact lie groups. Canad. Math. Bull. 50(1), 24–34 (2007)10. Esrafilian, E., Salimi Moghaddam, H.R.: Flag curvature of invariant Randers metrics on

homogeneous manifolds. J. Phys. A: Math. Gen. 39, 3319–3324 (2006)11. Esrafilian, E., Salimi Moghaddam, H.R.: Induced invariant Finsler metrics on quotient groups.

Balkan J. Geom. Appl. 11(1), 73–79 (2006)12. Deng, S., Hou, Z.: Invariant Finsler metrics on homogeneous manifolds. J. Phys. A: Math.

Gen. 37, 8245–8253 (2004)13. Deng, S., Hou, Z.: Invariant Randers metrics on homogeneous Riemannian manifolds. J. Phys.

A: Math. Gen. 37, 4353–4360 (2004)14. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry Vol. 2. Interscience

Publishers, John Wiley & Sons (1969)15. Milnor, J.: Curvatures of left invariant metrics on lie groups. Adv. Math. 21, 293–329 (1976)16. Nomizu, K.: Invariant affine connections on homogeneous spaces. Amer. J. Math. 76, 33–65

(1954)17. Püttmann, T.: Optimal pinching constants of odd dimensional homogeneous spaces. Invent.

Math. 138, 631–684 (1999)18. Shen, Z.: Lectures on Finsler Geometry. World Scientific (2001)


Block Toeplitz Determinants, Constrained KPand Gelfand-Dickey Hierarchies

M. Cafasso

Received: 26 November 2007 / Accepted: 19 February 2008 /Published online: 1 April 2008© Springer Science + Business Media B.V. 2008

Abstract We propose a method for computing any Gelfand-Dickey τ functiondefined on the Segal-Wilson Grassmannian manifold as the limit of blockToeplitz determinants associated to a certain class of symbols W(t; z). Alsotruncated block Toeplitz determinants associated to the same symbols areshown to be τ functions for rational reductions of KP. Connection withRiemann-Hilbert problems is investigated both from the point of view of inte-grable systems and block Toeplitz operator theory. Examples of applicationsto algebro-geometric solutions are given.

Keywords Block Toeplitz determinants · Integrable hierarchies ·Grassmannians · KP · Riemann-Hilbert problems

Mathematics Subject Classifications (2000) 37K10 · 47B35

1 Introduction

This paper deals with the applications of block Toeplitz determinants andtheir asymptotics to the study of integrable hierarchies. Asymptotics of blockToeplitz determinants and their applications to physics is a developing field ofresearch; in recent years it has been shown how to compute some physicallyrelevant quantities (e.g. correlation functions) studying asymptotics of someblock Toeplitz determinants (see [27–29]). In particular in [27] and [28] theauthors, for the first time, showed effective computations for the case of blockToeplitz determinants with symbols that do not have half truncated Fourier

M. Cafasso (B)SISSA-International School for Advanced Studies,Via Beirut 2/4, 34014 Grignano, Italye-mail: [email protected]

12 M. Cafasso

series. This is of particular interest for us as, with our approach, we will beable to do the same for certain block Toeplitz determinants associated toalgebro-geometric solutions of Gelfand Dickey hierarchies. Let us mentionsome theoretical results about (block) Toeplitz determinants we will use inthis paper. Given a function γ (z) on the circle we denote TN(γ ) the Toeplitzmatrix with symbol γ given by

TN(γ ) :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

γ (0) . . . . . . γ (−N)

γ (1) . . . . . . γ (−N+1)

. . . . . . . . . . . .

γ (N) . . . . . . γ (0)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

where γ (k) are the Fourier coefficients γ (z) =∑k γ (k)zk.We use the term block Toeplitz for the case of matrix-valued symbol γ (z). In

that case the entries γ (i− j ) of the above matrix are n × n matrices themselves.We denote

DN(γ ) := det TN(γ )

and we use the notation T(γ ) for the N × N matrix obtained letting N go toinfinity. The main goal of the theory of Toeplitz determinants is to computeDN(γ ) as N goes to infinity and find expressions for DN(γ ) as well as for itslimit in terms of Fredholm determinants.

First result is due to Szegö that in 1952 gave a formula for asymptotics ofDN(γ ) in the scalar case [5]. This result has been generalized by H. Widomin the 70’s ([6, 7] and [8]) for the matrix case; namely he proved that undersuitable analytical assumptions it exists the limit

D∞(γ ) := limN→∞

DN(γ )

G(γ )N= det

(T(γ )T

(γ −1

))

where G(γ ) is a normalizing constant and the operator T(γ )T(γ −1) is suchthat its determinant is well defined as a Fredholm determinant (see Section 3for the precise statement). Once the asymptotics had been computed the nextquite natural question was to find an expressions relating directly DN(γ ), andnot just its asymptotics, to certain Fredholm determinants. The problem wassolved many years later by Borodin and Okounkov in [9] for the scalar caseand generalized, in the same year, for matrix case by E. Basor and H.Widomin [10]. For matrix valued case Borodin-Okounkov formula reads

DN(γ ) = D∞(γ ) det(I − Kγ,N)

(here we assume G(γ ) = 1). The operator (I − Kγ,N) can be written explic-itly in coordinates knowing certain Riemann-Hilbert factorizations of γ . ItsFredholm determinant is well defined (see Section 3 for details). Now manyproofs of Borodin-Okounkov formula are known (for instance [11] containsanother proof of the same formula, see also the earlier paper [12]).

Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies 13

In this paper we apply block-Toeplitz determinants to the computation of τ

function of an (almost) arbitrary solution of Gelfand-Dickey hierarchy

∂L∂t j

=[(

Ljn

)+

, L].

(L differential operator of order n, j �= nk).More precisely to a given point

W = W(z)H(n)+

in the big cell of Segal-Wilson vector-valued Grassmannian we associate an × n matrix-valued symbol W(t; z) obtained by deforming W(z) (see for-mula (12)). In this way we define a sequence of N-truncated block Toeplitzdeterminants {τW,N(t)}N>0 which are shown to be solutions of certain rationalreductions of KP; this is our

First result: Every symbol W(t; z) defines through its truncated determinants asequence {τW,N(t)}N>0 of solutions for KP such that

τW,N(t) ∈ cKP1,nN ∩ cKPn,n ∀N > 0.

This result is stated in Theorem 5. Here we used the notation from [20]; given aτ function for KP with corresponding Lax pseudodifferential operator L we saythat τ ∈ cKPm,n iff Lm can be written as the ratio of two differential operatorsof order m + n and n respectively. This sequence admits a stable limit whichis shown to be equal to the Gelfand-Dickey τ function τW(t) associated to W;this quantity can be computed using Szegö-Widom’s theorem. This will give usthe remarkable identity

τW(t ) = det[PW(t;z)

](1)

where PW(t;z) is the Fredholm operator appearing in Szegö-Widom’s theorem(here we put t instead of t to remember that, when working with W ∈ Gr(n),times tnj multiple of n must be set to 0). Next step is the study of Riemann-Hilbert (also called Wiener-Hopf) factorization of symbol W(t; z) given by

W(t; z) = T−(t; z)T+(t; z) (2)

with T− and T+ analytical in z outside and inside S1 respectively andnormalized as

T−(∞) = I.

Here we assume that the symbol can be extended to an analytic function in aneighborhood of S1. Using Plemelj’s results [13] we show that T−(t; z) mustsatisfy the integral equation

PTW(t;z)

T−(t; z) = I (3)

and we write a solution of (2) in terms of wave function ψW(t; z) correspondingto W.

14 M. Cafasso

In this way we arrive to our second result:

Second result: Take W ∈ Gr(n) in the big cell and its corresponding τ functionτW(t). τW(t) is equal to the Fredholm determinant of the homogeneous integralequation associated to (3) which is related to Riemann-Hilbert problem (2).The solution of this Riemann-Hilbert problem is unique for every value ofparameters t that makes τW(t) �= 0 and can be computed by means of relatedwave function ψW(t; z).

Theorem 5 explains how τW(t ) can be written as a Fredholm determi-nant while the relation with the corresponding Riemann-Hilbert problem isdescribed in Proposition 14. Explicit factorization of the symbols is writtenin Theorem 7. At the end of the paper we consider a particular class ofsymbols W(t; z) corresponding to algebro-geometric solutions of Gelfand-Dickey hierarchies. We formulate an alternative Riemann-Hilbert problemequivalent to (2) and explain how to solve it using θ -functions. In this waywe give concrete formulas for a wide class of symbols that do not have halftruncated Fourier series. We think this is quite remarkable since concreteresults for non half truncated symbols were available, till now, just for theconcrete cases presented in [27] and [28].

The paper is organized as follows:

– Second section states some results about Segal-Wilson Grassmannian andrelated loop groups we will need in the sequel; proofs can be found in [1]and [2].

– Third section states Szegö-Widom’s theorem and related results obtainedby Widom in [6, 7] and [8] and the Borodin-Okounkov formula for blockToeplitz determinant [10].

– In the fourth section we introduce and study the sequence of truncateddeterminants {τW,N(t)}N>0 and its stable limit τW(t). We want to remarkthat the property of stability was stated for the first time in [15] (see also[16]) and our sequence is actually a subsequence of the stabilizing chainstudied in [17]; nevertheless, to our best knowledge, this is the first timethat block Toeplitz determinants enter the game and also the observationthat τW,N ∈ cKPn,n seems to be something new. The main results of thissection are stated in Theorem 5.

– Fifth section is devoted to establishing the connection between integralequations formulated by Plemelj in [13] and Fredholm operator appearingin Szegö-Widom’s theorem.

– In the sixth section we show how to write Riemann-Hilbert factorization ofW(t; z) in terms of wave function ψW(t; z); this result is stated in Theorem7. Of course relation between Gelfand-Dickey hierarchy and factorizationproblem is something known; our exposition here is closely related to[14]. Moreover, knowing Riemann-Hilbert factorization of W(t; z), we canapply Borodin-Okounkov formula to give an expression of any τW,N(t )as Fredholm determinant and a recursion relation to go from τW,N(t ) toτW,N+1(t ).


– Last section gives explicit formulas for symbols and τ functions associatedto algebro-geometric rank one solutions of Gelfand-Dickey hierarchies.Also we formulate an alternative Riemann-Hilbert problem equivalent to(2) in analogy with what has been done in [27] and [28]. We explain howto solve it using θ -functions.

2 Segal-Wilson Grassmannian and Related Loop Groups

Here we recall some definitions and results from [1] and [2] that will be usefulin the sequel.

Definition 1 Let H(n) := L2(S1, Cn) be the space of complex vector-valued

square-integrable functions. We choose a orthonormal basis given by

{eα,k := (0, . . . , zk, . . . , 0)T : α = 1 . . . n, k ∈ Z

}

and the polarization

H(n) = H(n)+ ⊕ H(n)

−

where H(n)+ and H(n)

− are the closed subspaces spanned by elements {eα,k} withk � 0 and k < 0 respectively.

In the sequel in order to avoid cumbersome notations we will write Hinstead of H(1).

Definition 2 ([2]) The Grassmannian Gr(H(n)) modeled on H(n) consists of thesubset of closed subspaces W ⊆ H(n) such that:

– the orthogonal projection pr+ : W → H(n)+ is a Fredholm operator.

– the orthogonal projection pr− : W → H(n)− is a Hilbert-Schmidt operator.

Moreover we will denote Gr(n) the subset of Gr(H(n)) given by subspaces Wsuch that zW ⊆ W.

It’s well known [1] that through Segal-Wilson theory we can associate asolution of nth Gelfand-Dickey hierarchy to every element of Gr(n); this is thereason why we are interested in them.

Lemma 1 ([1]) The map

� : H(n) −→ H

( f0(z), . . . , fn−1(z))T −→ f (z) := f0(zn) + . . . + zn−1 fn−1(zn)

16 M. Cafasso

is an isometry. Its inverse is given by

fk(z) = 1

n

∑ζ n=z

ζ−k f (ζ )

where the sum runs over the nth roots of z.

Proposition 1 Under the isometry � we can identify Gr(n) with the subset

{W ∈ Gr(H) : znW ⊆ W}

It is obvious that loop groups act on Hilbert spaces defined above bymultiplications. We want to define a certain loop group L1/2Gl(n, C) with goodanalytical properties acting transitively on Gr(n); in such a way we can obtainany W ∈ Gr(n) just acting on the reference point H(n)

+ with this group. Goodanalytical properties will be necessary as we want to construct symbols ofsome Toeplitz operators out of elements of this group and then apply Widom’sresults (see below). Given a matrix g we denote with ‖g‖ its Hilbert-Schmidtnorm

‖g‖2 =n∑

i, j=1

‖gi, j‖2

Definition 3 Given a measurable matrix-valued loop γ we define two norms‖γ ‖∞ and ‖γ ‖2,1/2 as

‖γ ‖∞ := ess sup‖z‖=1

‖γ (z)‖ ‖γ ‖2,1/2 :=∑

k

(|k| ‖γ (k)‖2

)1/2

where we have Fourier expansion

γ (z) =∞∑

k=−∞γ (k)zk.

Definition 4 L1/2Gl(n, C) is defined as the loop group of invertible measurableloops γ such that

‖γ ‖∞ + ‖γ ‖2,1/2 < ∞.

Proposition 2 ([2]) L1/2Gl(n, C) acts transitively on Gr(n) and the isotropygroup of H(n)

+ is the group of constant loops Gl(n, C).

Proof can be found in [2], here we just mention the principal steps necessaryto arrive to this result.

– We define a subgroup Glres(H(n)) of invertible linear maps g : H(n) → H(n)

acting on Gr(H(n)) (the restricted general linear group).– We prove that every element of Glres(H(n)) commuting with multiplication

by z must belong to L1/2Gl(n, C).


– We take an element W ∈ Gr(n) and a basis {w1, ..., wn} of the orthogonalcomplement of zW in W.

– Out of this basis, putting vectors side by side, we construct W and easilycheck that W = W(z)H(n)

+ .– We verify that multiplication by W belongs to Glres(H(n)); since it ob-

viously commutes with multiplication by z we conclude that W(z) ∈L1/2Gl(n, C).

3 Szegö-Widom Theorem for Block Toeplitz Determinants

In his work ([6, 7] and [8]) H. Widom expressed the limit, for the size goingto infinity, of certain block Toeplitz determinants as Fredholm determinantsof an operator P acting on H(n)

+ . Also he gave two different corollaries thatallow us to compute this determinant in some particular cases. In this sectionwe recall, without proofs, these results. Moreover we state Borodin-Okounkovformula as presented in [10] for matrix case.

We begin with some notations; given a loop γ ∈ L1/2Gl(n, C) we denotewith TN(γ ) the block Toeplitz matrix given by

TN(γ ) :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

γ (0) . . . . . . γ (−N)

γ (1) . . . . . . γ (−N+1)

. . . . . . . . . . . .

γ (N) . . . . . . γ (0)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

where we have the Fourier expansion γ (z) =∑k γ (k)zk.We denote DN(γ ) itsdeterminant. We use the notation T(γ ) for the N × N matrix obtained lettingN go to infinity.

Remark 1 It’s easy to see that, in the base we have chosen above for H(n), T(γ )

is nothing but the matrix representation of

pr+ ◦ γ : H(n)+ −→ H(n)

+

Theorem 1 (Szegö-Widom theorem, [8]) Suppose γ ∈ L1/2Gl(n, C) and

0�θ�2π

arg(

det(γ (eiθ )

)) = 0

Then it exists the limit

D∞(γ ) := limN→∞

DN(γ )

G(γ )N= det

(T(γ )T

(γ −1

))

where

G(γ ) = exp

(1/2π

∫ 2π

0log(

det γ (eiθ ))dθ

)

18 M. Cafasso

The proof of the theorem is contained in [8]; instead of rewriting it we sim-ply consider the operator T(γ )T(γ −1) and explain the meaning of “det” in thiscase.

Lemma 2 Consider γ1, γ2 ∈ L1/2Gln(n, C); we have

T(γ1γ2) − T(γ1)T(γ2) =⎡⎣∑

k�1

γ(i+k)1 γ

(− j−k)

2

⎤⎦

i, j�0

.

Proof The (i, j )-entry of left hand side reads

∞∑k=−∞

γ(i−k)1 γ

(k− j )2 −

∞∑k=0

γ(i−k)1 γ

(k− j )2 =

−1∑k=−∞

γ(i−k)1 γ

(k− j )2 =

∞∑k=0

γ(i+k+1)1 γ

(−k− j−1)

2 .

�

In particular choosing γ1 = γ and γ2 = γ −1 we obtain

I − T(γ )T(γ −1

) =⎡⎣∑

k�1

γ (i+k)(γ −1

)(− j−k)

⎤⎦

i, j�0

Definition 5

Pγ := T(γ )T(γ −1

) =⎡⎣δ

ji −

⎛⎝∑

k�1

γ (i+k)(γ −1

)(− j−k)

⎞⎠⎤⎦

i, j�0

(4)

Thanks to the fact that∑i�0

∑k�1

‖γ (i+k)‖2 =∑k�1

k‖γ (k)‖2 < ∞

the product we have written on the right of (4) is a product of two Hilbert-Schmidt operators. So Pγ differs from the identity by a nuclear operator.Hence its determinant is well defined (see for instance [30]). In our notationwe obtained the equality

D∞(γ ) = det(Pγ ) (5)

We will call Pγ Plemelj’s operator as it is related in a clear way with aRiemann-Hilbert factorization problem (see Section 5) already considered byJosip Plemelj in 1964 [13].

Unfortunately, in concrete cases, det(Pγ ) turns out to be really hard tocompute; nevertheless we can use some shortcuts also provided by Widom inhis works ([6, 7] and [8]).


Proposition 3 ([6]) Suppose that γ satisfies conditions imposed in Szegö-Widom theorem and, moreover, γ (i) = 0 for i � j + 1 or γ (i) = 0 for i � j + 1.

Then

D∞(γ ) = Dj(γ −1)G(γ ) j (6)

Proposition 4 ([8]) Suppose we have a symbol γ satisfying conditions imposedin Szegö-Widom theorem. Suppose moreover that γ depends on a parameterx in such a way that the function x → γ (x) is differentiable. If γ −1 admits twoRiemann-Hilbert factorizations

γ −1(z) = t+(z)t−(z) = s−(z)s+(z)

such that

t+(z) :=∑k�0

t(k)+ zk s+(z) :=

∑k�0

s(k)+ zk

t−(z) :=∑k�0

t(k)− zk s−(z) :=

∑k�0

s(k)− zk

Then

ddx

log(D∞(γ )) = i2π

∮trace

[((∂zt+)t− − (∂zs−)s+

)∂xγ

]dz. (7)

Also DN(γ ) can be expressed as a Fredholm determinant as pointed out forthe scalar case in [9] and generalized for matrix case in [10].

Theorem 2 (Borodin-Okounkov formula, [10]) Suppose that our symbol γ (z)

satisfying conditions of Szegö-Widom’s theorem admits two Riemann-Hilbertfactorizations

γ (z) = γ+(z)γ−(z) = θ−(z)θ+(z)

such that

γ+(z) :=∑k�0

γ(k)+ zk θ+(z) :=

∑k�0

θ(k)+ zk

γ−(z) :=∑k�0

γ(k)− zk θ−(z) :=

∑k�0

θ(k)− zk

and G(γ ) = 1. Then for every N

DN(γ ) = D∞(γ ) det(I − Kγ,N) (8)

where, in coordinates, we have

(Kγ,N)ij ={

0 if min{i, j} < N∑∞

k=1

(γ−θ−1

+)(i+k) (

θ−1− γ+

)(− j−k)otherwise.

20 M. Cafasso

Remark 2 One can easily verify that θ−1− γ+ is the inverse of γ−θ−1

+ so that,again, we deal with operators of type T(φ)T(φ−1) with φ = γ−θ−1+ . Also wewant to point out that the assumption G(γ ) = 1 is not necessary. The formulafor G(γ ) �= 1 is written in [11]; since in our case we will always have G(γ ) = 1we wrote the formula as it was given in [10].

4 τ Functions for Constrained KP and Gelfand-Dickey Hierarchiesas Block Toeplitz Determinants

In order to fix notations we state some basic facts about KP hierarchy andsome reductions of it. Standard references are [1] and [3]. For cKP reductionswe make reference to [18–21] and [22].

Given the pseudodifferential Lax operator

L := D +∞∑j=1

ujD− j

KP hierarchy is defined as compatibility conditions of equations{Lψ = zψ∂∂t j

ψ = (L j)+ψ j = 1 . . .∞ (9)

where (L j)+ denote the differential part of j th power of L. These compatibilityconditions are written in Lax form as

∂

∂t jL =

[(L j)

+ ,L]

and should be seen as differential equations for coefficients {uj} with respect tovariables {t j}. Equivalently one can introduce the dressing operator

S = 1 +∞∑j=1

sjD− j

such that

ψ := S(

e∑∞

j=1 tizi)

= e∑∞

j=1 tizi (1 + s1z−1 + s2z−2 + . . .

)

is a solution of (9). In this way KP hierarchy is rewritten in Sato form as{L = SDS−1

∂∂t j

S = − (L j)− S

(10)

where (L j)− = L j − (L j)+. The first equation gives expression of {uj} in termsof {sj} and the second one gives time evolution for {sj}.

Connection with Grassmannian goes this way: given W ∈ Gr one defines

W(t) = e∑∞

j=1 tiziW.


For every values of parameters {ti} such that the orthogonal projection

pr+: W → H+

is still Fredholm one defines

τW(t) := det(

pr+ : W → H+)

and KP hierarchy can be recast as a set of differential equations for τW (Hirotabilinear form). Actually we have the remarkable formula, due to Sato,

ψ(t) := τW (t − 1/[z])τW(t)

e∑∞

j=1 tizi

(here τW(t − 1/[z]) = τW(t1 − 1/z, t2 − 1/2z2, . . .)) that gives ψ (and then Sand L) in terms of τW .

Given the pseudodifferential symbol L and related tau function τW we say,using the notation of [20], that τ ∈ cKPm,n iff Lm can be written as the ratioof two differential operators of order m + n and n respectively. For n = 0 werecover the usual definition of mth Gelfand-Dickey hierarchy; already Segaland Wilson in [1] noticed that this reduction corresponds to considering pointsW ∈ Gr such that zmW ⊆ W, i.e. W ∈ Gr(m).

For n generic these reductions begun to be studied in 1995 by Dickey andKrichever ([18, 19]); a geometric interpretation of corresponding points in theGrassmannian has been given in [21] and [22]. Namely τW ∈ cKPm,n iff Wcontains a subspace W ′ of codimension n in W such that zmW ′ ⊆ W.

Now, given a subspace W ∈ Gr(n), we define the corresponding τW in adifferent way from the one used in [1]. Our approach generalizes what has beendone by Itzykson and Zuber in the study of Witten-Kontsevich τ function in[15] (see also [16] and [17]). This approach allows us to define not just τW butalso a sequence of {τW,N}N>0 approximating τW and such that

τW,N ∈ cKP1,nN ∩ cKPn,n ∀N.

Suppose we have an element W ∈ Gr(n); thanks to results stated in Section 2we can represent this element as

W =

⎛⎜⎜⎝

w11 . . . . . . wn1

. . . . . . . . . . . .

. . . . . . . . . . . .

w1n . . . . . . wnn

⎞⎟⎟⎠H(n)

+ = W(z)H(n)+

with W(z) ∈ L1/2Gl(n, C).Also we assume that the matrix W(z) = {wij(z)}i, j=1..n satisfies

⎧⎪⎨⎪⎩

wii = 1 + O (1/z)

wij = z (O (1/z)) , i > jwij = O (1/z) , i < j

22 M. Cafasso

This means that we restrict to the big cell, i.e. we assume that the orthogonalprojection

pr+ : W −→ H+

is an isomorphism. Infact we have a base for W ∈ Gr(n) given by{zswj : s ∈ N, j = 1 . . . n

}

where wj is the column vector (w1 j...wnj)T.

Using the isomorphim � : H(n) → H the corresponding base for W ∈ Gr isgiven by

{ωns+ j = zns�(wj): s ∈ N, j = 1 . . . n

}

and, as in Section 2, we have

[�(wj)](z) =n∑

i=1

zi−1w ji(zn)

This means that we obtain

ωns+ j(z) = zns+ j−1 (1 + O (1/z))

and from this equation follows that the orthogonal projection onto H+ is anisomorphism since every ωk projects to zk−1.

For these points W ∈ Gr(n) and vectors spanning them we define the stan-dard time evolution (KP flow) given by

ωns+ j(t; z) := exp

(∑i>0

tizi)

ωns+ j(z) = exp(ξ(t, z))ωns+ j(z)

Now we want to define the τ function associated to W as limit for N → ∞of some block Toeplitz determinants τW,N .

Definition 6 Take M = Nn a multiple of n.

τW,N(t) := det

[ ∮z−iω j(t; z)dz

]

1�i, j�M=Nn

(11)

Fist of all we want to prove that τW,N is a block Toeplitz determinant andwrite explicitly the symbol.

Lemma 3 For every j = 1 . . . n we have

wj(t; z) := �−1(ω j(t, z)) = exp(ξ(t, �))wj(z)


where we denote

� :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 . . . . . . . . . z

1 0. . .

. . . 0

0 1. . .

. . . 0. . .

. . .. . .

. . .. . .

0. . . 0 1 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

Proof We simply verify that multiplication by z on Gr corresponds to multi-plication by � on Gr(n) through the isomorphism �−1. �

Proposition 5 τW,N is the N-truncated (n × n)−block Toeplitz determinant withsymbol

W(t; z) := exp(ξ(t, �))W(z) (12)

Proof Take i, j � n and s, v � N; the (i + sn, j + vn)-entry of the matrix in theright hand side of (11) is given by

∮z−i−snωj+vn(t; z)dz =

∮z−i−snzvnωj(t; z)dz

=∮

z−i+(v−s)n∑

k∈Z,l=1..n

wjl(t)(k)znk+l−1dz = wji(t)(s−v)

so that the right hand side of (11) is the transposed of the N-truncated n × nblock Toeplitz matrix with symbol W(t; z). �

In the sequel of this paper we will call such symbols Gelfand-Dickey (GD)symbols.

Now generalizing what has been done by Itzykson and Zuber in [15] weexpand τW,N(t) in characters.

Proposition 6

τW,N(t) =∑

l1,...,lnN�0

(∏i

ω(−li+i−1)

i

)χl1,...,lnN (X)

where X = diag(x1, ..., xnN) is related to times {ti} through Miwa’sparametrization

tk := trace(

Xk/k)

24 M. Cafasso

and

χl1,...,lnN (X) :=det

⎛⎝

xl1+nN−11 xl2+nN−2

1 . . . xlnN1

. . . . . . . . . . . .

xl1+nN−1nN xl2+nN−2

nN . . . xlnNnN

⎞⎠

det

⎛⎝

xnN−11 xnN−2

1 . . . 1. . . . . . . . . . . .

xnN−1nN xnN−2

nN . . . 1

⎞⎠

Proof We start from determinant representation (11). The (i, j )-entry of thematrix will be

∑n

ω(−n)

j pn+i−1(t)

where for every n � 0

pn(t) := 1

2π i

∮exp(ξ(t, z))

zn+1dz

are the classical Schur polynomials and pn(t) = 0 for every negative n. Thenresumming everything we obtain

τW,N(t) =∑

k1,...,knN

⎛⎝∏

j

ω(−k j)

j

⎞⎠ det[pk j+i−1(t)]i, j=1...nN

with k j � 1 − j.Equivalently we write

τW,N(t) =∑

l1,...,lnN�0

⎛⎝∏

j

ω(−l j+ j−1)

j

⎞⎠ det[pl j− j+i(t)]i, j=1...nN.

On the other hand it’s well known that under Miwa’s parametrization thislast determinant can be written as χl1,...,lnN (X) (see for instance [15, 16]); thiscompletes the proof. �

We now assign degree 1 to every xi or, equivalently, degree m to tm forevery m. For every N the function τW,N is a formal series belonging to thegraded algebra C[[t1, t2, . . .]]. In general given A ∈ C[[t1, t2, . . .]] we define itsdegree as the minimal degree of its terms and we state the following definitionof stable limit for sequences in C[[t1, t2, . . .]].

Definition 7 Given a sequence of formal series

{AN(t) ∈ C[[t1, t2, . . .]], N = 0 . . .∞}we say that the sequence admits a stable limit A(t) iff

limN→∞

deg(AN(t) − A(t)) = ∞


We want to prove that the sequence {τW,N} admits stable limit. It’s easy tosee that

deg(χl1...lnN ) =M∑

i=1

li

From this easily verified property we obtain the following

Lemma 4 Suppose deg(χl1,...,lnN ) = Q � nN. Then, if the character is differentfrom zero, we have

χl1...lnN = χl1,...,lQ,0,...,0.

Proof Suppose l j �= 0, j > Q and li = 0 ∀ i > j.The j th column of the matrix [pl j− j+i(t)] has positive subscripts l1 + j −

1, l2 + j − 2, . . . , l j.On the other hand

∑li = Q; hence the sum of these subscripts is

Q +j−1∑r=0

r �j∑

r=0

r

hence two subscripts must be equal, then two lines of the matrix are equal. �

From this corollary it follows directly the following result.

Proposition 7 Up to degree Q the function τW,N(t) does not depend on N withN � Q.

Thanks to this proposition we deduce that it exists the stable limit

τW(t) := limN→∞

τW,N(t) (13)

On the other hand, in the sequel, we will prove that the symbol W(t) satisfiesSzegö-Widom’s condition for every values of ti so that the limit in (13) existpointwise in time parameters and can be written as a Fredholm determinant.

Now, following again [15], we write a differential operator W,N(t) asso-ciated to the function τW,N(t). In the sequel we will always write D for thepartial derivative with respect to t1. We will prove that for every N the pseudo-differential operator W,N(t)D−nN satisfies Sato’s equations for the dressingand we recover the usual relation between τ and wave functions.

Lemma 5 Define

fs,N(t) :=∑k>s

ω(−k)s pk+nN−1(t)

Then we have

τW,N(t) = Wr( f1,N(t), . . . , fnN,N(t)) := det[DnN− j fi,N(t)

]1�i, j�nN

26 M. Cafasso

Proof From Definition 6 the (i, j)-entry of matrix defining τW,N(t) is∑k> j

ω(−k)

j pk+i−1(t)

On the other hand we have

DnN− j fi,N(t) = DnN− j

(∑k>i

ω(−k)

i pk+nN−1(t)

)=∑k>i

ω(−k)

i pk+ j−1(t)

(using the equation Ds(pm(t)) = pm−s(t)). Hence we obtained the proof. �

Definition 8 We define the differential operator W,N of order N in D as

W,N( f ) := Wr( f, f1,N(t), . . . , fnN,N(t))Wr( f1,N(t), . . . , fnN,N(t))

where f ∈ H depends in a differentiable way on {ti}i�1.

Proposition 8 The following equations for time-derivatives of W,N holds:

∂

∂tiW,N =

(W,N Di−1

W,N

)+W,N − W,N(t)Di (14)

Proof It is enough to prove the equality of the two differential operatorswhen acting on f1,N(t), ... fnN,N(t) which are nN independent solutions of theequation

(W,N)( f (t)) = 0

But this amounts to proving[

∂

∂ti(W,N)

]f j,N(t) + W,N

∂ i

∂ti1

f j,N(t) = 0 ∀ j

which is true iff∂

∂ti(W,N f j,N(t)) = 0 ∀ j.

This equality is obviously satisfied. �

Multiplying W,N from the right with D−nN we found a pseudodifferentialoperator that, in fact, gives a solution of KP equations.

Definition 9

SW,N := W,N D−nN

Proposition 9 SW,N is a monic pseudo-differential operator of order 0 satisfyingSato’s equation

∂

∂tiSW,N = −

(SW,N DiS−1

W,N

)−

SW,N (15)


Hence the monic pseudo-differential operator of order 1

LW,N := (SW,N DS−1W,N

)(16)

satisfies the usual Lax system for KP

∂LW,N

∂tk=[(Lk

W,N

)+ ,LW,N

](17)

Proof It is obvious that SW,N is a monic pseudo-differential operator of order0 since W,N , which is of order nN, is normalized so that the leading term isequal to 1. Equation (15) follows directly from (14). The derivation of Laxsystem from Sato’s equations is well known: one has just to derive the relation

LW,N SW,N = SW,N D

for tk and use the obvious relation [LW,N,LkW,N] = 0 �

It remains to prove that τW,N(t) is really the τ function for these solutionsLW,N(t) of KP equations. We recall the usual relations between the dressing S,the wave function ψ and τ function given by

ψ(t; z) = S(t)(exp(ξ(t, z))) = exp(ξ(t, z))τ (t − 1/[z])

τ (t)

(we recall that the notation t − 1/[z] stands for the vector with ith componentequal to ti − 1

izi ) All we have to prove is the following

Proposition 10

ψW,N(t; z) := SW,N(exp(ξ(t, z))) = exp(ξ(t, z))τW,N (t − 1/[z])

τW,N(t)(18)

Proof Equivalently we prove that

(W,N) exp(ξ(t, z)) = exp(ξ(t, z))znN τW,N(t − 1/[z])τW,N(t)

Since we have

pn (t − 1/[z]) = pn(t) − z−1 pn−1(t)

the right hand side of the equality above can be written as

znNeξ(x,t)

det

⎛⎝

DnN−1 f1 − z−1 DnN f1 . . . f1 − z−1 Df1

. . . . . . . . .

DnN−1 fnN − z−1 DnN fnN . . . fnN − z−1 DfnN

⎞⎠

Wr( f1, . . . , fnN)

28 M. Cafasso

(here derivative is with respect to t1, we don’t write dependence on fi on t toavoid heavy notation) The left hand side can be written as

det

⎛⎜⎜⎝

znNeξ(t,z) . . . . . . eξ(t,z)

DnN f1 . . . . . . f1

. . . . . . . . . . . .

DnN fnN . . . . . . fnN

⎞⎟⎟⎠

Wr( f1, . . . , fnN).

It is easy to check that these two expressions are equal. �

We want now to study the structure of LW,N with more attention; ourinvestigation will lead us to discover that, actually, we are dealing with rationalreductions ([18, 19]) of KP.

First of all we recall a useful lemma (proof can be found for instance in [31]).

Lemma 6 Let {g1, ..., gm} be a basis of linearly independent solutions of adifferential operator K of order m. Then one can factorize K as

K = (D + Tm)(D + Tm−1) · · · (D + T1)

with

T j = Wr(g1, . . . , gm−1)

Wr(g1, . . . , gm).

We will state now properties of symmetry for fs,N that will be useful in thesequel.

Proposition 11 The following equalities hold:

fs,N+1(t) = fs−n,N(t) (19)

fs,N(t) = Dn fs+n,N(t) (20)

Proof We will use the equality

ω(−k)s = ω

(−k+n)s+n

which follows from the very definition of these coefficients.Then for (19) we have

fs,N+1(t) =∑

k

ω(−k)s pk+n+nN−1(t) =

∑k

ω(−k−n)s−n pk+n+nN−1(t)

=∑

k

ω(−k)s−n pk+nN−1(t) = fs−n,N(t)


For (20) we have

Dn( fs+n,N(t)) = Dn∑

k

ω(−k)s+n pk+nN−1(t) =

∑k

ω(−k)s+n pk+nN−1−n(t)

=∑

k

ω(k−n)s pk−n+nN−1(t) = fs,N(t) �

Theorem 3 For every N > 0 the pseudodifferential operator LW,N and its nth

power

LW,N = LnW,N

can be factorized as

• LW,N = L1,W,N(L2,W,N)−1

• LW,N = M1,W,N(M2,W,N)−1

where all the factors are differential operators and

• ord(L1,W,N) = nN + 1, ord(L2,W,N) = nN• ord(M1,W,N) = 2n, ord(M2,W,N) = n

Hence, for every N, τW,N ∈ cKP1,nN ∩ cKPn,n.

Proof The first factorization comes directly from the fact that

LW,N(t) = W,N(t)D(W,N(t))−1

For the second factorization we note that we have the factorization

LW,N(t) = W,N(t)Dn(W,N(t))−1

where the first operator W,N(t)Dn has order M + n while the second (i.e.W,N) has order nN. Moreover as follows from (20) we have

– W,N fi,N = 0 ∀i = 1, . . . , nN– W,N Dn fi,N = 0 ∀i = n + 1, . . . , nN

hence using Lemma 6 one can simplify factorization above as

LW,N = M1,W,N(M2,W,N)−1

where M2,W,N is given explicitly by the formula

M2,W,N = (D + Kn,N)(D + Kn−1,N)...(D + K1,N)

with

K j,N = D

[log

(Wr(

fn+1,N, . . . , fnN,N, f1,N, . . . , f j−1,N)

Wr(

fn+1,N, . . . , fnN,N, f1,N, . . . , f j,N))]

�

Theorem 4 The sequence {LW,N}N�1 satisfies recursion relation

LW,N+1 = TNLW,N(TN)−1 (21)

30 M. Cafasso

with

TN = (D + Tn,N)(D + Tn−1,N)...(D + T1,N)

T j = D log

(Wr(

f1,N, ..., fnN,N, f1,N+1, ... f j−1,N+1)

Wr(

f1,N, ..., fnN,N, f1,N+1, ... f j,N+1))

.

Proof We observe that thanks to (19)

– W,N fi,N = 0 ∀i = 1, . . . , nN– W,N+1 fi,N = 0 ∀i = n + 1, . . . , nN– W,N+1 fi,N+1 = 0 ∀i = 1, . . . , n

Hence using again (6) we obtain the recursion relation

W,N+1 = TNW,N

and from this last equation we recover the recursion relation for the Laxoperator. �

We want to point out that the first decomposition as well as the recursionformula are already known and, as pointed out in [20], come simply from thefact that we have a truncated dressing. Actually our sequence of {τW,N}N�1

is a part of a sequence already studied by Dickey in [17] under the name ofstabilizing chain; in that article Dickey already provided the recursion formulawritten above as well as some differential equations for coefficients of TN . Nev-ertheless, to our best knowledge, connection with block Toeplitz determinantsnever appeared before. Also the fact that τW,N ∈ cKPn,n is something new. Itcould be interesting to find recursion relations as well as differential equationsfor M1,W,N and M2,W,N ; we plan to do it in a subsequent work. Till now allwe can do is to infer from recursion formula for Lax operator the followingformula

(M1,W,N+1)−1TN M1,W,N = (M2,W,N+1)

−1TN M2,W,N. (22)

Now we want to go one step further and see what happens for N → ∞.Obviously thanks to the property of stabilization stated in Proposition 7 wecan define a pseudodifferential operator LW and a wave function ψW relatedto τW in the same way as for finite N and we will obtain a solution of KP aswell. Actually a stronger statement holds.

Proposition 12 Given W ∈ Gr(n) the functions τW, ψW and LW := (LW)n arerespectively the τ function, the wave function and the differential operator oforder n corresponding to a solution of nth Gelfand-Dickey hierarchy.

Proof It is known [1] that subspaces satisfying znW ⊆ W correspond to solu-tions of nth Gelfand-Dickey hierarchy. What we have to prove is that LW(t) =(LW(t))+.


From the usual relation

∂ψW

∂tn(t; z) = (LW)+ψW(t; z) (23)

we obtain immediately

∂SW

∂tn+ SW Dn = (LW)+SW

so that we have to prove that

∂SW

∂tn= 0

On the other hand

ψW(t; z) = exp(ξ(t, z))

(1 +

∞∑1=1

si(t)z−i

)

where

SW = 1 +∞∑

1=1

si(t)D−i

Using this explicit expression for the wave function and substituting in (23) weobtain

(LW)+ψW(t; z) − znψW(t; z) = exp (ξ(t, z))

∞∑i=1

∂si(t)∂tn

z−i

The left hand side of this equation lies on W(t) = exp(ξ(t, z))W for every t sothat multiplying both terms for exp(−ξ(t, z)) one obtains that they belong tosubspaces transverse one to the other (W and H−), hence both of them vanish.This means that ∂si

∂tn= 0 for every i. �

In virtue of this proposition, when computing τW associated to W ∈ Gr(n),we will always omit times t jn multiple of n. Setting {t jn = 0, j ∈ N} will beimportant in order to be able to apply Szegö-Widom’s theorem; in this casewe will write t instead of t.

Proposition 13 Take any W ∈ Gr(n) in the big cell of Gr(n) and a correspondingGD symbol W(t; z).

Then

τW(t) = det(PW(t;z)

). (24)

Proof All we have to prove is that conditions of Szegö-Widom’s theorem aresatisfied and G(W(t; z)) = 1. We observe that

W(t; z) ∈ L1/2Gl(n, C) ∀t

32 M. Cafasso

since we can always find W(z) ∈ L1/2Gl(n, C) such that W = W(z)H(n)+ and

exp(ξ(t, �)) is continuously differentiable (obviously when restricted to a finitenumber of times).

Moreover

det[exp(ξ(t, �))] = 1

since we deleted times multiple of n and det(W(z)) = 1 + O(z−1) by big cellassumption.

This implies that we have

0�θ�2π

det(W(t; eiθ)) = 0

and

G(W(t; z)) = 1. �

We are now in the position to state the main result of this paper.

Theorem 5 Given any point

W(z)H(n)+ = W ∈ Gr(n)

and corresponding GD symbol

W(t; z) = exp

( ∞∑i=1

ti�i

)W(z)

the following facts hold true:

– {τW,N(t) := DN(W(t; z))}0�N<∞is a sequence of τ functions for KP associated to wave function

ψW,N(t; z) = SW,N

(e∑∞

i=1 tizi)

and pseudodifferential Lax operator

LW,N = SW,N DS−1W,N.

The dressing is given by the formula

SW,N = W,N D−nN.

– For every N > 0 we have τW,N ∈ cKP1,nN ∩ cKPn,n.– The sequence admits stable limit

τW(t) = limN→∞

τW,N(t).

τW(t ) is a solution of the nth Gelfand-Dickey hierarchy and can be writtenas the Fredholm determinant

τW(t ) = det(PW(t;z)

).


Proof The expression of the dressing as well as the expression of LW,N aregiven in Proposition 9. Proposition 10 gives the expression of ψW,N and provesat the same time that τW,N is the corresponding τ function. The fact thatτW,N ∈ cKP1,nN ∩ cKPn,n is proven in Theorem 3 while the existence of thestable limit τW(t ) is given by Propostion 7; Proposition 12 and Proposition 13prove respectively that τW(t ) is a solution of the nth Gelfand-Dickey hierarchyand that it can be written as a Fredholm determinant. �

Remark 3 Also all the τW,N(t ) can be expressed as Fredholm determinants;in order to give explicit expressions we need a certain Riemann-Hilbertfactorization of symbol W(t; z). This factorization will be obtained in Section6 and it will be exploited to express τW,N(t ) as a Fredholm determinant.

5 Riemann-Hilbert Problem and Plemelj’s Integral Formula

It is evident from Proposition 4 that Riemann-Hilbert decomposition ofsymbol γ for a block Toeplitz operator plays an important role in computingD∞(γ ).

Here we will show that actually Plemelj’s operator itself enters in a integralequation (see [13]) giving solutions of Riemann-Hilbert problem

ϕ+(z) = γ T(z)ϕ−(z). (25)

Here ϕ+(z) and ϕ−(z) are respectively analytical functions defined inside andoutside the circle. In this section we consider a smaller class of loops; γ (z) willbe a matrix-valued function that extends analytically on a neighborhood of S1.For convenience of the reader we recall here main steps to arrive to Plemelj’sintegral formula [13].

Lemma 7 Suppose that f+(z), f−(z) are functions on S1 satisfying

| f (ζ2) − f (ζ1)| < |ζ2 − ζ1|μ C

for some positive constants μ, C and for every ζ1, ζ2 ∈ S1. Necessary and suffi-cient conditions for f+(z) and f−(z) to be boundary values of analytic functionsregular inside or outside S1 ⊆ C and with value c at infinity are respectively

1

2π i

∮f+(ζ ) − f+(z)

ζ − zdζ = 0 (26)

1

2π i

∮f−(ζ ) − f−(z)

ζ − zdζ + f−(z) − c = 0 (27)

We have to point out that here both ζ and z lies on S1 so that one has to becareful and define (26) and (27) as appropriate limits. Namely one proves thattaking ζ slightly inside or outside S1 along the normal and making it approachto the circle we obtain the same result which will be, by definition, the value ofour integral. Now suppose we want to find solutions of (25); we normalize the

34 M. Cafasso

problem requiring ϕ− taking value C at infinity. Taking an appropriate linearcombination of (26) and (27) and using (25) we find that ϕ−(z) must satisfy theequation

C = ϕ(z) − 1

2π i

∮ (γ T)−1

(z)γ T(ζ ) − I

ζ − zϕ(ζ )dζ (28)

Note that here we do not have to take any limit since the integrand iswell defined for every point of S1. We also want to consider the associatehomogeneous equation

0 = ϕ(z) − 1

2π i

∮ (γ T)−1

(z)γ T(ζ ) − I

ζ − zϕ(ζ )dζ (29)

as well as its adjoint

0 = ψ(z) + 1

2π i

∮γ (z)γ −1(ζ ) − I

ζ − zψ(ζ )dζ (30)

Obviously, as usual in Fredholm’s theory, the equations (29) and (30) eitherhave only trivial solution or they have the same number of linearly indepen-dent solutions.

Lemma 8 Consider two adjoint RH problems

ϕ+(z) = γ (z)Tϕ−(z) (31)

ψ+(z) = γ (z)ψ−(z) (32)

normalized as ψ−(∞) = ϕ−(∞) = 0.Any solution ϕ− of (31) is a solution of (29) as well as any solution ψ+ of (32)is a solution of (30).

Proof We just repeat computations made for non-homogeneous case. �

Now we introduce a new integrable operator acting on H(n)+ and prove that

it is actually equal to the Plemelj’s operator.

Definition 10 For every f ∈ H(n)+ we define

[Pγ ψ](z) := pr+

(ψ(z) + 1

2π i

∮γ (z)γ −1(ζ ) − I

ζ − zψ(ζ )dζ

)(33)

where pr+ denote the projection onto H(n)+ .

Proposition 14

Pγ = Pγ .

Proof We write Pγ in coordinates and verify we obtain the same as in (4). Todo so as in the definition of integrals (26) and (27) we compute (33) imposing


|ζ | < |z|; the formula will hold when ζ approach to S1 in the same way as in(26) and (27). For a consistency check we will prove we obtain the same resultimposing |ζ | > |z|. Let’s start with |ζ | < |z|; we have

ψ(z) + 1

2π i

∮γ (z)γ −1(ζ ) − I

ζ − zψ(ζ )dζ =

ψ(z) + 1

2π i

∮ ∑k�1

ζ k

zk

⎛⎝I −

∑p,q∈Z

γ (p)(γ −1

)(q)zpζ q

⎞⎠∑

s�0

ψ(s)ζ s dζ

ζ

Imposing k + q + s = 0 we get that this is equal to

ψ(z) +∑p∈Z

∑k�1

∑s�0

γ (p)(γ −1

)(−k−s)ψ(s)zp−k

= ψ(z) +∑t∈Z

∑k�1

∑s�0

γ (t+k)(γ −1

)(−k−s)ψ(s)zt

Taking the projection on H(n)+ we obtain exactly formula (4). Now for |ζ | > |z|

we have

ψ(z) + 1

2π i

∮γ (z)γ −1(ζ ) − I

ζ − zψ(ζ )dζ

= ψ(z) + 1

2π i

∮ ∑k�0

ζ k

zk

⎛⎝∑

p,q∈Z

γ (p)(γ −1

)(q)zpζ q − I

⎞⎠∑

s�0

ψ(s)ζ s dζ

ζ

Imposing q + s = k we arrive to∑

k,s�0

∑p∈Z

γ (p)(γ −1

)(k−s)ψ(s)zk+p =

∑k,s�0

∑t∈Z

γ (t−k)(γ −1

)(k−s)ψ(s)zt

Taking the projection on H(n)+ we obtain that this is equal to T(γ )T(γ −1) so

that the two computations for |ζ | < |z| and for |ζ | > |z| coincide in virtue ofLemma 2 �

Theorem 6 Suppose we are given a symbol γ (z) analytic in a neighborhood ofS1 and such that

D∞(γ ) �= 0

Then the Riemann-Hilbert problem

ϕ+(z) = γ (z)Tϕ−(z)

normalized as ϕ−(∞) = C admits (if existing) a unique solution.

Proof Suppose we have two distinct solutions (ϕ1−, ϕ1+) and (ϕ2−, ϕ2+); takingthe difference we obtain a non-trivial solution of (31). Then also (32) admits

36 M. Cafasso

non trivial solutions and the same holds for (30). But this means that we havea non zero ψ(z) ∈ H(n)

+ such that [Pγ ψ](z) = 0 which is impossible since

det(Pγ ) = D∞(γ ) �= 0

�

Existence of factorization will be treated in the next section for the specificcase of Gelfand-Dickey symbols. For a general treatment of the problem ofexistence see [13].

6 Factorization for Gelfand-Dickey Symbols

Here we will prove that for Gelfand-Dickey symbols we can write the uniquesolution of factorization (25) in terms of data LW(t ), ψW(t; z). We recall thatLW(t ) and ψW(t; z) are the stable limits of LW,N(t) and ψW,N(t; z). Theyrepresent the differential operator and the wave function associated to thesolution τW(t ). Our exposition here is closely related to [14]. At the end ofthe section we will use the factorization obtained to express any τW,N(t ) as aFredholm determinant. As we have written before in the proof of Proposition12 we have the relation

LW(t)ψW

(t; z) = znψW

(t; z)

(34)

where ψW(t; z)

admits asymptotic expansion

ψW(t; z) = exp

(ξ(t, z)) (

1 + O(z−1))

Now out of ψW we construct n time-dependent functions

ψW,i(t; z) := Di (ψW

(t; z)) : i = 0, . . . , n − 1

belonging to the subspace W ∈ Gr.

Definition 11

�W(t; z) :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

−1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

ψW,0(t; ζ1

)ψW,1

(t; ζ1

). . . ψW,n−1

(t; ζ1

)

ψW,0(t; ζ2

)ψW,1

(t; ζ2

). . . ψW,n−1

(t; ζ2

)

. . . . . . . . . . . .

ψW,0(t; ζn

)ψW,1

(t; ζn

). . . ψW,n−1

(t; ζn

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

where ζi is the ith root of z.

Proposition 15 The matrix �W(t; z) admits asympotic expansion

�W(t; �

) = exp(ξ(t; �

)) (I + O

(z−1))


Moreover under the isomorphism �−1 : H → H(n) we can write W ∈ Gr(n) as

W = �W(0, z)H(n)+ (35)

Proof One has to note that the ith column of matrix �W(t; z) is nothingbut �−1(ψW,i(t, z)) so that asymptotic expansion follows easily. Equation 35corresponds to the fact that {znsψW,i(0, z) : s ∈ Z} is a basis for W. �

Observe that, since we also have

W = W(z)H(n)+

we obtain

�W(0, z) = W(z)(I + O

(z−1))

.

From this equation and from Lemma 2 it follows that for every N > 0 we have

TN(W(t; z) (

I + O(z−1))) = TN

(W(t; z))

TN(I + O

(z−1))

.

Now since for every N

det(TN(I + O

(z−1))) = 1

we will assume, without loss of generality, that

�W(0, z) = W(z)

since this is true modulo an irrelevant term that does not affect values ofdeterminants we want to compute. We now want to define a matrix �W(t; z)

analytic in z near 0 and with similar properties as �W(t; z).

Definition 12 Let φW(t; z) be the unique solution of

LW(t)φW(t; z) = znφW

(t; z)

analytic in z = 0 and such that

(Diφ)(0, z) = zi : i = 0, . . . , n − 1

We define

�W(t; z) :=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

−1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

φW,0(t; ζ1

)φW,1

(t; ζ1

). . . φW,n−1

(t; ζ1

)

φW,0(t; ζ2

)φW,1

(t; ζ2

). . . φW,n−1

(t; ζ2

)

. . . . . . . . . . . .

φW,0(t; ζn

)φW,1

(t; ζn

). . . φW,n−1

(t; ζn

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

where as before ζi is the ith root of z and

φW,i(t) := Di (φW

(t; z)) : i = 0, . . . , n − 1.

38 M. Cafasso

Remark 4 �W(t; z) admits regular expansion in z = 0 and Cauchy initial val-ues we imposed on φW imply

�W(0; z) = I.

Proposition 16 �W(t; z)�−1W (t; z) does not depend on ti for any i.

Proof It is well known that equations

∂

∂tif =

(L

inW

)+

f

satisfied by φW and ψW can be translated into matrix equations

∂

∂tiF = F M

satisfied by �W(t; z) and �W(t; z)(one can write explicitly M in terms of co-

efficients of(L

inW

)+). Hence we have

∂

∂ti

(�W

(t; z)�−1

W

(t; z)) = �W

(t; z)

M�−1W

(t; z)

−�W(t; z)�−1

W

(t; z)�W

(t; z)

M�−1W

(t; z) = 0 �

Theorem 7 Given a Gelfand-Dickey symbol

W(t; z) = exp

(ξ(t, �

))W(z)

one can factorize it as

W(t; z) = [exp

(ξ(t, �

))�W

(−t, z)]

�−1W

(−t; z)

where the term inside the square bracket is analytic around z = ∞ and the otheris analytic around z = 0. For assigned values of t for which

τW(t ) �= 0

this is the unique solution of the factorization problem (25) normalized atinfinity to the identity.

Proof Using the previous proposition we have

W(t; z) = exp ξ

(t, �

)W(z) = exp

(ξ(t, �

))�W(0, z)

= exp(ξ(t, �

))�(−t; z

)�−1

W

(−t; z)�W(0; z)

= exp(ξ(t, �

))�(−t; z

)�−1

W

(−t; z)

Unicity of the factorization follows from Section 5. �

Corollary 1 For every N > 0

τW,N(t) = τW

(t)

det(I − KW(t;z),N

)


with

(KW(t;z),N

)ij =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

0 if min{i, j} < N

∞∑k=1

(�W

(−t; z))(i+k)

(�W

(−t; z)−1)(− j−k)

otherwise.

Proof It is enough to apply Borodin-Okounkov formula using factorizationobtained above. �

Corollary 2 For every N > 0

τW,N(t)

τW,N+1(t) = det

(T(�W

(−t; z))

T(�W

(−t; z)−1))

N,N

(observe that the right hand side of this equation is an ordinary n × n determi-nant, not a Fredholm determinant).

Proof

τW,N(t)

τW,N+1(t) = det

(I − KW(t;z),N

)

det(I − KW(t;z),N+1

) .

On the other hand the operator (I − KW(t;z),N+1)−1(I − KW(t;z),N) can be

written as a block matrix obtained taking the identity matrix and replacingthe Nth block column by the Nth block column of the matrix with (i, j)-entryequal to

∞∑k=1

(�W

(−t; z))(i+k)

(�W

(−t; z)−1)(− j−k)

.

Hence proof is obtained applying Lemma 2

7 Rank One Stationary Reductions and CorrespondingGelfand-Dickey Symbols

We want to describe, more explicitly, GD symbols corresponding to solutionsof Gelfand-Dickey hierarchies obtained by rank one stationary reductions. Inorder to emphasize that we are dealing with rank-one generic case insteadof the standard expression Krichever locus we will speak about Burchnall-Chaundy locus.

Definition 13 Given a point W ∈ Gr(n) we say that W stays in Burchnall-Chaundy locus iff the Lax operator LW of the corresponding solution satisfies

[LW, MW] = 0

40 M. Cafasso

for some differential operator MW of order m coprime with n. Without loss ofgenerality we also assume m > n.

The name we use is due to the fact that, already in 1923, Burchnall andChaundy were the first to study algebras of commuting differential operators in[23–25] where they stated this important proposition we will use in the sequel.

Proposition 17 ([23–25]) Given a pair of commuting differential operator L, Mwith relatively prime orders it exists an irreducible polynomial F(x, y) such that

F(x, y) = xm + ... ± yn

and F(L, M) = 0.

This proposition in particular allows us to associate to every Burchnall-Chaundy solution a spectral curve defined by polynomial relation existingbetween the pair of commuting differential operators. From the Grassmannianpoint of view one can define an action A of pseudodifferential operators invariable t1 on H by

A : �DO × H −→ H((t1)m ∂n

∂tn1

, ϕ(z)

)−→

(∂n

∂zn

) (zn)ϕ(z)

and, using this action, prove the following propostition

Proposition 18 ([4]) Given a point W in the Burchnall-Chaundy locus one has

znW ⊆ W (36)

b(z)W ⊆ W (37)

where LW and MW are of order n and m respectively and b(z) is a series in zwhose leading term is zm. Conversely, if W satisfies above properties, it stays inthe Burchnall-Chaundy locus.

Proof We just sketch the proof and make reference to Mulase’s article [4].Suppose we are given LW and MW ; under conjugation with the dressing SW(t )we have

S−1W

(t)

LW(t)

SW(t) = ∂n

∂tn1

Under the action A this gives invariance of W with respect to zn whileinvariance with respect to b(z) is obtained acting with

S−1W

(t)

MW(t)

SW(t)

Viceversa given W we reconstruct the dressing SW(t ); using it we defineLW(t ) and MW(t ) conjugating pseudodifferential operators corresponding tozn and b(z). In particular observe that also zn and b(z) will satisfy the samepolynomial relation as LW(t ) and MW(t ). �


Remark 5 Without loss of generality we can assume

1

2π i

∮b(z)

zns+1dz = 0 ∀s ∈ Z. (38)

Now suppose we are given an element W = W(z)H(n)+ ∈ Gr(n) in the

Burchnall-Chaundy locus. Using the explicit isomorphism � we can constructa matrix B(z) := b(�) such that

B(z)W ⊆ W. (39)

Proposition 19

C(z) := W−1(z)B(z)W(z)

has the following properties:

– C(z) is polynomial in z.– trace(C(z)) = 0– m = maxi( j − i + n deg Cij(z)) ∀ j = 1 . . . n– The characteristic polynomial pC(z)(λ) of C(z) defines the spectral curve of

the solution.

Proof Equation 39 can be equivalently written as

W−1(z)B(z)W(z)H(n)+ ⊆ H(n)

+and this means precisely that C(z) can’t have terms in z−k for any k > 0.

The other properties are satisfied if and only if they are equally satisfiedby B(z) so that we will prove them for B(z) instead of C(z). B(z) is tracelessthanks to equation (38) and thanks to the fact that

trace(�k) = 0 ∀k �= sn

The third properties is satisfied as B(z) = b(�) represents in H multiplicationby a series whose leading term is equal to m. For the last property we observethat if F(x, y) is the polynomial defining the spectral curve, i.e. F(LW, MW)=0,then we will have

F(diag(z, z, . . . , z), B(z)) = 0

as well; on the other hand thanks to Cayley-Hamilton theorem we have

pB(z)(B(z)) = 0.

Since F is irreducible and pB(z)(λ) has the same form

pB(z)(λ) = λn + . . . ± zm

we conclude that they are equal. �

Observe that since W(z) is defined modulo multiplication on the left byinvertible triangular matrices also C(z) is defined modulo conjugation byelements of the group of upper triangular invertible matrices. It was a

42 M. Cafasso

remarkable observation of Schwarz [26] that actually Burchnall-Chaundy locuscan be described by means of matrices with properties as in proposition 19modulo the action of . Here we adapt the results of [26] to our situation.Namely we explain how, given C(z), one can recover W(z) and the correspond-ing spectral curve.

Proposition 20 Given a matrix C(z) such that:

– C(z) is polynomial in z.– trace(C(z)) = 0– m = maxi( j − i + n deg Cij(z)) ∀ j = 1 . . . n

it exists a unique W = W(z)H(n)+ in Burchnall-Chaundy locus such that its spec-

tral curve is defined by pC(z)(λ).

In order to prove this proposition we need two lemmas.

Lemma 9 Given a polynomial matrix C(z) such that

m = maxi

(j − i + n deg Cij(z)

) ∀ j = 1 . . . n

(with m and n coprime) coefficients of characteristic polynomial

pC(z)(λ) := λn + c1(z)λn−1 + . . . + cn(z)

satisfy

n deg cs � ms ∀s = 1, . . . , n − 1

deg cn = m

Proof From

n deg Ci, j � m − j + i

and definition of determinant follows immediately that

n deg cs � ms ∀s = 1, . . . , n.

Strict inequality for s < n follows from the fact that m and n are coprime. Forthe equality

deg cn = deg(

det(C(z))) = m

we observe that in every line there is a unique element Cij(z) such that m = j −i + n deg Cij(z); taking this unique element for every line and multiplying themwe will obtain the leading term of determinant which will be of order m. �

Lemma 10 The equation

λn + c1(z)λn−1 + . . . + cn(z) = 0 (40)


with

n deg cs � ms ∀s = 1, . . . , n − 1

deg cn = m

and n, m coprime has n distinct solutions {λi = b(ζi), i = 1 . . . n} with

b(ζ ) = ζ m (1 + O(ζ−1))

(as usual ζi is the ith root of z).

Proof Imposing λi = ζ mi we have a solution of the equation

(ζ mi )n + c1(ζ

ni )(ζ m

i )n−1 + . . . + cn(ζni ) = 0

at the leading order mn. Then imposing λi = ζ mi (1 + l1ζ

−1i ) and plugging it into

the equation (40) one obtains

(ζ m

i + l1ζm−1i

)n + c1(ζ n

i

) (ζ m

i + l1ζm−1i

)n−1 + . . . + cn(ζ n

i

) = O(ζ mn

i

)

l1 can be found so that terms of order nm − 1 in the equation vanish; going onsolving the equation term by term we obtain

λi = ζ mi

(1 +

∑j<0

l jζ− ji

)

Clearly coefficients lj do not depend on the choice of the root ζi so that it existsb(λ) with stated properties. �

Now we can prove Proposition 20.

Proof We start computing the characteristic polynomial pC(z)(λ); thanks toLemmas 9 and 10 we find n distinct roots b(ζ1), . . . , b(ζn) with propertiesstated above.

The aim is to find W(z) such that

W(z)b(�)W−1(z) = C(z)

Since we have n distinct solutions {b(ζi), i = 1, . . . , n} of the equation

pC(z)(λ) = 0

it exists a matrix ϒ(ζ1, . . . , ζn) such that

ϒ(ζi)C(z)ϒ−1(ζi) =

⎛⎜⎜⎜⎝

b(ζ1) 0 . . . 00 b(ζ2) . . . 0

0 . . .. . . 0

0 . . . . . . b(ζn)

⎞⎟⎟⎟⎠

44 M. Cafasso

On the other hand it’s easy to observe that the matrix � can be diagonalizedas

� =

⎛⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎠

−1⎛⎜⎜⎜⎝

ζ1 0 . . . 00 ζ2 . . . 0

0 . . .. . . 0

0 . . . . . . ζn

⎞⎟⎟⎟⎠

⎛⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎠

and this means that multiplication by b(z) can be written in H(n)+ as multiplica-

tion by

⎛⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎠

−1⎛⎜⎜⎜⎝

b(ζ1) 0 . . . 00 b(ζ2) . . . 0

0 . . .. . . 0

0 . . . . . . b(ζn)

⎞⎟⎟⎟⎠

⎛⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎠

Hence we have

W(z) = ϒ−1(ζi)

⎛⎜⎜⎝

1 ζ1 . . . ζ n−11

1 ζ2 . . . ζ n−12

. . . . . . . . . . . .

1 ζn . . . ζ n−1n

⎞⎟⎟⎠

Note that W(z) is defined modulo the action of so that, by construction, C(z)

corresponds to a unique W ∈ Gr(n) such that

W = W(z)H(n)+ . �

Remark 6 As it was pointed out by Schwarz [26], matrices C(z) with propertiesstated above can be used to describe points in the Grassmannian describingstring solutions of Gelfand-Dickey hierarchies, i.e. solutions associated toreduction of type

[L, M] = 1

This class of solutions has not been treated in this article since they do not livein Segal-Wilson Grassmannian but just on Sato’s Grassmannian constructedon the space of formal series; this means that we cannot use any more Szegö-Widom theorem as the analytical requirements are not satisfied. Neverthelesssome results obtained in Section 4 still hold since the property of stability for{τW,N(t)} does not depend on analytical properties of the symbol W(z). Henceone can try to apply the approach used in this article to the study of these(much less studied) string solutions; perhaps results obtained by Okounkovand Borodin in [9] for formal series and a generalization to block case can playin the setting of formal theory the same role played by Szegö-Widom theoremin this paper.


Example 1 (Symmetric n-coverings) Take a symmetric n-covering C of P1

given by equation

λn = P(z) =nk+1∏

j=1

(z − a j

)(41)

For this particular type of curves, choosing in a appropriate way the divisoron the curve, we can write explicitly W(z), B(z) and C(z). We start to observethat for any W corresponding to this spectral curve we have b(z)W ⊆ W with

b(z) = P(zn)1/n

Then it’s easy to prove that the corresponding B(z) = b(�) can be written as

B(z) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 . . . 0 zn−1

n P(z)1/n

z−1/n P(z)1/n 0 . . . 0 0

0. . .

. . ....

......

. . .. . .

. . ....

0 . . . 0 z−1/n P(z)1/n 0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

Now we define n functions

wi(z) :=(

P(z)

z

) i−1n 1∏(i−1)k

j=1

(z − aj

) , i = 1, . . . , n.

We take

W := diag(w1(z), ..., wn(z))

It is easy to verify that the matrix

C(z) = W−1(z)B(z)W(z) =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0 0 . . . 0 zn−1

n P(z)1/n wn(z)

w1(z)

z−1/n P(z)1/n w1(z)

w2(z)0 . . . 0 0

0. . .

. . ....

......

. . .. . .

. . ....

0 . . . 0 z−1/n P(z)1/n wn−1(z)

wn(z)0

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

is polynomial in z. It is worth noticing that this example already gives all pos-sible double coverings; hence for any (possibly singular) hyperelliptc surfacewe found (assigning a particular divisor) the GD symbol of the correspondingalgebro-geometric rank one solution of KdV.

Example 2 (Rational solutions) As pointed out by Segal and Wilson [1], sub-space of Burchnall-Chaundy locus corresponding to rational curves are givenby W = W(z)H(n)

+ with W(z) rational in z. In particular the corresponding

46 M. Cafasso

Gelfand-Dickey symbol will satisfy hypothesis given in Proposition 3 so thatwe recover the following (known) result.

Proposition 21 Every rational solution of Gelfand-Dickey hierarchies can bewritten as a finite-size determinant.

For instance, for n = 2, taking

W(z) =(

1 − d2z−1 00 1 − c2z−1

)H2

+

the inverse of Gelfand-Dickey symbol is equal to

W−1(t; z)=⎛⎜⎜⎝

cosh(

z1/2(∑

i�0 t2i+1z2i))

−z1/2 sinh(

z1/2(∑

i�0 t2i+1z2i))

−z−1/2 sinh(

z1/2(∑

i�0 t2i+1z2i))

cosh(

z1/2(∑

i�0 t2i+1z2i))

⎞⎟⎟⎠

⎛⎜⎜⎝

zz−d2

0

0z

z−c2

⎞⎟⎟⎠

Simply taking the residue one obtains that the corresponding τ function willbe equal to

τW(t1, t3, ...) = det

⎛⎜⎜⎝

cosh(∑

i�0 t2i+1d2i+1)

−d sinh(∑

i�0 t2i+1d2i+1)

−c−1 sinh(∑

i�0 t2i+1c2i+1)

cosh(∑

i�0 t2i+1c2i+1)

⎞⎟⎟⎠

and recover 2-solitons solution for KdV.

We want to point out that, for algebro geometric solutions treated in thissection, the problem of factorization for Gelfand Dickey symbol can be easilytranslated into a Riemann-Hilbert problem on some cuts on the plane withconstant jumps. For simplicity we reduce to the case n = 2; the procedureused here is equivalent to the one used by Its, Jin and Korepin in [27] andgeneralized by Its, Mezzadri and Mo in [28]. Suppose we want to solve thefactorization problem

W(t; z) := exp

(ξ(t, �

))W(z) = T−

(t; z)

T+(t; z)

for our GD symbol with W(z) = diag(w1(z), w2(z)) as in Example 1; since itwill appear many times we denote A the matrix

A :=(

1√

z1 −√

z

)

Also we impose

P(z) :=2g+1∏j=1

(z − aj

)

with all aj having modulo less then 1 and

‖a1‖ < ‖a2‖ < . . . < ‖a2g+1‖


We denote l1, ...lg+1 the oriented intervals (a1, a2), (a3, a4), ...(a2g+1, ∞). In-stead of looking for T−(t; z) and T+(t; z) we define a new matrix S(t; z)

imposing

⎧⎪⎨⎪⎩

S(t; z) := A exp

(−ξ(t, �))

T−(t; z)

z � 1

S(t; z) := AW(z)T−1+

(t; z)

z � 1

Proposition 22 S(t; z) has the following properties:

– It has no jumps on S1

– It has jumps on intervals lj; precisely calling SL(t; z) and SR(t; z) the valuesof S(t; z) approaching from the left and approaching from the right theinterval we have

SL(t; z) :=

(0 11 0

)SR(t; z)

– It is invertible in any points but aj; there it has singular behaviour of type

S(t; z) ∼

(1 11 −1

)(z − aj)

⎛⎝1 0

0 ±1/2

⎞⎠

S j(t; z)

with Sj(t; z) invertible in aj; minus is for a1, . . . , ag, plus for the others.– At infinity it behaves as

S(t; z) ∼

⎛⎝exp

(− √

z(t1z + t3z + . . .)) √

z exp(

− √z(t1z + t3z + . . .)

)

exp(√

z(t1z + t3z + . . .))

−√z exp

(√z(t1z + t3z + . . .)

)⎞⎠

Proof Let’s call S+(t; z) and S−(t; z) the limiting values of S(t; z) approachingthe unit circle from inside and outside; we have

S−(t; z)

S−1+(t; z) = A exp

(−ξ(t, z))

T−(t; z)

T+(t; z)W−1(z)A−1

= A exp(−ξ

(t, z))

exp(ξ(t, z))W(z)W−1(z)A−1 = I

and this proves we haven’t any jumps on S1.

48 M. Cafasso

Writing explicitly S(t; z) as⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

S(t; z)=

⎛⎜⎜⎝

exp(−√

z(t1z + t3z+. . .)) √

z exp(−√

z(t1z+t3z+. . .))

exp(√

z(t1z+t3z+. . .))

−√z exp

(√z(t1z+t3z+. . .)

)

⎞⎟⎟⎠

T−(t; z)

z�1

S(t; z)=

⎛⎜⎜⎜⎜⎜⎝

1(P(z))1/2

∏gj=0(z−aj)

1− (P(z))1/2

∏gj=0(z−aj)

⎞⎟⎟⎟⎟⎟⎠

T−1+(t; z)

z � 1

we obtain almost immediately the other points of the proposition; the onlything we have to observe is that both T+(t; z) and T−(t; z) are invertible insideand outside the circle respectively. This is because we have

detW(t; z) = (P(z))1/2

∏gj=0

(z − aj

) = det(T+(t; z))

det(T−(t; z))

This condition combined with

limz→∞ det

(T−(t; z)) = 1

gives

det(T+(t; z)) = 1

det(T−(t; z)) = (P(z))1/2

∏gj=0

(z − aj

) .

�

The Riemann-Hilbert problem given by Proposition 22 is equivalent to theone proposed in Section 6. What can be done is to write explicitly the solutionS(t; z) using θ functions associated to the curve; this is what has been done in[27] and [28]. Actually comparing previous proposition with results obtainedin Section 6 we immediately realize that

S(t; z) =

⎛⎝

ψW,0(t; √

z)

ψW,1(t; √

z)

ψW,0(t; −√

z)

ψW,1(t; −√

z)

⎞⎠

so that all we have to do in our case is to write down Baker-Akhiezerfunction in terms of special functions. We can carry on the same procedurefor n arbitrary; the only difference will be that the jump matrices will remain


constant but more complicated; in any case the solution of this Riemann-Hilbert problem with constant jumps will be

S(t; z) =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

ψW,0(t; ζ1

)ψW,1

(t; ζ1

). . . ψW,n−1

(t; ζ1

)

ψW,0(t; ζ2

)ψW,1

(t; ζ2

). . . ψW,n−1

(t; ζ2

)

. . . . . . . . . . . .

ψW,0(t; ζn

)ψW,1

(t; ζn

). . . ψW,n−1

(t; ζn

)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

Explicit formulas involving special functions can be used here to apply Propo-sition 4 to our case. For instance taking the elliptic curve C given by equation

w2 = 4z3 − g2z − g3

with uniformization given by the Weierstass ℘ function

(z, w) = (℘ (u), ℘ ′(u))

one can write wave function as

ψ(x, t, u) := σ(u − c − x)σ (c)σ (u − c)σ (x + c)

exp(xζ(u) − 1/2t℘ ′(u)

)

(here ζ and σ are Weierstrass ζ and σ function respectively, x and t correspondto the first and the third time). With some tedious computations, makingthe change of variables u = u(z), the right hand side of equation (7) can beobtained. It turns out that the only relevant factorization is the one given by

W−1(x, t; u) = [W−1(u)� (−x, −t; u)] [

�−1(−x, −t; u) exp(−x� − t�3

)]

where as before (we just wrote z as a function of u) we have

�(x, t; u) :=(

1 (℘ (u))1/2

1 −(℘ (u))1/2

)−1 (ψ(x, t, u) ∂xψ(x, t, u)

ψ(x, t, −u) ∂xψ(x, t, −u)

)

Plugging into equation (7) we obtain

ddx

τ(x, t) = Kt + 2ζ(−c) − 2ζ(x − c)

(here K is some constant); taking another derivative we obtain elliptic solutionof KdV as expected.

Acknowledgements I am grateful to A. Its for a fruitful discussion in which he pointed out thatour Riemann-Hilbert problem can be reduced to a problem with constant jump as in [27, 28] andgave me some interesting hints about possible developments of this work.

J. van de Leur suggested me to verify if vector-constrained reductions of KP had some relationswith block Toeplitz determinant, I am grateful to him for this really useful suggestion.

50 M. Cafasso

Moreover I wish to thank my advisor B.Dubrovin for his constant support and suggestions hegave me during many hours spent together discussing the preparation of this paper.

This work is partially supported by the European Science Foundation Programme ‘Methodsof Integrable Systems, Geometry, Applied Mathematics’ (MISGAM), the Marie Curie RTN‘European Network in Geometry, Mathematical Physics and Applications’ (ENIGMA), and bythe Italian Ministry of Universities and Researches (MUR) research grant PRIN 2006 ‘Geometricmethods in the theory of nonlinear waves and their applications’.

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Degree Complexity of a Family of Birational Maps

Eric Bedford · Kyounghee Kim ·Tuyen Trung Truong · Nina Abarenkova ·Jean-Marie Maillard

Received: 13 November 2007 / Accepted: 31 March 2008 /Published online: 3 June 2008© Springer Science + Business Media B.V. 2008

Abstract We compute the degree complexity of a family of birational map-pings of the plane with high order singularities.

Keywords Birational mappings · Degree complexity

Mathematics Subject Classifications (2000) 37F99 · 32H50

Research of Eric Bedford was supported in part by the NSF. Research of Nina Abarenkovasupported in part by a Russian Academy of Sciences/CNRS program.

E. Bedford (B) · T. T. TruongDepartment of Mathematics, Indiana University, Bloomington, IN 47405, USAe-mail: [email protected]

T. T. Truonge-mail: [email protected]

K. KimDepartment of Mathematics, Florida State University, Tallahassee, FL 32306, USAe-mail: [email protected]

N. AbarenkovaLaboratory of Mathematical Problems of Physics, Petersburg Department,Steklov Institute of Mathematics, 27, Fontanka, 191023, St. Petersburg, Russiae-mail: [email protected]

J.-M. MaillardLab. de Physique Théorique et de la Matière Condensée, Université de Paris 6,Tour 24, 4ème étage, case 121, 4, Place Jussieu, 75252 Paris Cedex 05, Francee-mail: [email protected]

54 E. Bedford et al.

1 Introduction

Birational maps have been found to arise in lattice statistical mechanics, forinstance in vertex models or in spin-edge models [17, 28, 36]. These arefundamental non-linear symmetries of the parameter space that arise fromnatural (geometrical) symmetries of the lattice, combined with the so-calledinversion relation [29–31]. In the case of the Yang-Baxter integrability, theanalysis of these symmetries can lead efficiently to a parameterization ofthe Yang-Baxter equations in terms of selected algebraic varieties [27, 32].More generally, beyond Yang-Baxter integrability, these birational maps haveto be compatible with the phase diagram of the model [33], and, for instance,with the renormalization group. It is important to note that for non-Yang-Baxter-integrable lattice models, these birational maps can still be integrable[14]. The connection between birational mappings and lattice statistical me-chanics is discussed in greater length, for instance, in [9, 15] (Maillard, unpub-lished manuscript). Furthermore, and far beyond lattice statistical mechanics,one even can consider these birational transformations, per se, since theynaturally correspond to a very important class of discrete dynamical systems,namely the reversible [18, 34, 35] discrete dynamical systems.

One such map gives rise to a family ka,b of birational maps of the plane (see[2, 20–22]). Dynamical properties of this family have been studied in a numberof works [1–9, 11, 17, 23]. Recall the quantity

δ(k) := limn→∞

(deg

(kn)) 1

n ,

which is the exponential rate of growth of the iterates of k. This is variouslyknown as the degree complexity, the dynamical degree, or the algebraicentropy of k. When b �= 0 and a is generic, δ(ka,b ) is the largest root of thepolynomial x3 − x2 − 2x − 1. When b = 0 and a is generic, δ(ka,0) is the largestroot of x2 − x − 1. The form of a map can change radically under birationalequivalence: a simpler form for ka,0 which was obtained in [19] made it moreaccessible to detailed analysis (see [10, 12]).

A basic property is that k is reversible in the sense that k = j ◦ ι is acomposition of two involutions. In this case, j corresponds to lattice symmetry,and ι corresponds to matrix inversion. In this paper we give (a birationallyequivalent version of) k as a composition of involutions in a new way. Thisshows how ka,b fits naturally into a larger family of maps. Namely, for anyrational function F(y) = P(y)/Q(y), we define the involutions

jF(x, y) = (−x + F(y), y), ι(x, y) =(

1 − x − x − 1

y, −y − 1 − y

x − 1

),

and the family of birational maps is given by kF = jF ◦ ι. When F is constant,the family kF is birationally equivalent to ka,0, and when F is linear, kF

Degree complexity of a family of birational maps 55

is equivalent to ka,b . In this paper we determine the structure and degreecomplexity for the maps kF :

Theorem 1 Let p (resp. q) denote the degree of P (resp. Q). If p < q, thenfor generic parameters δ(kF) = q + 1. Otherwise, if p − q � 0 is even, then forgeneric parameters δ(kF) is the largest root of the polynomial x2 − (p + 1)x −(q + 1). If p − q � 0 is odd, then for generic parameters δ(kF) is the largest rootof x3 − px2 − (p + q + 1)x − (q + 1).

When kF is not generic, the growth rate δ(kF) decreases (i.e. F �→ δ(kF) islower semicontinuous in the Zariski topology). One of the interesting thingsabout the family is to know which parameters are not generic as well as thecorresponding values of δ(kF) is decreased. The exceptional values of a for thefamily ka,0, as well as the corresponding values of δ(ka,0), were found by Dillerand Favre [24]. Similarly, the exceptional values of (a, b) are given in [11].Here we look at the maximally exceptional parameters for the case where F iscubic. These are the cubic maps with the slowest degree growth and give a 2complex parameter family of maps which are (equivalent to) automorphisms:

Theorem 2 If F(y) = ay3 + ay2 + b y + 2, a �= 0, then kF is an automorphismof a compact, complex surface Z . Further, the degrees of kn

F grow quadratically,and kF is integrable.

We will analyze the family kF by inspecting the blowing-up and blowing-down behavior. That is, there are exceptional curves, which are mapped topoints; and there are points of indeterminacy, which are blown up to curves. Aswas noted by Fornæss and Sibony [25], if there is an exceptional curve whoseorbit lands on a point of indeterminacy, then the degree is not multiplicative:(deg(kF))n �= deg(kn

F). The approach we use here is to replace the originaldomain P2 by a new manifold X . That is, we find a birational map ϕ : X → P2,and we consider the new birational map k = ϕ ◦ kF ◦ ϕ−1. There is a welldefined map k∗ : Pic(X ) → Pic(X ), and the point is to choose X so that theinduced map k satisfies (k∗)n = (kn)∗. By the birational invariance of δ (see[16] and [24]) we conclude that δ(kF) is the spectral radius of k∗. This methodhas also been used by Takenawa [37–39]. The general existence of such a mapk when δ(k) > 1 was shown in [24]. We comment that the construction of Xand k can yield further information about the dynamics of k (see, for instance,[13] and [11]).

The bulk of this paper is devoted to proving Theorem 1. After a division,we may rewrite F(y) = an yn + · · · + a1 y + a0 + P(y)/Q(y), where deg(P) <

deg(Q). In our treatment below, we first do the cases where F(y) is a polyno-mial, which is in some sense the most singular and most difficult case becauseit involves iterated blowups to depth n. We give general properties of themap kF in Section 2. In Section 3, we describe the iterated blowup process


in some detail. In Section 4 we carry out the blowup process to regularizekF in the case where F is a polynomial of even degree n. What we do hereis to determine the action of the induced map k∗

F on Pic(X ); and δ(kF) isthe spectral radius of k∗

F . The case n odd is distinct, and we carry it out inSection 5. In Section 6 we handle the case where F(y) = P(y)/Q(y) withdeg(P) � deg(Q). That is, we present the blowup procedure, and we determinek∗

F . We will see in Section 6 that the blowup process for the cases q = 0 andq � p are essentially independent, since the blowup operations are performedin different places. After Sections 2–6, it is not hard to put these separateanalyses together to cover the general case. The Picard group in the generalcase is generated by the elements produced in the independent cases, and theinduced linear transformation k∗

F maps them the same way it does when theyare independent. Thus it is just a matter of bookkeeping to combine the twocases. Since Sections 2–5 and Section 6 are the two parts that need to be puttogether, and since it would be repetitive to do them simultaneously, we omitthe details.

The exceptional cases are also of considerable interest, but many cases arise,and it is not easy to handle them efficiently, so we do not address this issue here.As an example, however, we treat in Section 7 the case where the coefficientsare as non-generic as possible when p = 3, q = 0. This leads to a proof ofTheorem 2, which gives a family of automorphisms for which the degrees growquadratically.

2 The Maps

Let us set F(z) = ∑nj=0 a jz j with n � 2 and an �= 0. The map k = jF ◦ ι is the

composition of the two involutions defined above. The map k = [k0 : k1 : k2]is given in homogeneous coordinates as

k0 = (x0x1 − x2

0

)nx2

k1 = xn−10 (x1 − x0)

n+1(x2 + x0) + x2

n∑

j=0

a j(x0x1 − x2

0

)n− j (x2

2 − x0x1 − x1x2) j

k2 = x2(x0x1 − x2

0

)n−1 (x2

2 − x0x1 − x1x2). (1)

Each coordinate function has degree 2n + 1, which means that deg(k) =2 deg(F) + 1. Since the jacobian of this map is x3n−3

0 (x0 − x1)3n−1x2

2

(x2

0−x0x1 − x1x2

)we have four exceptional curves :

C1 :={x0 = 0}, C2 :={x0 =x1}, C3 :={x2 =0}, C4 :={−x20 + x0x1 + x1x2 =0

}.

When n � 2 and a0 �= 2, the exceptional hypersurfaces are mapped as:

k : C4 �→ [1 : −1 + a0 : 0] ∈ C3 and C1 ∪ C2 ∪ C3 �→ e1. (2)

The points of indeterminacy for k are

e1 := [0 : 1 : 0], e2 := [0 : 0 : 1], and e01 := [1 : 1 : 0].


Fig. 1 n � 2. Exceptionalcurves and points ofindeterminacy

Figure 1 shows the relative position of the points of indeterminacy (dots withcircles around them), exceptional curves, and the critical images (big dots).The information that C1, C2, C3 → e1 is not drawn for lack of space.

The sort of singularity that will be the most difficult to deal with arisesfrom the exceptional curve C1 �→ e1 ∈ C1. In local coordinates near e1, thislooks like

k[t : 1 : y] =[

tn + · · ·an(−y)n + · · · : 1 : tn−1 + · · ·

an(−y)n−1 + · · ·]

. (3)

For this, we will perform the iterated blowups described in Section 3.The inverse map k−1 = [

k−10 : k−1

1 : k−12

]is given as

k−10 = xn

0 x2

(F − xn−1

0 (x0 + x1))

k−11 = (x0 + x2)

⎛

⎝n∑

j=0

a jxn− j0 x j

2 − xn−10 (x0 + x1)

⎞

⎠

2

k−12 = xn−1

0 x2

(xn−1

0

(x2

0 + x0x1 + x1x2) − (x0 + x2)F

)

where F = xn0 F(x2/x0) = ∑n

j=0 a jxn− j0 x j

2. The jacobian for the inverse map is

x3n−30 x2

2

(xn

0 + xn−10 x1 − F

)2 (xn+1

0 − (x0 + x2)(

xn0 + xn−1

0 x1 − F))

The exceptional curves for k−1 are C′j, 1 � j � 4, where

C′1 = C1, C′

2 :={

xn0 + xn−1

0 x1 − F = 0}

, C′3 = C3,

C′4 :=

{xn+1

0 − (x0 + x2)(

xn0 + xn−1

0 x1 − F)

= 0}

.

k−1 : C′1 ∪ C′

3 �→ e1, C′2 �→ e2, and C′

4 �→ e01 ∈ C′3, (4)


3 Blowups and Local Coordinate Systems

In this section we discuss iterated blowups, and we explain the choices of localcoordinates which will be useful in the sequel. Let π : X → C2 denote thecomplex manifold obtained by blowing up the origin e = (0, 0); the space isgiven by

X = {((t, y), [ξ : η]) ∈ C2 × P ; tη = yξ

},

and π is projection to C2. Let E := π−1(e) denote the exceptional fiber overthe origin, and note that π−1 is well defined over C2 − e. The closure in Xof the y-axis (π−1({t = 0} − e)) corresponds to the hypersurface {ξ = 0} ⊂ X.On the complement {ξ �= 0} set u = t and η = y/t. Then (u, η) defines acoordinate system on X \ {t = 0}, with a point being given by ((t, y), [1 :y/t]) = ((u, uη), [1 : η]). We will use the notation (u, η)L. On the set t �= 0, thecoordinate projection π is given in these coordinates as

πL(u, η)L = (u, uη) = (t, y) ∈ C2. (5)

Figure 2a illustrates this blowup with emphasis on the relation between thepoint e and the lines t = 0 and y = 0 which contain it. The space X is drawntwice to show two choices of coordinate system; the dashed lines show whereeach coordinate system fails to be defined. The left hand copy of X shows theu, η-coordinate system in the complement of t = 0. The right hand side showsa different choice of coordinate; we would choose this coordinate system towork in a neighborhood of the point p1 := E ∩ {t = 0}.

In the u, η coordinate system (on the upper left side of Fig. 2a), theη-axis (u = 0) represents the exceptional fiber E ∼= P1. The line γη = {(s, η)L :s ∈ C} projects to the line {y = ηt} ⊂ C2, and (0, η)L = E ∩ γη. It follows thatE ∩ {y = 0} = (0, 0)L in this coordinate system.

On the upper right side of Fig. 2a, we define a (ξ, v)-coordinate system onthe complement of t-axis (y = 0):

πR : (ξ, v)R = (t/y, y) → (vξ, v) ∈ C2. (6)

The exceptional fiber E is given by ξ -axis (v = 0). Next we blow up p1 = E ∩{t = 0} = {ξ = v = 0} = (0, 0)R. Let P1 denote the exceptional fiber over p1.The choice of a local coordinate system depends on the center of next blowup.Suppose the third blowup center is an intersection of two exceptional fibersp2 := E ∩ P1. For this we are led to the (u, η)- coordinate system, as on theleft side of Fig. 2a. Thus we have a local coordinate system on the complementof {t = 0} ∪ {y = 0};

(u1, η1)1 = (t/y, y2/t

) → (u1, u1η1)R → (u2

1η1, u1η1) ∈ C2. (7)

This (u1, η1)-coordinate system is defined only off the axes (t = 0) ∪ (y = 0);the new exceptional fiber P1 is given by the η1-axis.

Now we define a sequence of iterated blowups which will let us deal withthe singularity (3). We start with the blowup space X as in Fig. 2b, and wecontinue inductively for 2 � j � n by setting pj := E ∩ Pj−1 and letting Pj be


Fig. 2 a Two choices of localcoordinate systems. b Blowupof p1 in (u1, η1)-coordinates.c n-th iterated blowup η ξ

ππ

a

η

b

c

the exceptional fiber. For each 2 � j � n, we use the left-hand coordinatesystem of Fig. 2a, which corresponds to (5). Thus we have the coordinateprojection πj : Pj → C2:

π j : (u, η)j → (u j+1η, u jη

) = (t, y) ∈ C2, π−1j (t, y) = (u, η) = (

t/y, y j+1/t j ) .

(8)


This coordinate system is defined off of {y = 0} ∪ {t = 0} ∪ P1 ∪ · · · ∪ Pj−1. Apoint (0, η = c)j ∈ Pj is the landing point of the curve u �→ (u, c) j as u → 0,which projects to the curve u �→ (t(u) = u j+1c, y(u) = u jc) ∈ C2. In Fig. 2c, theexceptional fibers Pj, 1 � j � n are drawn with their fiber coordinates y j+1/t j.

4 Mappings with q = 0 and n = Even

We define a complex manifold πX : X → P2 by blowing up points e1, q,

p1, . . . , pn−1 in the following order:

(a) blow up e1 = [0 : 1 : 0] and let E1 denote the exceptional fiber over e1,(b) blow up q := E1 ∩ C4 and let Q denote the exceptional fiber over q,(c) blow up p1 := E1 ∩ C1 and let P1 denote the exceptional fiber over p1,(d) blow up pj := E1 ∩ Pj−1 with exceptional fiber Pj for 2 � j � n − 1.

The iterated blow-up of p1, . . . , pn−1 is exactly the process described inSection 3, so we will use the local coordinate systems defined there. That is, ina neighborhood of Q we use a (ξ1, v1) = (t 2/y, y/t) coordinate system. For E1

and Pj, 1 � j � n − 1 we use local coordinate systems defined in (6–8). We usehomogeneous coordinates by identifying a point (t, y) ∈ C2 with [t : 1 : y] ∈ P2.Let kX : X → X denote the induced map on the complex manifold X . In thenext few lemmas, we will show that kX maps the exceptional fibers as shownin Fig. 3.

Lemma 1 Under the induced map kX , the blowup fibers E1 and Pn−1 aremapped to themselves:

kX : E1 � ξ �→ −ξ/(ξ + 1) ∈ E1

Pn−1 � ηn−1 �→ ηn−1/(1 + anηn−1) ∈ Pn−1. (9)

Proof First let us work on E1. We use the local coordinate system defined in(6), so a point in the exceptional fiber E1 is (ξ, 0)R. To see the forward imageof E1 we consider a nearby point (ξ, v)R → (vξ, v) with small v and we havekX (ξ, 0)R = limv→0 kX (ξ, v)R. By (1) we see that

k [vξ : 1 : v] = [vξ + · · · : 1 + · · · : −v(ξ + 1) + · · · ]

Fig. 3 The space X and theaction of kX


where we use · · · to indicate the higher order terms in v. As in Fig. 2a,the coordinate of the landing point in E1 is given by the ratio of t- and y-coordinates. Thus we have

kX |E1 : ξ �→ limv→0

k0/k2 = limv→0

(vξ + · · · )/(−v(ξ + 1) + · · · ) = −ξ/(ξ + 1).

Now we determine the behavior of kX on Pn−1. A fiber point (0, ηn−1) ∈Pn−1 is the landing point of the arc u �→ (u, ηn−1) as u → 0. To show that kXmaps Pn−1 to Pn−1, we need to evaluate:

limu→0

kX (u, ηn−1) = limu→0

π−1n−1 ◦ k ◦ πn−1(u, ηn−1).

Using the formulas for πn−1 and π−1n−1 in (8), we obtained the desired limit. ��

Now we may use similar calculations to show that kX : Pj → Pn−1; we fix apoint (0, η j) ∈ Pj and show the existence of the limit

limu→0

kX(u, η j

) = limu→0

π−1n−1 ◦ k ◦ π j

(u, η j

).

Doing this, we find that the line C1 and all blowup fibers Pj, j = 1, . . . , n − 2are all exceptional for both kX and k−1

X . And C2 is exceptional for kX :

kX : C1, C2, P1, · · · , Pn−2 �→ 1/an ∈ Pn−1

k−1X : C1, P1, · · · , Pn−2 �→ (−1)n−1/an ∈ Pn−1 (10)

Combining (9–10) it is clear that the indeterminacy locus of kX consists ofthree points

e2, e01, and (−1)n−1/an ∈ Pn−1.

Lemma 2 If n is even, then the orbits of the exceptional curves C1, C2,

P1, . . . , Pn−2 are disjoint from the indeterminacy locus.

Proof By Lemma 1, the orbit of 1/an in Pn−1 is {1/an, 1/(2an), 1/(3an), . . . } ⊂Pn−1. This is disjoint from the indeterminacy locus since it does not containpoint −1/an in Pn−1. ��

A computation as in the proof of Lemma 1 shows that kX maps Q ↔ C3

according to:

kX : Q � ξ1 �→ [1 : a0 − ξ1 : 0] ∈ C3,

C3 � [x0 : x1 : 0] �→ −x1/x0 ∈ Q. (11)

Lemma 3 If a0 �= 2/m for all m > 0 then the indeterminacy locus of kX andthe forward orbit of C4 under the induced map kX are disjoint. If a0 = 2/mfor some m > 0, we have k2m−1

X C4 = e01.

Proof Since the forward image of C4 is [1 : −1 + a0 : 0] ∈ C3, using (11)we have that k2m−1

X C4 = [1 : ma0 − 1 : 0] ∈ C3. Since the unique point of


indeterminacy in C3 is e01, for C4 to be mapped to a point of indeterminacy,a0 must satisfy ma0 − 1 = 1 for some m � 0. ��

The following theorem comes directly from previous lemmas.

Theorem 3 Suppose that n is even and a0 �= 2/m for all integers m � 0. Thenno orbit of an exceptional curve contains a point of indeterminacy.

Let us recall the Picard group Pic(X ), which is the set of all divisors inX , modulo linear equivalence, which means that D1 ∼ D2 if D1 − D2 is thedivisor of a rational function. Pic(P2) is 1-dimensional and generated by theclass of any line (hyperplane) H, and a basis of Pic(X ) is given by the classof a general hyperplane HX := π∗ H, together with all of the blowup fibersE1, Q, P1, . . . , Pn−1. If r is a rational function on X , then the pullback k∗

X r :=r ◦ kX is just the composition. To pull back a divisor, we just pull back itsdefining functions. This gives the pullback map k∗

X : Pic(X ) → Pic(X ). Thusfrom (9–10) we see that the pullback of E1 is E1 and the pulling back of mostof basis elements are trivial, that is k∗

X Pj = 0 for all j = 1, . . . , n − 2.Next we pull back HX . Since k has degree 2n + 1 we have k∗ H = (2n + 1)H

in Pic(P2). Now we pull back by π∗X to obtain:

(2n + 1)HX = π∗X (2n + 1)H = π∗

X (k∗ H). (12)

A line is given by {h := α0x0 + α1x1 + α2x2 = 0}, so k∗ H is the divisor definedby h ◦ k = ∑

j α jk j. To write this divisor as a linear combination of basiselements HX , E1, Q, P1, . . . , Pn−1, we need to check the order of vanishingof h ◦ k at all of these sets. Let us start with the coordinate system πX (ξ, v) =[vξ : 1 : v] near E1, defined in Section 3. Using the expression for k given inSection 2 we see that α0k0 + α1k1 + α2k2 vanishes to order n in v. It follows thatπ∗X k∗ H vanishes at E1 with multiplicity n. Similar computations for all other

basis elements gives us π∗X k∗ H = k∗

X HX + nE1 + (n + 1)Q + (n + 1)∑

j jPj.Combining with (12) we have

k∗X HX = (2n + 1)HX − nE1 − (n + 1)Q − (n + 1)

n−1∑

j=1

jPj. (13)

Similarly, we obtain:

k∗X : Q �→ HX − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1

Pn−1 �→ 2HX − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1. (14)

Theorem 4 q = 0 and n = even. Suppose F(z) = ∑nj=1 a jz j is an even degree

polynomial associated with jF. If a0 �= 2/m for any positive integer m, thenthe degree complexity is the largest root of the quadratic polynomial x2−(n + 1)x − 1.

Proof Since P1, . . . , Pn−2 are mapped to 0 under the action on cohomol-ogy, it suffices to consider the action restricted to HX , E1, Q, and Pn−1.


By (13, 14) the matrix representation of k∗X , restricted to the ordered basis

{HX , E1, Q, Pn−1}, is⎛

⎜⎜⎝

2n + 1 0 1 2−n 1 −1 −1

−n − 1 0 −1 −1−n2 + 1 0 −n + 1 −n + 1

⎞

⎟⎟⎠ .

The characteristic polynomial is x(x − 1)(x2 − (n + 1)x − 1). ��

5 Mappings with q = 0 and n = Odd

Let us start with the space X from Section 4. When n is odd, we see from (10)that the image of all exceptional lines of kX coincide with a point of indeter-minacy in pn ∈ Pn−1. Let πY : Y → P2 be the complex manifold obtained byblowing up X at the point pn, and let Pn denote the exceptional fiber over pn.In the un−1, ηn−1 coordinate system, pn has coordinate (0, 1/an)n−1. Thus, atPn, we use the coordinate projection:

πn : Y � (u, η)n → (un(uη + 1/an), un−1(uη + 1/an)) ∈ C2.

Most computations in the previous section remain valid for n odd. ThusLemma 3, (9) and (11) are still valid for the induced map kY : Y → Y . UnderkY curves C1, C2, P1, . . . , Pn−3 are still exceptional:

kY : C1, C2, P1, . . . , Pn−3 �→ −an−1/a2n ∈ Pn

k−1Y : C1, P1, . . . , Pn−3 �→ (an−1 − (n − 1)an)/a2

n ∈ Pn. (15)

The blowup fibers Pn and Pn−2 form a two cycle, kY : Pn ↔ Pn−2 and Pn−1 ismapped to itself as before. It follows that the points of indeterminacy for kYare e2, e01 and (an−1 − (n − 1)an)/a2

n ∈ Pn. For all m � 0, we have

k2mY : Pn � −an−1/a2

n �→ (2man − (2m + 1)an−1)/a2n ∈ Pn (16)

The induced action of k on Y is pictured in Fig. 4. As a consequence of (15)and (16) we have:

Fig. 4 The space Y and theaction of kY


Lemma 4 If n is odd, and if

(2m + 2)an−1 �= (2m + n − 1)an (17)

for all m � 0, then the forward orbits of C1, C2, P1, . . . , Pn−3 under kY do notcontain any point of indeterminacy.

Combining Lemmas 3 and 4 we have

Theorem 5 Suppose that n is odd, a0 �= 2/m for all m > 0, and an−1 �= (n −1)an/2. Then the forward orbits of exceptional curves do not contain any pointsof indeterminacy.

To determine kY , we use the basis {HY , E1, Q, P1, . . . , Pn} for Pic(Y).Now the exceptional lines C1, C2, P1, . . . , Pn−2 are mapped to Pn. Let {C1} ∈Pic(Y) denote the class of the strict transform of C1, i.e., the closure in Yof π−1

Y (C1 − centers of blowup). (The curve C2 does not pass through anycenter of blowup, so with the same notation we have {C2} = HY ∈ Pic(Y).) Inorder to write {C1 = (x0 = 0)} in terms of our basis, we note first that π−1

Y C1 =C4 ∪ E1 ∪ Q ∪ P1 ∪ · · · ∪ Pn−1, i.e., the pullback function x0 ◦ πY vanishes onall of these curves. Thus we have to compute the multiplicities of vanishing. AtPn−1, for instance, we consider the (un−1, ηn−1) coordinate system defined in(8), and we see that k∗

Yx0 vanishes to order n at Pn−1 = (un−1 = 0). Similarlywe can compute the multiplicities for E1, Q, P1, . . . , Pn−2 and Pn, so

HY = π∗YC1 = {C1} + E1 + Q + 2P1 + 3P2 + · · · + nPn−1 + nPn.

It follows that

k∗Y Pn = {C1} + {C2} +

n−2∑

j=1

Pj = 2HY − E1 − Q −n−2∑

j=1

jPj − nPn−1 − nPn.

For the rest of basis entries we have

k∗Y : HY �→ (2n + 1)HY − nE1 − (n + 1)Q − (n + 1)

n−1∑

j=1

jPj − n2 Pn

Q �→ HY − E1 − Q − P1 − 2P2 − · · · − (n − 1)Pn−1 − (n − 1)Pn,

E1 �→ E1, Pn−2 �→ Pn, and Pn−1 �→ Pn−1.

Theorem 1: q = 0 and n = odd. If a0 �= 2/m for all m > 0, then the degreecomplexity is the largest root of the cubic polynomial x3 − nx2 − (n + 1)x − 1.

Proof The classes of the exceptional fibers P1, · · · , Pn−3 are all mapped to 0,and exceptional fibers E1 and Pn−1 are simply interchanged. It follows thatto get the spectral radius of k∗

Y we only need to consider 4 × 4 matrix withordered basis {HX , Q, Pn−2, Pn} and the spectral radius is given by the largestroot of x3 − nx2 − (n + 1)x − 1. ��


6 Mappings with p �� q

Now we consider the case F(w) = P(w)/∏q

�=1(w − β�), where the degree of Pis no greater than q. In this case we have a limit λ0 = limw→∞ F(w), and λ0 �= 0if p = q, and λ0 = 0 if p < q. We see that E( jF) = ⋃q

�=1{y = β�}. In case p � q,we have

E(kF) = C2 ∪ C3 ∪ C4 ∪q⋃

�=1

D�

where D� = ι{y = β�}. This is different from the previous case (in Section 2) inseveral ways: (1) C1 is no longer exceptional; (2) C2 is mapped to e2 instead ofe1, and (3) D�, 1 � � � q, are exceptional.

As before, we start by blowing up e1 to create an exceptional fiber E1, and asbefore, C3 maps to a point q ∈ E1, which is indeterminate. So we also blow up qto obtain an exceptional fiber Q. Finally, we blow up the indeterminate point e2

to create an exceptional fiber E2. Figure 5b shows the mapping of exceptionalcurves under the induced map k at this stage. We note that the intersection

λ

a

λ

c

b

d

Fig. 5 a p � q. Exceptional curves and points of indeterminacy. b p = q. Mapping of theexceptional fibers after first blowups. c p = q. Mapping of the exceptional fibers at the secondstage. d The case p < q


E2 ∩ C4 ∩ D� is indeterminate, and it corresponds to fiber coordinate equal tozero. We have drawn the case λ0 �= 0, in which case the exceptional image ofC2 never encounters a point of indeterminacy because λ0 ↔ 1 is a 2-cycle. Asbefore, we see that the orbit of C4 never encounters the point of indeterminacyC2 ∩ C3 ∩ C4 if the parameters are generic. Specifically, as in Section 4, weneed F(0) �= 2/m for any positive integer m.

It remains to look at the orbits of the curves D�. If we use the coordinatesystem (s, ξ) �→ [s : 1 : sξ ] for E1, then the induced map on E1 maps to E1 andis the involution ξ �→ −(ξ + 1). The image of D� is then ξ = β� ∈ E1, which ismapped to −(1 + β�). On the other hand,

k−1F : {y = β�} → {ξ = −(β� + 1)} ∈ E1,

which means that −(β� + 1) is indeterminate for kF . So now at the secondstage, we blow up the points β� and −(β� + 1) in E1, 1 � � � q. The exceptionalcurves for the induced map are now C4, C2, and D�, 1 � � � q. Figure 5cshows how the exceptional fibers map at this stage. We see that for genericparameters, these orbits do not meet the indeterminacy locus.

Now we will describe the behavior of k∗F on the Picard group in terms of the

ordered basis L, E1, Q, E2, A�, B�, 1 � � � q. We see that k∗F acts as:

E1 ↔ E1, E2 ↔ E2, Q → C3, B� → A� → D� + B�. (18)

To make use of (18) we must write C3 and D� in terms of our basis. In P2, wehave L = C3. Now if we move “up,” taking the pullback π∗ as we make thevarious blowups, we find that at the second stage we have

L = C3 + E1 + Q +q∑

�=1

(A� + B�),

and this gives us C3 in terms of our basis. Similarly, we start with D� = 2L inP2 since D� has degree 2. After the first stage of blowups, we have

D� + E1 + E2 + Q = 2L.

For the second stage of blowups, we see from Fig. 5b that one of the centers ofblowup (the one that produces B�) belongs to both E1 and D�. Thus we havean “extra” B�:

D� + B� + E1 + E2 +q∑

s=1

(Bs + As) + Q = 2L.

This gives D� in terms of our basis.Finally, we need to express k∗

F L in terms of our basis. We have {k−1L} =(2q + 3)L in P2 because kF has degree 2q + 3. Now when we blow up e1,for instance, we will obtain a fiber E1 with multiplicity. To determine themultiplicity we work in local coordinates (s, η) → [sη : 1 : s] near E1 = {s = 0}.We write a generic line as L = {∑

a jx j}, so k−1

{∑a jx j

} = {∑a jk j = 0

}. In


the (s, η) coordinates, we have∑

a jk j[sη : 1 : s] = sq+1ϕ(s, η), where ϕ(0, η) isnot identically zero. Thus the multiplicity of E1 is q + 1, or

{k−1

F L} + (q + 1)E1 = (2q + 3)L

at the next level. Repeating this argument for the various blowup fibers,we find

{k−1L

} + (q + 1)E1 + (q + 1)E2 + (q + 2)Q ++

∑

�

((q + 1)A� + (q + 2)B�) = (2q + 3)L, (19)

which gives us {k−1F L} in terms of our basis.

Using (18–19), we may write k∗F on the Picard group, and we find that its

characteristic polynomial is x2 − (q + 1)x − (q + 1). This proves Theorem 1 inthe case p = q.

Next we consider the case where p < q, which means that λ0 = 0. Thus afterthe second stage of blowups, we see in Fig. 5c that C2 now maps to the pointof indeterminacy 0 ∈ E2. Now we blow up 0 ∈ E2, creating a new fiber G. Wefind that on our new manifold, the curve C2 is no longer exceptional and mapsonto G. Thus we add G to our ordered basis in the Picard group. The action ofk∗

F is changed in the following ways. First, we now have

E2 → E2, G → C2 = L − E2 − G.

Next, there is a change in D�. Since G was obtained by blowing up the(transversal) intersection point of D� and E2, the expression D� = 2L−E2 − · · · is changed to D� = 2L − E2 − 2G − · · · . Last, we subtract an extra(2q + 2)G from k∗

F(L). With this new expression for k∗F , we obtain the charac-

teristic polynomial

−(x − q − 1)(x − 1)3(x + 1)x2q,

and this proves Theorem 1 in the case p < q.If n = 1, we have F(w) = aw + P(w)/Q(w), where deg(P) � deg(Q). The

situation is like what we have just done in this section, with the added fact that

C2 → [0 : a : 1] → e2 ∈ I(kF).

We blow up [0 : a : 1] and e2, creating new fibers M and E2. The induced mapbehaves like

C2 → ∗ ∈ M ↔ E2.

For generic parameters, the orbit of C2 does not encounter the indeterminacylocus. To finish the proof, now, we go back and repeat the earlier parts ofSection 6.

Proof of Theorem 1 In order to prove Theorem 1 in general, we first doSection 6, which covers the cases n = 0 and n − 1. If n � 2, we go back toSection 4 or Section 5, according to whether n is even or odd. The associatedPicard group will be larger because of the iterated blowups over the point


0 ∈ E1. However, the fibers arising from the iterated blowup are disjoint fromthe blowups in Section 6 and so they still map the same as in Sections 4, 5,and the multiplicities of the pullback of a general line are the same. This givesus k∗

F in this case. Finding the characteristic polynomial in this case gives usTheorem 1.

7 Degree 3: A Family of Automorphisms

Let us consider the 2 parameter family of maps k = jF ◦ ι where F(z) = az3+az2 + bz + 2 with a �= 0. We consider the complex manifold πZ : Z → P2

obtained by blowing up 6 points e2, e01, p4, p5, p6, r in the complex mani-fold Y constructed in Section 5. As we construct the blowups, we will letE2, E01, P4, P5, P6 and R denote the exceptional fibers over e2, e01, p4, p5, p6,

and r respectively. Specifically, we blow up e2 and e01 and then:

p4 := −1/a ∈ P3, p5 := (2 − b)/a ∈ P4,

p6 := (2b − 2 − a)/a2 ∈ P5, and r := 0 ∈ E2 ∩ {x1 = 0}.

We define the local coordinate system in a similar way we define localcoordinates in Section 3. Using these local coordinates we can easily verifythat under the induced map kZ we have

C1 → P4 → C1, E2 → P5 → E2, C4 → E01 → C′4, and C2 → P6 → R → C′

2

and all mappings are dominant and holomorphic.For example, let us consider E2. We may use coordinates w, ζ which are

mapped by πE2 : (x, ζ ) → [w : wζ : 1] ∈ P2. Thus E2 = (w = 0) is given byζ -axis in this coordinate system and by considering limw→0 π−1

P5◦ k ◦ πE2(w, ζ )

we find:

kZ : E2 � ζ �→ (2b − a − ζ − 1)/a2 ∈ P5.

The mapping among the exceptional fibers is shown in Fig. 6. What is notshown is that R → C′

2 and E01 → C′4

Theorem 6 Suppose F(z) = az3 + az2 + bz + 2 with a �= 0. Then the inducedmap kZ is biholomorphic.

Proof Since kZ and k−1Z have no exceptional hypersurface, indeterminacy

locus for kZ is empty. It follows that kZ is an automorphism of Z . ��


Fig. 6 The space Z andaction of fZ

Repeating the argument in previous two sections, we have that k∗Z acts on

each basis element as follows:

HZ �→ 7HZ − 3E1 − 4P1 − 8P2 − 9P3 − 10P4 − 10P5 −−10P6 − 3E2 − 6R − 4Q − 4E01,

E1 �→ E1, P1 �→ P3 �→ P1, and P2 �→ P2,

P4 �→ HZ − E1 − 2P1 − 3P2 − 3P3 − 3P4 − 3P5 − 3P6 − E2 − R − Q,

P5 �→ E2, P6 �→ HY − E2 − R − E01, E2 �→ P5, and E01 �→ P6,

Q �→ HZ − E1 − P1 − 2P2 − 2P3 − 2P4 − 2P5 − 2P6 − Q − E01,

E01 �→ 2HZ − E1 − P1 − 2P2 − 2P3 − 2P4 − 2P5 − 2P6 −−E2 − 2R − 2Q − E01.

Theorem 7 Suppose F(z) = az3 + az2 + bz + 2 with a �= 0. Then the degree ofkn = k ◦ · · · ◦ k grows quadratically, and k is integrable.

Proof All the eigenvalues of the characteristic polynomial of k∗Z have modulus

one. The largest Jordan block in the matrix representation of k∗Z is a 3 × 3

block corresponding to the eigenvalue 1. Thus the growth rate of the powersof the matrix is quadratic.

Integrability follows from more general results: Gizatullin [26] showed thatif the growth rate is quadratic, then there is an invariant fibration by ellipticcurves. In this case, we can give an explicit invariant. If we define φ = φ1/φ2 tobe the quotient of the following two polynomials;

φ1[x0 : x1 : x2] = x20x2

2,

φ2[x0 : x1 : x2] = −2x40 + 4x3

0x1 − (2 + a)x20x2

1 + 2ax1x22(x0 + x2) −

−2b(x30x2 − x2

0x1x2),

then φ ◦ k = φ. ��


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Phase Vortex: A Dynamical System Approach

Luis Fernando Mello · Denis de Carvalho Braga

Received: 22 February 2008 / Accepted: 8 April 2008 /Published online: 20 May 2008© Springer Science + Business Media B.V. 2008

Abstract In this paper we study the flow lines defined by integral curves of thecurrent field Im ψ∗∇ψ associated to a complex scalar field ψ near a generalphase vortex. This study naturally leads to the center–focus problem. Sufficientconditions for spiral behavior of the flow lines near a vortex in terms of theLyapunov numbers are given.

Keywords Phase vortex · Flow lines · Center–focus problem ·Lyapunov numbers · Degenerate center

Mathematics Subject Classifications (2000) 34D99 · 34C07 · 34C60

1 Introduction

Consider a complex scalar function (wave function)

ψ(x, y) = ρ(x, y) exp(iχ(x, y)

) = u(x, y) + iv(x, y), (1)

with ψ(0, 0) = 0 and assume that ψ does not depend on time. For a given ψ , itis well defined the vector field (current field)

J (x, y) = Im ψ∗∇ψ = ρ2 ∇χ = u(x, y)∇v(x, y) − v(x, y)∇u(x, y). (2)

L. F. Mello (B)Instituto de Ciências Exatas, Universidade Federal de Itajubá,Avenida BPS 1303, Pinheirinho, CEP 37.500-903, Itajubá, Minas Gerais, Brazile-mail: [email protected]

D. de Carvalho BragaInstituto de Sistemas Elétricos e Energia, Universidade Federal de Itajubá,Avenida BPS 1303, Pinheirinho, CEP 37.500-903, Itajubá, Minas Gerais, Brazile-mail: [email protected]

74 L.F. Mello, D. de Carvalho Braga

The origin is a singularity (equilibrium point) of J and it is called a vortex.In the context of quantum mechanics J is the probability current. For moredetails of interpretation of ψ as well as of J see the recent paper [5] andreferences therein. There is a large amount of work devoted to the vortexmodels within the two main areas where such objects are studied: the Bose–condensate physics with vortices and the field theory of superfluid media.Concerning vortices in condensates, the reader is referred to [10, 18] whileconcerning vortices in superfluid media, the reader is referred to [3, 7, 17].

The aim of this paper is the study of the generic behavior of the flow lines,which are the integral curves of J , near the vortex at the origin from thedynamical system point of view. The vortex is called a center if there exists aneighborhood V of the origin filled by closed flow lines and it is called a focusif the flow lines spiral in or out of the vortex in V.

This paper is based on the recent article of M. Berry [5] and answers thefollowing two questions that are not clear in the Berry’s paper:

1. What are the hypotheses under which the flow lines of the current fieldcirculate the vortex?

2. In the case where the Berry’s analysis fails (K = 0 in equation (9) of [5]),what can be said about the behavior of the flow lines near the vortex?

The results of the present paper extend in a different direction the analysisin [5]. Our main goal is the study of isolated degenerate vortices. In Section 2we show that the vortex at the origin is either a center or a focus under ageneric condition. The geometrical behavior of the flow lines near the vortexis studied in Section 3 via Lyapunov numbers. Some concluding remarks arepresented in Section 4. An overview on the Lyapunov numbers is presented inthe Appendix.

2 Flow Lines near the Vortex

In this section we will study the trajectories of the planar differential equation

X ′ = J (X ), (3)

near the vortex at the origin. Here X = (x, y). As ψ(0, 0) = 0 one has u(0, 0) =v(0, 0) = 0. This implies that the sets A = u−1(0) and B = v−1(0) have inter-section at the origin. The description of the vortex in terms of u(x, y) = 0 andv(x, y) = 0 is essentially the Clebsch–variable representation of vortices [14].

The following general hypothesis is fundamental for what follows, which weassume henceforth:

T: The sets A and B are regular curves that meet transversally at the origin.

Lemma 1 Under the hypothesis T the current field (2) has a vortex at the originwhich is either a center or a focus.

Phase vortex: a dynamical system approach 75

Proof Write the Taylor expansion of the functions u and v near the vortex atthe origin

u(x, y) = ax + by + O(||(x, y)||2) , v(x, y) = cx + dy + O

(||(x, y)||2) .

From (2) the current field is given by

J (x, y) = ((bc − ad)y + O

(||(x, y)||2) , (ad − bc)x + O(||(x, y)||2)) .

Therefore the Jacobian matrix of J (x, y) at the origin has the form

DJ (0, 0) =⎛

⎝0 bc − ad

ad − bc 0

⎞

⎠

and its eigenvalues are λ1,2 = ±iω0, where ω0 = ad − bc. But ∇u(0, 0) =(a, b) and ∇v(0, 0) = (c, d) are linearly independent (condition (T)). Thusω0 �= 0 and the origin is either a center or a focus. ��

If u(x, y) = ax + by and v(x, y) = cx + dy are linear with ad − bc �= 0 thenJ (x, y) = ((bc − ad)y, (ad − bc)x). This current field has a vortex at the originwhich is a center under the hypothesis T. Therefore the possibility to finda vortex of focus-type is due to the presence of the higher-order terms, orequivalently due to the nonlinearity of the current field.

Without the validity of the general hypothesis T the flow lines of thecurrent field not necessarily circulate the vortex. As an example, take thewave function ψ(x, y) = y + i(x2 + y2). The current field given by J (x, y) =(2xy, y2 − x2) has an isolated vortex at the origin called dipole. The flow linesof this current field are illustrated in Fig. 1. Another example is given by the

Fig. 1 Flow lines ofthe current fieldJ (x, y) = (

2xy, y2 − x2)


wave function ψ(x, y) = (y − x2) + iy whose current field J (x, y) = (2xy, −x2)

has whole the y-axis as singular points.The problem of to distinguish whether an equilibrium point of a planar

vector field with pure imaginary eigenvalues is a center or a focus is calledthe center-focus problem. For analytic vector field it was solved by Lyapunov,who gave a set of polynomial conditions on the coefficients of the vector fieldin order to have an equilibrium point of center type. These expressions arecalled Lyapunov numbers (constants). There is a large literature on algorithmsto compute these constants and the reader is referred to the paper of Gasulland Torregrosa [9] and references therein.

Without loss of generality take ω0 = 1. Consider (3) in the form

x′ = −y + P(x, y),

y′ = x + Q(x, y), (4)

where P and Q have Taylor expansions at the origin beginning with quadraticterms at least. Differential equation (4) has the form

drdφ

=∞∑

n=2

Rn(φ)rn, (5)

in polar coordinates (r, φ), where Rn(φ) are homogeneous trigonometric poly-nomials [9]. Denote by r(φ, r0) the solution of (5) by r = r0 at φ = 0. Thus

r(φ, r0) = r0 +∞∑

n=2

αn(φ)rn0,

for r small, where αn(0) = 0 for all n � 2. The Poincaré map (first return map)can be defined as

P(r0) = r(2π, r0) = r0 +∞∑

n=2

αn(2π)rn0 . (6)

The first n such that ln = αn(2π) �= 0 is always odd (see [9], p. 164), thusn = 2k + 1 and l2k+1 is called the k-th Lyapunov number of (4). In this case thevortex is called a weak focus of order k. The sign of the Lyapunov number l2k+1

quantifies the spiralling at the vortex: outwards when l2k+1 > 0 and inwardswhen l2k+1 < 0. If αk(2π) = 0 for all k � 2 then the origin is a center.

The next well–known lemma gives a simple sufficient condition for thevortex to be a center.

Lemma 2 If the wave function ψ is either holomorphic or antiholomorphic inthe complex sense then the vortex at the origin is a center.


Proof The divergence of J is given by

∇ · J (x, y) = ∇ · (u(x, y)∇v(x, y) − v(x, y)∇u(x, y)

)

= u(x, y)v(x, y) − v(x, y)u(x, y) ≡ 0,

since u and v are harmonic functions. The lemma follows from Liouville’sTheorem [2] with says that the area of the phase plane is preserved by theflow of a vector field whose divergence vanishes. ��

In [5] Berry gives a method for the calculation of the first Lyapunov numberl3 (denoted by K) in terms of the Taylor expansion of the wave function atthe vortex (see [5], equation (11) p. L747). In the next section we extend theanalysis of Berry for other Lyapunov numbers.

3 Phase Vortex and Lyapunov Numbers

In the Appendix we give some explicit expressions for the calculation of theLyapunov numbers used in this section.

Consider the class of wave functions of the form

ψ(r, φ) = exp(imφ) rm f(r2

), (7)

which represent waves with a well–defined orbital angular momentum quan-tum number m [5]. Here f

(r2

)is a smooth function of r2. The case where m = 1

and f(r2

) = (1 + i br2

), b �= 0, was analyzed in [5], resulting that if b > 0 (resp.

b < 0) then the vortex is a repelling (resp. attractor) focus.The following proposition gives an example of wave function with a vortex

at the origin which is a weak focus of arbitrary order.

Proposition 1 Fix a positive integer k � 1. If m = 1 and

f(r2

) = 1 + ia(r2

)k,

where a �= 0, then the current field J associated to the wave function (7) has avortex at the origin which is a weak focus of order k.

Proof It is immediate that the wave function (7) has the form

ψ(x, y) =(

x − ay(x2 + y2

)k)

+ i(

y + ax(x2 + y2

)k)

.

The current field J defines the following differential equations

x′ = −y + 2akx(x2 + y2)k − a2 y

(x2 + y2)2k

,

y′ = x + 2aky(x2 + y2

)k + a2x(x2 + y2

)2k. (8)


In polar coordinates the above differential equations have the form

r′ = 2ak r2k+1,

φ′ = 1 + a2r4k.

Thus differential equation (5) can be written as

drdφ

= 2akr2k+1

1 + a2r4k.

Therefore we have

l3 = 0, . . . , l2k−1 = 0, l2k+1 = 4πak �= 0.

The proposition is proved.In terms of the notations in the Appendix the proposition can be proved as

follows. Let w = x + iy. Define the functions

s(x, y) = (x2 + y2)k

,

f (w, w) = u((w + w)/2, (w − w)/2i

), g(w, w) = v

((w + w)/2, (w − w)/2i

),

h(w, w) = s((w + w)/2, (w − w)/2i

), G(w, w) = J

((w + w)/2, (w − w)/2i

).

It follows that

G(w, w) = 2 f (w, w)∂g∂w

(w, w) − 2g(w, w)∂ f∂w

(w, w).

By a long but simple calculation one has

G(w, w) = i(1 + a2|w|2k) w + 2ka w|w|2k. (9)

From (19) and (9) the proposition is proved since the differential equation (3)can be written as w′ = G(w, w). In this case we have

l3 = 0, . . . , l2k−1 = 0, l2k+1 = 2ak �= 0. ��

By induction on m Proposition 1 can be extended to the following one.

Proposition 2 Fix a positive integer k � 1. If m � 1 and

f(r2

) = 1 + ia(r2

)k,

where a �= 0, then the current field J associated to the wave function (7) has avortex at the origin which is a weak focus of order k.

Propositions 1 and 2 give examples of wave functions whose current fieldshave vortices at the origin which are weak foci of order k. As a consequencethe flow lines spiral more and more slowly in (a < 0) or out (a > 0) the vortex


Fig. 2 Flow lines of thecurrent field (8) for a = −1and k = 2

as k increases and the distance between successive windings decreases as r2k+1

near the vortex. The vortex is a center if and only if a = 0. The case studiedby Berry [5] can be obtained from Proposition 1 taking k = 1. The flow linesof the current field (8) near the vortex at the origin for a = −1 and k = 2 areillustrated in Fig. 2. Since a < 0 the vortex is a weak attractor focus of order 2.

In Fig. 3 are depicted the flow lines of the current field defined by thewave function (7) with m = 2 and f

(r2

) = 1 + i(r2

)2, that is, a = 1 and k = 2.According to Proposition 2 the vortex at the origin is a weak repelling focus oforder 2.

Fig. 3 Flow lines of thecurrent field induced by(7) with m = 2 andf(r2

) = 1 + i(r2

)2


Consider a long solenoid of small transverse cross section endowed witha magnetic flux �. The limit configuration when the cross section becomesvanishingly small while the magnetic flux enclosed is kept constant is calleda magnetic string. The magnetic field vanishes everywhere except inside themagnetic string and, as there is no Lorentz force, charged particles around thestring are not affected by it, according to the classical mechanics point of view.Nevertheless, there is a nontrivial quantum scattering due to the Aharonov–Bohm effect. See references [13] and [16].

The nontrivial scattering mentioned above can be studied from the follow-ing family of current fields

J (x, y) = �kM

(−1 + δ

ky

x2 + y2, − δ

kx

x2 + y2

), (10)

where 0 � δ = e�/(2π�c) � 1/2 is the flux parameter, M and e are the massand the charge of the particle, respectively, and 0 � k < ∞ is associated to theenergy �

2k2/2M for a stationary state. See [12, 13, 16] for more details.Consider the modified current field

J (x, y) = −M�δ

(x2 + y2)J (x, y) =(

−y + kδ

(x2 + y2

), x

), (11)

obtained from the current field (10) as δ �= 0. It follows that the currentfield J in (10) has a vortex of center–type at the origin if and only if themodified current field J in (11) has a vortex of center–type at the origin. Directcalculation leads to l3 = l5 = l7 = 0 (see the Appendix), and the vortex at theorigin is a center for all 0 < δ � 1/2. See Fig. 4.

Fig. 4 Flow lines of thecurrent field (10) for0 < δ � 1/2


Fig. 5 Flow lines of thecurrent field J (x, y) =(−y, x + k(x2 + y2))

for k �= 0

For δ = 0 there is no vortex at the origin and the flow lines of the currentfield (10) are parallel.

These vortices were predicted 30 years ago and observed in an analogueexperiment with water waves [4]. Also, the conclusion that the vortex at theorigin is of center-type for 0 < δ � 1/2 can be obtained from the vanishing ofthe divergence of the current field.

Another example of physical interest is the wave function

ψ(x, y) = (x + iy) exp(iky), k ∈ R,

of a simple dislocated wave which is an approximation to a solution of aHelmholtz equation [15]. The current field is given by J (x, y) = (−y, x +k(x2 + y2)). Direct calculation leads to l3 = l5 = l7 = 0 (see the Appendix), andthe vortex at the origin is a center for all k ∈ R. If k = 0 whole R

2 − {0, 0} isfilled by closed flow lines. But if k �= 0 the neighborhood of the vortex filled byclosed flow lines is bounded. See Fig. 5.

Theorem 1 Consider the wave function

ψ(x, y) = −y − ak

2x

(x2 + y2

)+aG(x, y) +

+i(

x− ak

2y

(x2 + y2) + aH(x, y)

), (12)

where G and H are analytic functions having Taylor expansions at the originbeginning with terms of degree 4 at least, k � 1 is an integer number and a ∈ R.Then the first Lyapunov number is l3 = 4ak and the vortex at the origin is acenter if and only if a = 0.


Proof The differential equations defined by the current field obtained fromthe wave function (12) are given by

x′ = −y + akx(x2 + y2

) + aP(x, y, a),

y′ = x + ak y(x2 + y2

) + aQ(x, y, a), (13)

where

P(x, y, a) = −1

4a2k−1 y

(x2 + y2

)2 + (1 − akxy

)G(x, y) +

+ 1

2ak(3x2 + y2)H(x, y) −

(x − 1

2ak y

(x2 + y2

)) ∂G∂x

(x, y) −

−(

y + 1

2akx

(x2 + y2

)) ∂ H∂x

(x, y) + aG(x, y)∂ H∂x

(x, y) −

− aH(x, y)∂G∂x

(x, y),

Q(x, y, a) = 1

4a2k−1x

(x2 + y2

)2 + (1 + akxy)H(x, y) −

− 1

2ak(x2 + 3y2)G(x, y) −

(x − 1

2ak y

(x2 + y2

)) ∂G∂y

(x, y) −

−(

y + 1

2akx

(x2 + y2)

)∂ H∂y

(x, y) + aG(x, y)∂ H∂y

(x, y) −

− aH(x, y)∂G∂y

(x, y).

With the notations of the Appendix one has

A =(

0 −11 0

), q =

(i1

), p =

(i/21/2

), B(x, x) =

(00

),

C(x, x, x) =(

2ak(3x1 y1z1 + x2 y2z1 + x2 y1z2 + x1 y2z2

)

2ak(x2 y1z1 + x1 y2z1 + x1 y1z2 + 3x2 y2z2

)

)

.

Therefore G21 = 〈p, C(q, q, q)〉 which gives G21 = 8ak. From (20) the firstLyapunov number is l3 = 4ak. ��


In the above example l3 = 4ak �= 0 if a �= 0 but the number of the limit cyclesbifurcating from the vortex can be high. In fact the number of the limit cyclesbifurcating from the vortex is at most k − 1. See the recent paper Gasull andGiné (unpublished manuscript).

4 Concluding Remarks

A vortex is a monodromic vortex if there is no flow line tending to the vortexwith definite tangent at this point. When the current field is analytic and thevortex is monodromic then it is either a center or a focus. For a generalvortex the following two open problems generalize the two initial questionsin Section 1:

1. Monodromic Problem: determine hypotheses (on the wave function) un-der which the isolated vortex of the current field is monodromic;

2. Stability Problem: once is known that the vortex is monodromic, decidewhether it is a center or a focus.

In this paper we give partial answers for the above two problems. We haveshown that the study of the flow lines of the current field near a phase vortexunder the general hypothesis T naturally leads to the center–focus problem.Here we have analyzed the case where the linear part of the current field atthe vortex is nondegenerate. One possible direction of research is the study ofthe flow lines of the current field near a vortex when the hypothesis T is notsatisfied, but the flow lines circulate the vortex. For example, the case wherethe linear part of the current field at the vortex is degenerate can be of interest.

In this paper we have studied only planar wave functions and flow lines thatcirculate the vortex (the center–focus problem). Of course wave functions in3–dimensional space as well as other types of vortex are of interest, particularlythe knotted vortices [6].

Appendix: An Overview on Lyapunov Numbers

The beginning of this section is a review of the projection method given in[11] for the calculation of the Lyapunov numbers l3 and l5. The method iseasily adapted to the calculation of any Lyapunov number. The expressionsof the Lyapunov numbers l7 and l9 can be found in [19] and [20]. Otherequivalent definitions and algorithmic procedures to write the expressions forthe Lyapunov numbers for two dimensional systems can be found in Andronovet al. [1], Gasull and Torregrosa [9] and Farr et al. [8], among others.

Consider the differential equations

x′ = F(x), (14)


where x ∈ R2 and F is of class C∞. Suppose (14) has an equilibrium point

x = x0 where the Jacobian matrix A has a pair of purely imaginary eigenvaluesλ1,2 = ±iω0, ω0 > 0. Denoting the variable x − x0 also by x, one has

F(x) = Ax + 1

2B(x, x) + 1

6C(x, x, x) + 1

24D(x, x, x, x)

+ 1

120E(x, x, x, x, x) + 1

720K(x, x, x, x, x, x)

+ 1

5040L(x, x, x, x, x, x, x) + O

(||x||8) , (15)

where, for i = 1, 2,

Bi(x, y) =2∑

j,k=1

∂2 Fi(ξ)

∂ξ j ∂ξk

∣∣∣∣∣∣ξ=0

x j yk, Ci(x, y, z) =2∑

j,k,l=1

∂3 Fi(ξ)

∂ξ j ∂ξk ∂ξl

∣∣∣∣∣∣ξ=0

x j yk zl,

and so on for Di, Ei, Ki and Li. Let p, q ∈ C2 be vectors such that

Aq = iω0 q, A p = −iω0 p, 〈p, q〉 =2∑

i=1

pi qi = 1, (16)

where A is the transposed matrix. Any vector y ∈ R2 can be represented as

y = wq + wq, where w = 〈p, y〉 ∈ C. The phase plane can be parameterized byw, w, by means of x = H(w, w), where H : C

2 → R2 has a Taylor expansion of

the form

H(w, w) = wq + wq +∑

2� j+k�7

1

j!k! h jkwjwk + O(|w|8), (17)

with h jk ∈ C2 and h jk = hkj. Substituting this expression into (14) we obtain the

following differential equation

Hww′ + Hww′ = F(H(w, w)

). (18)

The complex vectors hij are obtained solving the system of linear equationsdefined by the coefficients of (18), taking into account the coefficients of F, sothat system (18) writes as follows

w′ = iω0w + 1

2G21w|w|2 + 1

12G32w|w|4 + 1

144G43w|w|6 + O

(|w|8),(19)

with G jk ∈ C.The first Lyapunov number l3 is defined by

l3 = 1

2Re G21, (20)

where

G21 = ⟨p, C

(q, q, q

) + B(q, (2iω0 I2 − A)−1 B(q, q)

) − 2B(q, A−1 B(q, q)

)⟩,


and I2 is the unit 2 × 2 matrix. Defining H32 as

H32 = 6B(h11, h21

) + B(h20, h30

) + 3B(h21, h20

) + 3B(q, h22

) + 2B(q, h31

) ++ 6C

(q, h11, h11

) + 3C(q, h20, h20

) + 3C(q, q, h21

) + 6C(q, q, h21

) ++ 6C

(q, h20, h11

) + C(q, q, h30

) + D(q, q, q, h20

) + 6D(q, q, q, h11

) ++ 3D

(q, q, q, h20

) + E(q, q, q, q, q

) − 6G21h21 − 3G21h21,

the second Lyapunov number l5 is given by

l5 = 1

12Re G32, (21)

where G32 = 〈p,H32〉. The third Lyapunov number l7 is defined by

l7 = 1

144Re G43, (22)

where G43 = 〈p,H43〉 and H43 is given by

H43 = 12B(h11, h32

)+6B(h20, h32

)+3B(h20, h41

)+18B(h21, h22

) ++ 12B

(h21, h31

)+ 4B(h30, h31

)+B(h30, h40

)+4B(q, h33

)+3B(q, h42

) ++ 36C

(h11, h11, h21

) + 36C(h11, h20, h21

) + 12C(h11, h20, h30

) ++ 3C

(h20, h20, h30

) + 18C(h20, h20, h21) + 36C(q, h11, h22

) ++ 12C

(q, h20, h31

)+12C(q, h20, h31

)+36C(q, h21, h21

)+4C(q, h30, h30) +

+ 6C(q, q, h32

) + 12C(q, q, h32

) + 24C(q, h11, h31

) + 18C(q, h20, h22

) ++ 3C

(q, h20, h40

) + 18C(q, h21, h21

) + 12C(q, h21, h30

) + 3C(q, q, h41

) ++ 24D

(q, h11, h11, h11

) + 36D(q, h11, h20, h20

) + 36D(q, q, h11, h21

) ++ 6D

(q, q, h20, h30

) + 18D(q, q, h20, h21

) + 4D(q, q, q, h31

) ++ 18D

(q, q, q, h22

) + 72D(q, q, h11, h21

) + 36D(q, q, h20, h21

) ++ 12D

(q, q, h20, h30

) + 12D(q, q, q, h31

) + 36D(q, h11, h11, h20

) ++ 9D(q, h20, h20, h20

) + 12D(q, q, h11, h30

) + 18D(q, q, h20, h21

) ++ D

(q, q, q, h40

) + 12E(q, q, q, h11, h20

) + E(q, q, q, q, h30

) ++ 12E

(q, q, q, q, h21

) + 36E(q, q, q, h11, h11

) + 18E(q, q, q, h20, h20

) ++ 18E

(q, q, q, q, h21

) + 36E(q, q, q, h11, h20

) + 4E(q, q, q, q, h30

) ++ 3E

(q, q, q, h20, h20

) + 3K(q, q, q, q, q, h20

) + 12K(q, q, q, q, q, h11

) ++ 6K

(q, q, q, q, q, h20

) + L(q, q, q, q, q, q, q

) −− 6

(2G32h21 + G32h21 + 3G21h32 + 2G21h32

).


Acknowledgements The first author developed this work under the project CNPq 473747/2006-5.The second author is supported by CAPES. This work was finished while the first author visitedUniversitat Autònoma de Barcelona, supported by CNPq grant 210056/2006-1. The authors thankthe referee for the comments and suggestions which allowed them to improve the presentation ofthis paper.

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tions. arXiv:0709.3949v1 [math.DS] (2007)

Algebraic Theory of Linear ViscoelasticNematodynamics

Arkady I. Leonov

Received: 8 April 2008 /Accepted: 8 April 2008 /Published online: 3 June 2008# Springer Science + Business Media B.V. 2008

Abstract This paper consists of two parts. The first one develops algebraic theoryof linear anisotropic nematic “N-operators” build up on the additive group oftraceless second rank 3D tensors. These operators have been implicitly used incontinual theories of nematic liquid crystals and weakly elastic nematic elastomers.It is shown that there exists a non-commutative, multiplicative group N6 of N-operators build up on a manifold in 6D space of parameters. Positive N-operators,which in physical applications hold thermodynamic stability constraints, do notgenerally form a subgroup of group N6. A three-parametric, commutativetransversal-isotropic subgroup S3 � N6 of positive symmetric nematic operators isalso briefly discussed. The special case of singular, non-negative symmetric N-operators reveals the algebraic structure of nematic soft deformation modes. Thesecond part of the paper develops a theory of linear viscoelastic nematodynamicsapplicable to liquid crystalline polymer. The viscous and elastic nematic componentsin theory are described by using the Leslie–Ericksen–Parodi (LEP) approach forviscous nematics and de Gennes free energy for weakly elastic nematic elastomers.The case of applied external magnetic field exemplifies the occurrence of non-symmetric stresses. In spite of multi-(10) parametric character of the theory, the useof nematic operators presents it in a transparent form. When the magnetic field isabsent, the theory is simplified for symmetric case with six parameters, and takes anextremely simple, two-parametric form for viscoelastic nematodynamics withpossible soft deformation modes. It is shown that the linear nematodynamics isalways reducible to the LEP-like equations where the coefficients are changed forlinear memory functionals whose parameters are calculated from original viscositiesand moduli.

Keywords Liquid crystals . Nematodynamics . Nematic operators .

Transversal isotropy . Polymers . Viscoelasticity

Math Phys Anal Geom (2008) 11:87–116DOI 10.1007/s11040-008-9041-z

A. I. Leonov (*)Department of Polymer Engineering, The University of Akron, Akron, OH 44325-0301, USAe-mail: [email protected]

Mathematics Subject Classification (2000) SC: 08A62 . 20E32 . 47B99 . 74A20

1 Introduction

The properties of low molecular weight liquid crystals (LMW LCs) have been wellinvestigated experimentally and theoretically [3]. Nematic orientational order of theirmolecules, with the lowest, axial symmetry of molecular groups is the most commonfor LMW LC. On the macroscopic level, it is convenient to describe this order by theunit vector �n called director, with new degree of freedom, internal rotations. Theequilibrium elastic properties of LMW nematics, with the Frank stress proportionalto the space gradient of director, �r�n, as well as their flow properties are welldescribed on macroscopic level by the Leslie–Ericksen–Parodi (LEP) theory callednematodynamics [3].

In spite of 25 years of study, the properties of liquid crystalline polymers (LCP’s)have not been well understood as their LMW counterparts. Common LCP’s consistof very rigid macromolecules. As in case of low molecular weight liquid crystals(LMW LCs), there are two classes of LCP: (1) lyotropic, LCP solutions in LMWsolvents, and (2) thermotropic, LCP melts. Commercial thermotropic LCP’s, such asTitan (Eastman) with melting temperature Tm=330°C and Zenith 6000 (Dupont),Tm=345°C, are the random polyesters with rigid main-chain mesogenic groups.These LCP’s have very narrow time and temperature intervals between beginningthe crystal melting and onset of polymer degradation, where the liquid crystallinephase exists. Even in the case of rigid LCP macromolecules, their contour length ishigher (and commonly much higher) than the macromolecule end-to-end distance.Therefore the low molecular flexibility, always existing in LCP’s gives rise to their“molecular elasticity” [10] because of possible variation in the end-to-end distanceof macromolecules. At low polymer concentrations in lyotropic LCP’s the effect ofmolecular elasticity can be neglected and these LCP’s could be treated as molecularsuspensions of rigid rods [4]. The majority of experimental data for LCP’s wereobtained from rheological experiments in simple shearing with the absence of theexternal magnetic and electrical fields [10]. The degradation of commercialthermotropic LCP’s is the main problem for their rheological studies. Therefore tomake easy rheological experiments some model thermotropic LCP’s weresynthesized with introducing either flexible spacers in the main chain or mesogenicside groups. Such LCP’s have been widely used in rheological experiments (e.g. see[8, 19]). In these more flexible model LCP’s the effects of Frank elasticity occur atthe end of relaxation with forming a specific “texture” during very long post-relaxation time. Some new reliable results have also been recently obtained inrheological studies of Titan and Zenith. It should also be mentioned that there alsoexists a class of highly deformable nematic cross-linked elastomers where rigidchemical fragments were inserted into very flexible silicon rubber chains [21].

The concept of space–time evolution of director is also commonly utilized todescribe the nematic order and internal rotations of LCP rigid macromolecules ortheir rigid chemical fragments. Nevertheless, this theory is not much developed. Oneof the main questions is whether the molecular elasticity is important for all the typesof LCP. If it is, complicated viscoelastic phenomena should be involved in the

88 A.I. Leonov

theory, describing along with director evolution also the contribution of anisotropic(molecular) elastic and viscous forces. Papers [17, 18] developed a thermodynamictheory for linear anisotropic viscoelasticity for LC polymers, using along with statevariables also their space gradients, which created an awkward, almost no testabledescription. Several molecular or semi-phenomenological theories were alsodeveloped to model the lyotropic LCP’s, when employing the same state variables,

�n, �r�n and T as in case of LMW LC’s. The papers [6, 10, 15] typically used andelaborated the Doi’s long rigid rod statistical approach. To describe LCP’s anextended Poisson–Bracket approach [1] was also elaborated and reduced to the Doitheory in the homogeneous (mono-domain) limit. To take into account a flexibilityof LCP chains in model thermotropic LCP’s nematic Rouse-like molecular theories[13, 20] were developed. Currently these theories could be used for description oflinear viscoelastic LCP properties, however, they have not been experimentallytested. In spite of many simplifying physical assumptions, the theories [13, 20] arestill very complicated, so their extension even to a weakly nonlinear case seemsimprobable.

The Doi rigid rod theory has been extensively tested [10]. For lyotropic LCP’s ofsmall and moderate concentrations these tests demonstrated a success of Doi theory.However, both the modeling and industrial thermotropic LCP’s are not described bythe Doi theory. Surprisingly, the Doi rigid rod theory is incapable to describe therheological behavior of real “rigid rod” polymers whose contour length is closed tothe end-to-end macromolecular distance (e.g. see [5, 16]).

An unusual effect of soft deformation modes with ideally no resistance in certaindirections was predicted in paper [7] for anisotropic elastic solids and observed forcross-linked nematic elastomers as predicted by a particular equilibrium theory [21].A possible fundamental reason for occurrence of these soft modes is that largefluctuations typical for nematics in equilibrium, move these systems almost to theboundary of their thermodynamic stability where the free energy is effectivelyminimized not only with respect to the state variables but also with respect tomaterial parameters [14]. Following this idea the “marginal stability” concept hasbeen introduced and formally used in papers [11, 12] for describing the soft modesfor both viscous LMW LC’s and nematic elastomers.

It is now well recognized that the continual theories of viscous nematody-namics and nematic elasticity describe well the respective behavior of LMWLC’s and nematic elastomers. It seems that the theoretical description of LCPdynamics is also needed a similar general continual (or “field”) theory, whichcould consistently describe their specific viscoelastic nematic properties at leastin weakly nonlinear limit. Developing such a theory with many sets of nematicrelaxation modes seems currently unrealistic. Even in the simplest Maxwellapproximation, similar to nematic dumbbell theory in statistical treatment, withone set of nematic relaxations, such a theory has not been developed. It seemsthat the awkward common tensor/matrix formulation of operations in existingviscous and elastic nematic theories is a primary reason for the lack of such atheory. This common formulation does not allow displaying a simple algebraicstructure of these theories and makes difficult (if possible) developing a generalviscoelastic theory for describing linear or nonlinear dynamic behavior ofLCP’s.

Algebraic theory of linear viscoelastic nematodynamics 89

The paper consists of two parts. The first part reveals the algebraic structureof the nematic theories and presents it in a simple form. The second partdevelops a continuum theory of linear viscoelastic nematodynamics of Maxwelltype with a single set of anisotropic relaxations in mono-domain case. Unlikethe dumbbell limit in theories [13, 20], using the algebraic approach makespossible to treat this problem in all generality. Such a general approach also allowsus to employ the concept of marginal stability and implement the algebraic softmode analysis that highly simplifies the results. This approach also makes possibleto develop a weakly nonlinear theory, which could model dynamic behavior ofLCP’s.

2 Algebra of Nematic Operators

2.1 Definitions and General Properties

Consider the additive group^

X of traceless 3D second rank Cartesian tensors x ¼xij� � 2 ^

X : trx ¼ 0 defined on the field of real numbers. This group can bedecomposed in the sum,

^

X ¼ ^

X s þ ^

X a,^

X sT ^

X a ¼ 0ð Þ of two additive subgroupsof symmetric

^

X s and asymmetric^

X α tensors, so that 8x 2 ^

X : x ¼ xsþ x

a, x

s2 ^

X s

and xa2 ^

X a.We introduce on

^

X linear, axially symmetric operations^

X ! ^

X characterized bya given unit vector �n (director). A linear operation invariant relative totransformation �n ! ��n, is called nematic operation (or simply N-operator). Theimplicit definition of N-operation in the common tensor presentation is:

ys¼ r0xs þ r1 �n�n � xs þ x

s� �n�n� 2nn x

s: �n�n

� �h iþ r2 �n�n� δ

.3

� �xs: �n�n

� �þ r3 �n�n � xa � x

a� �n�n

� �ya¼ r4 �n�n � xs � x

s� �n�n

� �� r5 �n�n � xa þ x

a� �n�n

� �:

ð1:1Þ

Hereafter �n�n� �

ij¼ njnj, the symbol · means the tensor (matrix) multiplication, the

symbol: denotes the trace operation, d is the unit tensor, and rk are six independentordered basis parameters, characterizing the operation in (1.1), denoted as: r = (r0,r1,..., r5). Because of (1.1) y

s2 ^

X s, ya2 ^

X a, which justifies that operation in (1.1)

transforms^

X ! ^

X . Relations of (1.1) type were first introduced in the vector formin paper [2] for weak elastic gels and have been used in the tensor form in papers[11, 12] as constitutive equations for viscous and weakly elastic nematic cases. Thepossible non-nematic term ∼x

awas excluded from (1.1) because of physical

arguments (e.g. see [11, 12]).We now denote the N-operator as Nr �n

� �, and symbolically present (1.1) as

y ¼ Nr �n� � � x. A particular but physically significant Onsager N-operator (or ON-

operator) is defined as:

Nor �n� � � Nr �n

� �r4¼�r3j :

90 A.I. Leonov

For any N-operator Nr �n� �

one can introduce on^

X the quadratic form, a scalar P,defined as:

P � x � Nr �n� � � x � r0 xs

�� 2 þ 2r1�n�n : x2sþ r2 � 2r1ð Þ �n�n : x

s

� �2� 4r�3 �n�n : x

s� x

a

� �� 2r5�n�n : x2

a:

r�3 � r3 � r4ð Þ=2:ð1:2Þ

In case of ON operator Nor �n� �

, when P→Po, the quadratic form Po has potentialproperties: 2y

s¼ @Po

.@x

s, 2y

a¼ @Po

.@x

a. Operator Nr �n

� �is said to be positive if

8x 2 ^

X : P � x � Nrð�nÞ � x > 0. The same holds for ON operator Nor �n� �

.We now show that (1) N operator Nr �n

� �is positive (P>0) iif:

r 2 eRþ6 : r0 > 0; r0 þ r1 > 0; 3=2r0 þ r2 > 0; r0 þ r1ð Þr5 > r3 � r4ð Þ2

.4 � r*23 ;

ð1:31; 2; 3; 4Þand that (2) any positive N-operator Nr �n

� �has inverse, N�1

r �n� �

.Using an orthogonal transformation, one can choose a coordinate system whose

axis 1 is directed along the director. In this coordinate system (1.1) and (1.2), writtenin component form, are reduced to:

ys11¼ðr0 þ 2=3r2Þx11; ys22¼r0x22�x11r2=3; ys33¼r0x33 � x11r2=3; y

s23 ¼ r0x23 ð1:11aÞ

ys1k ¼ ðr0 þ r1Þxs1k þ r3xa1kya1k ¼ �r4xs1k þ r5xa1k

ðk ¼ 2; 3Þ

ð1:11bÞ

P ¼ 3=2r0 þ r2ð Þx211 þ 2r0 x22 þ x11=2ð Þ2 þ x2s23

h iþ 2

Xk¼2;3

r0 þ r1ð Þx2s1k þ 2r*3 xa1kxs1k þ r5x2a1k

h i ð1:21Þ

Here the traceless condition, x11 þ x22 þ x33 ¼ 0, has been used to exclude x33 from(1.2). Demanding P>0 and using independence of terms in (1.2) yields inequalities(1.31,2,3,4). Note that the inequality (1.34) results in the inequality:

r0 þ r1ð Þr5 > �r3r4: ð1:35ÞWhen (1.31,2,3,5) holds for equations (1.1), there is a unique linear dependence of xijon yij, which means the existence of a unique inverse operation N�1

r �n� �

. Note thatinequalities (1.31,2,3,4,5) hold for ON-operators No

r �n� �

when r*3 ¼ r3 P ! Poð Þ.Resolving equations (1.1) does not, however, necessitate that the parameters r 2

R6 belong to the manifold eRþ6 defined by inequalities (1.31,2,3,4). The necessary and

sufficient conditions for this resolution are: r0≠0, 3=2r0 þ r2 6¼ 0, r0 þ r1ð Þr5þr3r4 6¼ 0. Nevertheless, it is more convenient to use the positive conditions of theresolution:

r 2 Rþ6 :r0 > 0; 3=2r0 þ r2 > 0; r0 þ r1ð Þr5 þ r3r4 > 0: ð1:31; 3; 5Þ

(1.3,1,2,3,4)

(1.31,3,5)

(1.11a)

(1.11b)

(1.21)

(1.3.5)


N-operator is called N+-operator if its basis parameters satisfy the inequalities(1.31,3,5). Evidently, eRþ

6 � Rþ6 , i.e. N

+-operators are not necessarily positive, whereasany positive N-operator is N+- operator.

2.2 Basis N-operators and their Multiplicative Properties

The tensor/matrix presentations of basis N-operators ak �n� �

k ¼ 0; 1; . . . ; 5ð Þ in (1)are explicitly defined via fourth rank numerical tensors (or simply 4-tensors)ak nð Þf gijab as:

a0f gijαβ¼ a 0ð Þijαβ ¼ 1

2 δiαδjβ þ δiβδjα � 2

3δijδαβ

� � ð1:41Þ

a1 nð Þf gijαβ¼a 1ð Þijαβ¼1

.2 δ?iαnjnβþδ?jαninβ þ δ?iβnjnαþδ?jβninα� �

δ?ij ¼δij � ninj ð1:42Þ

a2 nð Þf gijαβ¼ a 2ð Þijαβ ¼ ninj � 1

3δij

� �nαnβ � 1

3δαβ

� � ð1:43Þ

a3 nð Þf gijαβ¼ a 3ð Þijαβ ¼ 1

2 δiαnjnβ þ δjαninβ � δiβnjnα � δjβninα� � ð1:44Þ

a4 nð Þf gijαβ¼ a 4ð Þijαβ nð Þ ¼ 1

2 δiαnjnβ þ δiβnjnα � δjαninβ � δjβninα� � ¼ a3 nð Þf gαβij

ð1:45Þ

a5 nð Þf gijαβ¼ a 5ð Þijαβ nð Þ ¼ 1

2 δiβnjnα � δiαnjnβ þ δjαninβ � δjβninα� � ð1:46Þ

The basis four-tensors in (1.41–1.46) are traceless with respect to the first and thesecond pairs of indices, i.e. ak �n

� �� iiαβ¼ ak �n

� �� ijαα¼ 0 k ¼ 0; ::; 5ð Þ.

The following symmetry properties hold for the basis four-tensors:

ak �n� ��

ijαβ¼ ak �n� ��

jiαβ¼ ak �n� ��

ijβα¼ ak �nÞ� �

αβij k ¼ 0; 1; 2ð Þn

ð1:51Þ

a3 �n� ��

ijαβ¼ a3 �n� ��

jiαβ¼ � a3 �n� ��

ijβα ð1:52Þ

a4 �n� ��

ijαβ¼ � a3 �n� ��

jiαβ¼ a3 �n� ��

ijβα ð1:53Þ

a5 �n� ��

ijαβ¼ � a5 �n� ��

jiαβ¼ � a5 �n� ��

ijβα¼ a5 �n� ��

αβij: ð1:54Þ

(1.41)

(1.42)

(1.43)

(1.44)

(1.45)

(1.46)

(1.52)

(1.54)

(1.51)

(1.53)

92 A.I. Leonov

The symmetry properties (1.51–1.54) of the basis four-tensors ak �n� �

k ¼ 0; ::; 5ð Þshow that they represent irreducible set of traceless 4-th rank tensors.

The products of the basis tensors of different ranks are defined as:

as nð Þ&ar nð Þ ) a sð Þijαβa

rð Þβανγ ; ar nð Þ&x ) a rð Þ

ijαβxβα; etc: ð1:6Þ

This defines the sense of operation • symbolically used in “Section 2.2”. Theproducts ai nð Þ& aj nð Þ, established directly are presented in Table 1. It is seen thatexcept i,j=0,1,2, the multiplication of basis tensors is non-commutative, e.g.

a0& a3 ¼ a3 6¼ a3& a0 ¼ 0

2.3 Multiplicative Group of N-operators

Using basis operators ak �n� �

, (1.1) can be rewritten in the operator form:

y ¼ Nr nð Þ& x; Nr nð Þ �X5k¼0

rkak nð Þ; ð1:71Þ

or equivalently as:

ys¼X2k¼0

rkak nð Þ& xsþ r3a3 nð Þ& x

a; y

a¼ r4a4 nð Þ& x

sþ r5a5 nð Þ& x

a: ð1:72Þ

The product of two N-operators is defined in the common way:

Np nð Þ � Nq nð Þ&Nr nð Þ ¼X5k;m¼0

qkrmak nð Þ& am nð Þ ¼X6k¼0

pkak nð Þ: ð1:8Þ

With the use of multiplicative Table 1, the basic scalars pk for resulting operation arefound from the fundamental equation:

p0 ¼ q0r0; p1 ¼ q0r1 þ q1r0 þ q1r1 � q3r4; p2 ¼ q2r0 þ q0r2 þ 2=3q2r2p3 ¼ q0 þ q1ð Þr3 þ q3r5; p4 ¼ q4 r0 þ r1ð Þ þ q5r4; p5 ¼ �q4r3 þ q5r5

ð1:9Þ

Table 1 Products of basis tensors aiaj

j 0 1 2 3 4 5l

0 a0 a1 a2 a3 0 01 a1 a1 0 a3 0 02 a2 0 (2/3) a2 0 0 03 0 0 0 0 −a1 a34 a4 a4 0 −a5 0 05 0 0 0 0 a4 a5

(1.72)

(1.71)


Even in the Onsager case, when Nq nð Þ ¼ Noq nð Þ, Nr nð Þ ¼ No

r nð Þ i.e. q4=−q3 andr4=−r3, generally p4≠−p3. It means that Np nð Þ 6¼ No

p nð Þ, i.e. the product of two ON-operators is not generally an ON-operator.

We now show that the set of N+-operators Nr �n� �

, whose basis parameters r 2 Rþ6

satisfy inequalities (1.31,3,5), constitute a non-commutative, multiplicative, six-parametric group N6, with the fundamental equations (1.9).

Using the Table 1, one can define the unit N-operator I(n) and its basic propertiesas:

I nð Þ ¼ a0 þ a5 nð Þ; Nr nð Þ � I nð Þ ¼ I nð Þ � Nr nð Þ ¼ Nr nð Þ ð1:10ÞBecause of (1.2, 1.21) the unit N-operator is positive, and therefore it is a N+-operator.

If Nr nð Þ is N+-operator, its inverse, N�1r nð Þ � Nr nð Þ should satisfy the common

condition, Nr nð Þ&Nr nð Þ ¼ Nr nð Þ&Nr nð Þ ¼ I nð Þ which for parameters in (1.9) yields:

p0 ¼ 1; p1 ¼ p2 ¼ p3 ¼ p4 ¼ 0; p5 ¼ 1: ð1:11ÞThen using (1.9 and 1.11) yields the expressions for basis parameters rk of inverse

N-operator Nr �n� �

as:

r0 ¼ 1

r0; r1 ¼ � r3r4 þ r1r5ð Þ=r0

r5 r0 þ r1ð Þ þ r3r4; r2 ¼ �r2=r0

r0 þ 2=3r2

r3 ¼ �r3r5 r0 þ r1ð Þ þ r3r4

; r4 ¼ �r4r5 r0 þ r1ð Þ þ r3r4

; r5 ¼ r0 þ r1r5 r0 þ r1ð Þ þ r3r4

ð1:12ÞUsing inequalities (1.31,2,3,4 and 1.12) one can directly check that N�1

r nð Þ is a N+-operator and if Nr nð Þ is positive, N�1

r nð Þ is positive too. Additionally, any positiveON operator has inverse one, which is a positive ON operator.

Finally, we show that the product Np nð Þ ¼ Nr nð Þ&Nq nð Þ of two N+ operators isN+ operator, because 8r; q 2 Rþ

6 : p 2 Rþ6 . This follows from the direct calculations

with the use of (1.9):

p0 ¼ q0r0 > 0; 3=2p0 þ p2 ¼ 3=2 q0 þ 2=3q2ð Þ r0 þ 2=3r2ð Þ > 0p5 p0 þ p1ð Þ þ p4p3 ¼ q5 q0 þ q1ð Þ þ q4q3½ � r5 r0 þ r1ð Þ þ r4r3½ � > 0

: ð1:13Þ

These properties prove that the N+ operators constitute a multiplicative group N6.Note that due to (1.9), p0 þ p1 ¼ q0 þ q1ð Þ r0 þ r1ð Þ � q3r4; p5 ¼ q5r5 � q4r3.

Therefore the product of two positive N+-operators is positive only under additionalconstraints: q3r4, q4r3<0. In particular, the product of two positive ON-operators isgenerally non-positive N+ operator, and is positive only under additional constraint,q3r3>0.

Since the N+-operators also constitute the additive group with the groupoperation: Nq nð Þ þ Nr nð Þ ¼ Nqþr nð Þ, they form an associative ring Ń6 relative toboth, additive and multiplicative operations. In the following we are interested onlyin the multiplicative properties of N-operators.

Consider as an example the dual N-operations defined as: z ¼ Nq nð Þ&y ¼Nr nð Þ&x. If both Nr nð Þ and Nq nð Þ are N+ operators, the dual operations uniquely

94 A.I. Leonov

determine the direct and reciprocal dependences, y ¼ Np nð Þ&x and x ¼ Np nð Þ&y.Here

Np nð Þ ¼ N�1q nð Þ&Nr nð Þ; Np nð Þ ¼ N�1

p nð Þ ¼ N�1r nð Þ&Nq nð Þ: ð1:14Þ

Formulae (1.9 and 1.12) express the basis scalars p and p of dual N+-operatorsvia given basis scalars r and q. E.g. in case Np nð Þ ¼ No�1

q nð Þ&Nor nð Þ, where No

r nð Þand No

q nð Þ are positive ON-operators, the parameters p are:

p0 ¼ r0q0

; p1 ¼ r0 þ r1ð Þq5 � r3q3q5 q0 þ q1ð Þ � q23

� r0q0

; p2 ¼ r2q0 � r0q2q0 q0 þ 2=3q2ð Þ ;

p3 ¼ r3 q0 þ q1ð Þ � q3 r0 þ r1ð Þq5 q0 þ q1ð Þ � q23

; p4 ¼ r5q3 � r3q5q5 q0 þ q1ð Þ � q23

; p5 ¼ r5 q0 þ q1ð Þ � r3q3q5 q0 þ q1ð Þ � q23

ð1:15Þ

Due to (1.14) respective formulae for the basis parameters p of inverse dual N+

operation are obtained from (1.15) by substitution r $ q. Note that N-operatorNp nð Þ with basis scalars defined in (1.15), is generally neither Onsager nor positive.The evident sufficient condition for Np nð Þ to be positive is: r3q3<0.

We consider in the following the case of non-degenerating (NG) Nr nð Þ operators,rk 6¼ 0 k ¼ 0; 1; 2; 3; 4; 5ð Þ: ð1:16Þ

2.4 Spectral Properties of N-operators

The spectral problem for a NG operator Nr nð Þ is formulated in the standard way:

Nr nð Þ&x ¼ nx; or Nr nð Þ � nI nð Þ½ �&x ¼ 0: ð1:171Þ

Here Nr nð Þ ¼ P5k¼0

rkak nð Þ, I nð Þ ¼ a0 þ a5 nð Þ, ν being a generally complex

eigenvalue, and x νð Þ 2 ^

X is a respective “eigentensor”. If an eigenvalue ν is known,it is convenient to search instead of eigentensor x nð Þ, the N-“eigenoperator”

Q n; nð Þ ¼ P5k¼0

qk nð Þak from the equation:

Nr nð Þ&Q n; nð Þ � nQ n; nð Þ ¼ Nr n; nð Þ&Q n; nð Þ ¼ 0 where

Nr n; nð Þ � Nr nð Þ � nI nð Þ:ð1:172Þ

Evidently, the tensor x ¼ Q n; nð Þ&x0where x

0is a given tensor, is the eigentensor,

because it identically satisfies (1.172).A common analytical continuation n ! n is used to find the eigenvalues of

problem (1.171) for any NG operator Nr nð Þ. Substituting r0 ! r0 � n andr5 ! r5 � n, and using (1.12) results in formal finding the basis parametersrk n; rkð Þ of operator N�1

r n; nð Þ. The eigenvalues are then found as singular points

(1.171)

(1.172)


n ¼ n for the basis parameters rk n; rkð Þ of the inverse operator N�1r n; nð Þ. Using this

procedure, the eigenvalues are expressed via basis parameters rk as:

ν1 ¼ r0; ν2 ¼ r0 þ 2=3r2; ν3 ¼ 1=2 r0 þ r1 þ r5 þ dð Þ; ν4 ¼ 1=2 r0 þ r1 þ r5 � dð Þd2 ¼ r0 þ r1 þ r5ð Þ2 � 4 r5 r0 þ r1ð Þ þ r3r4½ � � r0 þ r1 � r5ð Þ2 � 4r3r4;

ð1:18ÞTo find the solution (1.172) one can use (1.9) with r0 ! r0 � n, r5 ! r5 � n.

Demanding here due to (1.172) p ¼ 0 yields:

r0 þ νð Þq0 ¼ 0; r0 þ r1 � νð Þq1 þ r1q0 � r4q3 ¼ 0; r0 þ 2=3r2 � νð Þq2 þ r2q0 ¼ 0q0 þ q1ð Þr3 þ q3 r5 � νð Þ ¼ 0; r0 þ r1 � νð Þq4 þ r4q5 ¼ 0; �r3q4 þ r5 � νð Þq5 ¼ 0

For each value of ν from (1.18), the parameters q nð Þ of eigenoperator Q n; nð Þ arefound from the above equations as:

q ν1ð Þ ¼ c1 1;�1;�3=2; 0; 0; 0ð Þ; q ν2ð Þ ¼ c2 0; 0; 1; 0; 0; 0ð Þ; q ν3ð Þ ¼ 0; c3; 0;λ1c3; c4;�λ1c4ð Þq ν4ð Þ ¼ 0; c5; 0; λ2c5; c6;�λ2c6ð Þ λ1 ¼ r0 þ r1 � ν3ð Þ=r5; λ2 ¼ r0 þ r1 � ν4ð Þ=r5f g

ð1:19ÞHere c1, c2,..., c6 are arbitrary constants. Additional solutions that use specificrelations between parameters rk have been rejected because they are not robust.Using (1.19), the “eigenoperators” Q nk ; nð Þ are presented as:

Q ν1; �n� � ¼ c1 a0 � a1 � 3=2a2ð Þ; Q ν2; �n

� � ¼ c2a2; Q ν3; �n� � ¼ c3 a1 � λ1a3ð Þ þ c4 a4 þ λ1a5ð Þ

Q ν4; �n� � ¼ c5 a1 � λ2a3ð Þ þ c6 a4 þ λ2a5ð Þ λ1 ¼ r0 þ r1 � ν3ð Þ=r4; λ2 ¼ r0 þ r1 � ν4ð Þ=r4f g

ð1:20ÞArbitrary parameters ck in (1.15) can be established from various additionalconditions. One of possible physical condition is:X4

i¼1

Q ni; nð Þ ¼ I nð Þ ¼ a0 þ a5 nð Þ: ð1:21Þ

Using (1.20 and 1.21) yields:

c1 ¼ 1; c2 ¼ 3=2; c3 ¼ r0 þ r1 � ν4ð Þ=d; c4 ¼ �r4=d; c5 ¼ ν3 � r0 � r1ð Þ=d; c6 ¼ r4=d

ð1:22ÞHere parameters rk, ν3, ν4, and d have been defined in (1.18).

Easy analysis reveals various cases of behavior of eigenvalues νk in (1.18): (1) ingeneral case of NG operators Nr nð Þ, the eigenvalues ν1, ν2 are real but generallyhave arbitrary signs, the eigenvalues ν3, ν4 being generally complex and conjugated;(2) in case of N+ operators when Nr nð Þ 2 N6, the eigenvalues ν1, ν2 are positive, theeigenvalues ν3, ν4 being generally complex and conjugated; (3) in case of positiveN+ operators, the eigenvalues ν1, ν2 are positive, the eigenvalues ν3, ν4 beinggenerally complex and conjugated, with Re(ν3, ν4)>0, and (4) all eigenvalues νk in(1.18) are real positive if r3r4<0, as in particular case of positive ON operatorswhere r4=−r3.

96 A.I. Leonov

There is also a remarkable feature of Onsager positive operators, which has animportant application to the viscoelastic nematodynamics. As shown in “Appendix”,a dual, generally not a positive operator Np �n

� �constructed due to (1.14 and 1.15)

from two ON positive operators has real positive eigenvalues.

2.5 Symmetric N-operators: Transversal Isotropy (TI)

2.5.1 General Properties

This section briefly describes the nematic operations on the additive subgroup^

X s �^

Xof traceless second rank symmetric tensors x

s2 ^

X s : trxs¼ 0. Therefore the lower

index “s” is omitted here. The linear symmetric N-operator Sr �n� �

on^

X s is defined as:

y ¼ Sr nð Þ&x ¼X2k¼0

rkak nð Þ&x; or Sr nð Þ ¼X2k¼0

rkak nð Þ ð1:23Þ

Here ak nð Þ are the basis tensors defined in (1.41–1.46), and rk are the real-valuedbasic scalar ordered parameters {r}, characterizing operation. The common tensorpresentation of symmetric operation (1.1), and corresponding quadratic form Ps are:

y ¼ r0xþ r1 nn & xþ x & nn� 2nn x : nn� �h i

þ r2 nn� δ.3

� �x : nn� �

: ð1:24Þ

Ps � x&Sr nð Þ&x � r0 xs�� 2þ2r1nn : x2

sþ r2 � 2r1ð Þ nn : x

s

� �2: ð1:25Þ

Equations (1.23 and 1.24) could also be obtained from (1.1) when ya≡0, using thenormalizing procedure [11, 12]. In this case the second equation in (1.1) is used forexpressing xa via xs and �n. Substituting so found dependence xa ¼ xa xs; �n

� �into (1)

results in the (1.23 and 1.24). As seen, this procedure does not violate the non-degenerating conditions (1.16). Relations (1.23 and 1.24) show that symmetric N-operators are transversally isotropic. Therefore they are called TI-operators.

2.5.2 Multiplicative Group

Due to the Table 1, the products of TI-operators are commutative:

Sp nð Þ ¼ Sr nð Þ&Sq nð Þ ¼ Sq nð Þ&Sr nð Þ: ð1:26ÞHere the basis parameters pk are found from the fundamental equation:

p0 ¼ r0q0; p1 ¼ r0q1 þ r1q0 þ r1q1; p2 ¼ r0q2 þ r2q0 þ 2=3r2q2: ð1:27ÞA TI-operator is called positive if 8x 2 ^

X the quadratic form Ps ¼ x & Sr nð Þ & x > 0.Using the same approach as in the general case of N operators, one can proof that

TI operator Sr �n� �

is positive iif

r0 > 0; r0 þ r1 > 0; r0 þ 2=3r2 > 0: ð1:28ÞDirect calculations show that the unit TI operation is I=a0, so

8Sr �n� �

:I &Sr �n� � ¼ Sr �n

� �& I ¼ Sr �n

� �Sr �n� �

:


Direct calculations also show that the product (1.26) of two positive TI-operators ispositive. The basis scalar parameters rk of inverse positive TI-operator S�1

r �n� � �

Sr �n� �

are found from the relation S�1r �n� � � Sr �n

� � ¼ Sr �n� � � Sr �n

� � ¼ a0 �n� �

as:

r0 ¼ 1

r0; r1 ¼ � r1=r0

r0 þ r1; r2 ¼ � r2=r0

r0 þ 2=3r2: ð1:29Þ

Due to (1.28 and 1.29), any positive TI-operator Sr �n� �

has inverse positive.All these properties of positive TI operators show that they constitute

commutative three parametric TI group S3 ⊂ N6.The dual linear transformations z ¼ Sq �n

� � � y ¼ Sr �n� � � x with positive TI

operators Sq �n� �

and Sr �n� �

define the direct y ¼ Sp �n� � � x and inverse x ¼ Sbp �n

� � � ylinear relations, where

Sp �n� � ¼ S�1

q �n� �

&Sr �n� �

; Sp �n� � ¼ S�1

p �n� � ¼ S�1

r �n� �

&Sq �n� �

; ð1:30ÞHere the parameters p are:

p0 ¼ r0q0

; p1 ¼ r1q0 � q1r0q0 þ q1

; p2 ¼ r2q0 � q2r0q0 þ 2=3q2

: ð1:311Þ

Parameters p of inverse operation found by substitution q $ r are:

p0 ¼ q0r0

; p1 ¼ q1r0 � r1q0r0 þ r1

; p2 ¼ q2r0 � r2q0r0 þ 2=3r2

: ð1:312Þ

2.5.3 Eigenvalue Problem

The formulation of the eigenvalue problem, similar to (1.171) is:

Sr �n� �� νI

� ��x¼0; or Sr �n� �

&Q ν; �n� ��νQ ν; �n

� � � eSr ν; �n� �

&Q ν; �n� �¼0: ð1:32Þ

Here Sr �n� � ¼ P2

k¼0rkak �n

� � 2 S3, I = a0, Q n; �n� � ¼ P2

k¼0qk nð Þak �n

� �and x ¼ x nð Þ 2 ^

X s.

Using the same methods as in “Section 2.4” it is easy to show that 8Sr �n� � 2 S3,

the spectral points of problem (1.32) are:

ν1 ¼ r0; ν2 ¼ r0 þ r1; ν3 ¼ r0 þ 2=3r2: ð1:33Þ

Here due to (1.28), vk>0. The corresponding eigentensors x nkð Þ are found as:x νkð Þ ¼ Q νk ; �n

� �& x

0with x

0being a given tensor, and the “eigenoperators”

Q nk ; �n� �

given by:

Q ν1; �n� � ¼ c1 a0 � a1 � 3=2a2ð Þ; Q ν2; �n

� � ¼ c2a1; Q ν3; �n� � ¼ c3a2: ð1:34Þ

If the arbitrary parameters ck in (1.34) are found from the physical condition,

X3i¼1

Q ni; �n� � ¼ I �n

� � ¼ a0; ð1:35Þ

(1.311)

(1.312)

98 A.I. Leonov

their values are:

c1 ¼ c2 ¼ 1; c3 ¼ 3=2: ð1:36Þ

2.5.4 Singular TI Operators

Consider now the limiting, marginal situation when some inequalities in (1.34) turnout to be equalities. If once again non-degeneration conditions rk 6¼ 0 k ¼ 0; 1; 2ð Þwith r0>0 are valid, there might be only two independent marginal conditions:

r0 þ r1 ¼ 0; r0 þ 2=3r2 ¼ 0 ð1:371ÞWhen one of these conditions is satisfied, TI operator is called partially soft. Whenboth of them are satisfied, TI operator is called completely soft. In both the partiallyor complete soft cases, the quadratic form Ps is positively semi-definite.

The nearly marginal situations are defined as those that reduce (1.371) to:

r0 þ r1 ¼ r0δ; r0 þ 2=3r2 ¼ 2=3r0κ 0 < δ;κ << 1ð Þ;r ¼ r0 1; δ � 1;κ� 3=2ð Þ:

ð1:372Þ

Formulae (1.372) mean that the corresponding TI operator Sr �n� � � Sd;k �n

� �is

positive.Due to (1.371) there is one-parametric marginal family of completely soft TI

operators, S �n� � ¼ r0a �n

� �r0 > 0ð Þ where:

! �n� � ¼ a0 �n

� �� a1 �n� �� 3=2a2 �n

� �; ð1:38Þ

The parameters of the marginal family are: r ¼ r0 1;�1;�3=2ð Þ. It is seen that theoperator a �n

� �is singular, i.e. a�1

�n� �

does not exist, and that:

a �n� �

&a �n� � ¼ a �n

� �: ð1:39Þ

Consider now a pair Sr �n� � ¼ Sd1;k1 �n

� �and Sq �n

� � ¼ Sd2;k2 �n� �

of positive, nearlymarginal TI operators, with parameters: r ¼ 1; δ1 � 1;κ1 � 3=2ð Þ andq ¼ 1; δ2 � 1;κ2 � 3=2ð Þ, where 0< δ1, κ1 <<1, 0< δ2, κ2 <<1. EvidentlySδ1;κ1 �n

� �! Sδ2;κ2 �n� �! a �n

� �when (δ1, κ1)→(δ2, κ2)→(0, 0). One can directly

show that when the two independent limits exist:

δ ¼ limδ1;2!0

δ2=δ1ð Þ; κ ¼ limν1;2!0

κ2=κ1ð Þ; 0 < δ;κ < 1ð Þ ð1:40Þ

there exist two positive limiting TI operators,

aδ;κ �n� � � lim

δ1;2;κ1;2!0S�1δ1;κ1 �n

� �& Sδ2;κ2 �n

� � ¼ a0 �n� �þ δ � 1ð Þa1 �n

� �þ 32 κ� 1ð Þa2 �n

� �ð1:411Þ

bδ;κ �n� � � lim

δ1;2;κ1;2!0S�1δ2;κ2 �n

� �& Sδ1;κ1 �n

� � ¼ a0 �n� �þ δ�1 � 1

� �a1 �n� �þ 3

2 κ�1 � 1� �

a2 �n� �

:

ð1:412Þ

(1.371)

(1.372)

(1.411)

(1.412)


Using direct calculations one can confirm that the TI operators defined in (1.411 and1.412) have the following properties:

bδ;κ �n� � ¼ a�1

δ;κ �n� � ¼ a1=δ;1=κ �n

� �; ð1:421Þ

aδ;κ �n� �

&a �n� � ¼ bδ;κ �n

� �&a �n� � ¼ a �n

� �; ð1:422Þ

a1;1 �n� � ¼ b1;1 �n

� � ¼ a0 �n� �

: ð1:423ÞUsing (1.421) it is easy to prove that when either δ→0 and/or .→0, or δ→∞ or/

and .→∞, there are only two non-trivial cases of singular limiting behavior of TIoperators aδ;κ �n

� �and bδ;κ �n

� �:

(i) α-type described by singular TI operators a0;κ �n� �

or aδ;0 �n� �

, ora0;0 �n

� � ¼ a �n� �

, when the operators bδ;κ �n� �

do not exist; and(ii) β-type described by singular TI operators b1;κ �n

� �or bδ;1 �n

� �, or

b1;1 �n� � ¼ a �n

� �, when the operators aδ;κ �n

� �do not exist.

The cases a0;0 �n� � ¼ a �n

� �and b1;1 �n

� � ¼ a �n� �

describe, respectively, thelimiting super-soft behaviors of either α- or β-types.

The eigenvalue problem for the singular operator S �n� � ¼ r0a �n

� �where a �n

� �is

presented in (1.38), has an easy solution. Due to (1.33) the eigenvalues are:

n1 ¼ r0; n1 ¼ n2 ¼ 0; ð1:43Þwhereas the “eigenoperators” Q nk ; �n

� �are given by (1.34). Note that the formulae

(1.35 and 1.36) are still valid in this case.

3 Linear Nematic Viscoelasticity

3.1 Linear Kinematical Relations

Bearing in mind that the main objective of this part is establishing a simple case ofnematic viscoelasticity, we omit discussing here the well-known general relations forthe balance of moment of momentum and related rotational inertia effects. One canfind their derivations in the texts [3, 9, 21].

We use in the following the linear viscoelastic kinematics based on the commondecomposition of full infinitesimal strain gradient tensor g into elastic (transient)γe and inelastic γ

p parts. Each of these tensors consists of infinitesimal strain

"; "e; "

p

� �and infinitesimal “body” rotation tensors, 4b;4b

e;4b

p

� �. Then decom-

position γ; γe; γ

p

�¼ "; "

e; "

p

� �þ 4b;4b

e;4b

p

� �is presented as γ ¼ γ

eþ γ

p.Differentiating this equation with respect to time and separating equations fordeformation and rotations, yields the kinematical rate equations:

:εeþ e

p¼ e ð2:1Þ

:4b

eþ ωb

p¼ ωb ð2:21Þ

(1.421)

(1.422)

(1.423)

(2.21)

100 A.I. Leonov

Here overdots denote time derivatives, e ¼ ð1=2Þ rvþ rv� �T� �

and, ωb ¼ð1=2Þ rv� rv

� �T� �are, respectfully, the full strain rate and “body” vorticity, and

v is the velocity vector. So :+ � rv ¼ eþ ωb, e

p¼ :ε

pand ωb

p¼ :4

b

p are, respectfully,the irreversible strain rate and vorticity. Following the papers [17, 18], we nowintroduce in (2.1 and 2.21) the additional rate equation describing internal rotationsfor nematic continuum: :

4ieþ ωi

p¼ ωi: ð2:22Þ

Here the total tensor of internal vorticity wi was decomposed into the rate ofreversible rotation

:4i

eand irreversible vorticity wi

p. Extracting now (2.22) from (2.21)

results in the equation describing the linear kinematics of relative rotations innematic continuum:

:4

eþ ω

p¼ ω 4

e¼ 4b

e� 4i

e;ω

p¼ ωb

p� ωi

p;ω ¼ ωb � ωi

� �: ð2:23Þ

The evolution of director, generally characterized by kinematical equation,:

�n ¼ wi n ¼ �wi � n, can be considered in the linear nematic theory as theevolution of director disturbance dn near the state with a known constant value ofdirector, n ¼ const:

d δ�n� �

dt ¼ �ωi � n: ð2:3ÞWe finally introduce the kinematical equation and related kinematical variables,

convenient for characterizing a combined effect of viscoelastic deformations andrelative rotations, as follows:

:*eþ γ

p¼ γ *

e¼ "þ 4

e; γ

p¼ e

pþ ω

p; γ ¼ eþ ω

�: ð2:4Þ

Only incompressible case is analyzed below, when all the strains and their rates aretraceless.

3.2 Thermodynamics and Constitutive Relations

3.2.1 Non-symmetric Theory

To describe the quasi-equilibrium effects in weak nematic viscoelasticity we will usethe de Gennes type [2, 11] of potential (Helmholtz’s free energy density):

f ¼ 1=2G0 "�� 2 þ G1�n�n : "

2 þ G2 �n�n : "� �2

�2G3�n�n : " �Ωe

� �� G5�n�n :4

2e

ð2:5Þ

Here Gk are the elastic moduli, and the lower index in "eis missed for simplicity.

The expression for the dissipation D in the system,

D � TPs Tj ¼ σs : eþ σa :ω� df Tj =dt;

(2.22)

(2.23)


can be represented with the aid of (2.5) in the form:

D ¼ σsp: eþ σa

p:ωþ σs

e: e

pþ σa

e:ω

p:

Here T is the temperature, Ps is the entropy production, ss and sa are the symmetricand asymmetric parts of the extra stress tensor, respectively, and

σsp� σs � σs

e; σa

p� σa � σa

e; σs

e¼ @f

.@"; σa

e¼ @f

.@4

e: ð2:6Þ

Here ssessp

� �and sa

esap

� �are the equilibrium (non-equilibrium) symmetric and

asymmetric parts of the extra stress.Keeping in mind possible applications to LCP we consider below only the

simplest (Maxwell-like) liquid case ssp¼ 0; sa

p¼ 0 with the dissipation presented

as:

D ¼ σs : epþ σa :ω

p: ð2:7Þ

This case describes “instant” elastic response of LCP’s to the applied forces.Due to (2.5) the symmetric ss and asymmetric sa parts of extra stress traceless

tensor are defined as:

σs ¼ @f.@" ¼ G0"þ G1 �n�n � "þ " � �n�n� 2nn " : �n�n

� �h iþ 2 G1 þ G2ð Þ

�n�n� δ.3

� �" : �n�n� �

þ G3 �n�n � 4e� 4

e� �n�n

� �; ð2:81Þ

σa ¼ @f.@Ω

e¼ �G4 �n�n � "� " � �n�n

� �þ G5 �n�n �Ωe

þΩe� �n�n

� �: G4 � �G3ð Þ ð2:82Þ

Note that due to (2.5) the Onsager relation G4≡−G3 is automatically fulfilled.Among several sources of stress asymmetry, such as inertial effects of internal

rotations, orientation (Frank) elasticity and the Cosserat/Born isotropic couples, themost important for LCP is the action of external magnetic field, H . The commonassumption used below is that the magnetic field is potential, with the potentialfunction < presented as: < ¼ �1=2χ �n

� �: H H . Here χ �n

� � ¼ χ?δ þ χa�n�n is thesusceptibility tensor, χk and χk are the susceptibilities parallel and perpendicular tothe director, with χa ¼ χk � χ? being the magnetic anisotropy. Because of a typicalweak magnetization of the considered diamagnetic liquid, the effect of magneticfield on constitutive parameters Gk in (2.5) can be neglected. In this case, the bodycouple (“effective magnetic field”) is defined as:

�h ¼ �@<@�n ¼ χa �H �n � �HÞ:�

ð2:9ÞWhen the inertial effects of internal rotations are negligible, the equilibrium equationfor internal torques in magnetic field is:

σa ¼ 12 �h �n� �n�h� �

; �Σ � σa � �n ¼ 12 �h � �n �h � �n

� �� ¼ 12χa �n � �H

� ��H � �n �n � �H

� �� :

ð2:10Þ

(2.81)

(2.82)

102 A.I. Leonov

Due to (2.7), the constitutive relations between the irreversible kinematicalvariables and stress are of the LEP type:

σs ¼ η0ep þ η1 �n�n � ep þ ep� �n�n� 2�n�n �n�n : e

p

� �h iþ 2 η1 þ η2ð Þ

�n�n� δ.3

� ��n�n : ep

� �þ η3 �n�n � ωp

� ωp� �n�n

� �; ð2:111Þ

σa ¼ �η4 �n�n � ep � ep� �n�n

� �þ η5 �n�n � ωp

þ ωp� �n�n

� �η4 ¼ �η3ð Þ: ð2:112Þ

Here we used the Onsager relation: η4=−η3. Under common assumptions, weconsider the kinetic coefficients ηk, as well as the parameters Gk, being independentof �H . Also, we further assume that Gk≠0 and ηk≠0 in order to avoid thedegeneration of CE’s.

Demanding the elastic potential (2.5) to be thermodynamically stable results inthe necessary and sufficient stability conditions [11], imposed on the parameters Gk:

G0 > 0 ; G5 > 0 ; G0 þ G1 > 0 ; 3=4G0 þ G1 þ G2 > 0;

G0 þ G1ð ÞG5 > G23 �

ð2:121Þ

The thermodynamic stability conditions for dissipation in (2.81 and 2.82) inincompressible case are the same as in (2.121) with substitution Gk→ηk [12]:

η0 > 0 ; η5 > 0; η0 þ η1 > 0 ; 3=4η0 þ η1 þ η2 > 0 ;

η0 þ η1ð Þη5 > η23ð2:122Þ

Substituting now CE’s (2.9 and 2.10) into the expression for dissipation (2.7)reduces it to the quadratic form:

D ¼ η0 ep

�� 2þ2η1�n�n : e2pþ 2η2 �n�n : ep

� �2�4η3�n�n : e

p� ω

p

� �� 2η5�n�n : ω2

p: ð2:13Þ

Due to the stability constraints (2.121–2.123) the quadratic form (2.13) is positivelydefinite.

3.2.2 Symmetric Theory

As follows from (2.10), in the absence of magnetic field �H ¼ 0ð Þ when �Σ ¼σa � n ¼ 0 and sa ¼ 0, stress tensor is symmetric. So (82 and 92) yield thekinematical relations,

4e¼ λ1 " � �n�n� �n�n � "

� �; ω

p¼ λ2 e

p� �n�n� �n�n � ep

� �λ1 ¼ G3=G5; λ2¼η3=η5ð Þ:

ð2:141; 2Þ

(2.111)

(2.112)

(2.121)

(2.122)

(2.141,2)


Here 1 1 and 1 2 are sign indefinite parameters. Substituting (2.141,2) back into (82and 92), and in (5) yields the reduced (or “renormalized”) formulation of theseequations with symmetric stress as:

f ¼ 1=2G0 "�� 2 þ Gr

1�n�n : "2 þ Gr

2 �n�n : "� �2

Gr1 ¼ G1 � G2

3

G5; Gr

2 ¼ G2 þ G23

G5

� � ð2:15Þ

σ ¼ @f.@" ¼ G0"þ Gr

1 �n�n � "þ " � �n�n� 2�n�n ":�n�n� �h i

þ 2 Gr1 þ Gr

2

� � " : �n�n� �

�n�n� 1.3δ

� � ð2:161Þ

σ ¼ 1

2

@D

@ep

¼ η0ep þ ηr1 �n�n � ep þ ep� �n�n� 2�n�n e

p: �n�n

� �h iþ 2 ηr1 þ ηr2� �

ep: �n�n

� ��n�n� δ

.3

� �ð2:162Þ

The dissipation D ¼ TPs Tj ¼ s : epis presented in the form:

D=2 ¼ η0 ep

�� 2þηr1�n�n : e2p

þ ηr2 �n�n : ep

� �2ηr1 ¼ η1 � η23

η5; ηr2 ¼ η2 þ η23

η5

� � ð2:17Þ

Equation (2.161) is the Ericksen CE. It is seen that D/2 there plays the role ofRaleigh dissipative function.

The (necessary and sufficient) conditions of thermodynamic stability [11] are:

G0 > 0; G0 þ Gr1 ¼ G0 þ G1 � G2

3

G5 > 0;

3=4G0 þ Gr1 þ Gr

2 ¼ 3=4G0 þ G1 þ G2 > 0ð2:181Þ

η0 > 0; η0 þ ηr1 ¼ η0 þ η1 � η23η5 > 0;

3=4η0 þ ηr1 þ ηr2 ¼ 3=4η0 þ η1 þ η2 > 0:ð2:182Þ

One can see complete similarity in inequalities (2.181 and 2.182). When analyzingthe symmetric case the notations for simplicity will be changed as: Gr

k $ Gk ,hrk $ hk k ¼ 1; 2ð Þ.

One can expect that excluding the variables wpand e

pfrom the final formulation

in non-symmetrical case will result in coupled equations for evolution of the hiddenthermodynamic parameters Ω

eand ". In symmetrical case only e

pshould be

(2.161)

(2.162)

(2.181)

(2.182)

104 A.I. Leonov

excluded to obtain evolution equation for ". These common awkward operations areeasy with the use of N-operators.

3.3 N-operators in Viscoelastic Nematodynamics

Table 2 establishes the correspondence between the continual equations in“Section 3.2” and algebraic relations in “Section 2.1”. The constitutive parametersGk and ηk in Table 2 are:

Gk ¼ Gk k ¼ 0; 1; 3; 4; 5ð Þ; G2 ¼ 2G1 þ 2G2; G4 ¼ �G3 ð2:191Þ

ηk ¼ ηk k ¼ 0; 1; 3; 4; 5ð Þ; η2 ¼ 2η1 þ 2η2; η4 ¼ �η3: ð2:192Þ

Only non-degenerating conditions Gk≠0, ηk≠0 are considered below. Using Table 2it is easy to establish direct and inverse relations for CE’s described by ON-operators.

3.3.1 N-operator Presentation of Non-symmetric Theory

1. N-operator presentation of “elastic” and “viscous” CE’s (2.181, 2.182 and 2.9):

σ � G �n� �

&*e¼ P5

k¼0Gkak �n

� �&*

e, σs ¼ P2

k¼0Gkak �n

� �& "þ G3a3 �n

� �&4

e;

σa ¼ G4a4 �n� �

& "eþ G5a5 �n

� �&Ω

e

σ ¼ h �n� �

& γp¼ P5

k¼0hkak �n

� �& γ

p, σs ¼ P2

k¼0hkak nð Þ & e

pþ η3a3 �n

� �&ω

p;

σa ¼ h4a4 �n� �

& epþ η5a5 �n

� �&ω

p

ð2:2012Þ

Here G �n� � � Ro

G �n� �

and ) �n� � � Ro

η �n� �

are the ON-operators of moduli andviscosity.

Table 2 Correspondence between algebraic and physical variables/parameters/equations

Algebraic Elastic Nematic Viscous Nematic

xs……………….. (1.1)

symmetric tensor (variable)" ……………….. (2.81 and 2.82)elastic strain tensor

ep…………….. (2.111 and 2.112)

inelastic strain rate tensorxa……………….. (1.1)

asymmetric tensor variableW

e……………… (2.81 and 2.82)

elastic relative rotationwp…………….. (2.111 and 2.112)

inelastic relative vorticityys………………. (1.1)

symmetric tensor, functionss ……………… (2.81)symmetric extra stress

ss …………….. (2.111)symmetric extra stress

ya………………. (1.1)

asymmetric tensor, functionsa ……………… (2.82)asymmetric extra stress

sa …………… (2.112)asymmetric extra stress

rk (k=0,.., 5) ……… (1.1)parameters of N-operators

Gk ……….. (2.81 and 2.82, 2.161)nematic elastic moduli

ηk … (2.111 and 2.112, 2.162)nematic viscosities

P ………………… (1.2)quadratic form

2 f ……………… (2.5)nematic free energy

D ………………. (2.13)dissipation

(2.191)

(2.192)

(2.201,2)


2. N-operator presentations of free energy and dissipation:

f ¼ 12X5k¼0

GkΓe� ak �n� ��Γ

e; D � TPs Tj ¼

X5k¼0

ηkγp� akðnÞ� γ

pð2:211; 2Þ

3. Inverse relations expressing kinematic variables via stresses:

*e¼ G�1

�n� �

&σ � J �n� �

&σ; J �n� � ¼ P5

k¼0Jkak �n

� �;

γ ¼ )�1�n� �

&σ � F �n� �

&σ; F �n� � ¼ P5

k¼08kak �n

� �:

ð2:221; 2Þ

Here J �n� � � No

J �n� �

and F �n� � � No

8 �n� �

are the ON-operators of complianceand fluidity, respectively. Their basis scalars, compliances Jk (dimensionality ofinverse modulus) in (2.221,2), and fluidities 8k (dimensionality of inverseviscosity) in (2.221,2), are:

J0 ¼ 1G0

; J1 ¼ G23�G1G5ð Þ=G0

G5 G0þG1ð Þ�G23; J2 ¼ � 3=2 G1þG2ð Þ=G0

3=4G0þG1þG2

J3 ¼ �J4 ¼ �G3

G5 G0þG1ð Þ�G23; J5 ¼ G0þG1

G5 G0þG1ð Þ�G23;

ð2:231Þ

80 ¼ 1η0; 81 ¼

η23�η1η5ð Þ=η0η5 η0þη1ð Þ�η23

; 82 ¼ � 3=2 η1þη2ð Þ=η03=4η0þη1þη2

83 ¼ �η3η5 η0þη1ð Þ�η23

; 84 ¼ �83; 85 ¼ η0þη1η5 η0þη1ð Þ�η23

ð2:232Þ

4. Expressions of gp¼ e

pþ w

pvia Γ

e¼ "

eþΩ

eand inverse from the dual

equations (2.221,2):

gp¼ s nð Þ&*

es nð Þ ¼

X5k¼0

skak nð Þ !

;

*e¼ E nð Þ&g

pE nð Þ ¼

X5k¼0

θ kak nð Þ ! ð2:241; 2Þ

Here s nð Þ ¼ )�1 nð Þ&G nð Þ ¼ F nð Þ&G nð Þ and E nð Þ ¼ G�1 nð Þ&) nð Þ ¼ J nð Þ&) nð Þare the N-operators of relaxation frequencies and relaxation times, respectively.Using (2.231 and 2.232), their basis scalar parameters sk and qk are calculated as:

s0 ¼ G0η0; s1 ¼ η5 G1η0�G0η1ð Þþη3 G0η3�G3η0ð Þ

η0 η5 η0þη1ð Þ�η23½ � ; s2 ¼ 32 � G1þG2ð Þη0�G0 η1þη2ð Þ

η0 3=4η0þη1þη2ð Þ

s3 ¼ G3 η0þη1ð Þ�η3 G0þG1ð Þη5 η0þη1ð Þ�η23

; s4 ¼ G5η3�G3η5η5 η0þη1ð Þ�η23

; s5 ¼ G5 η0þη1ð Þ�G3η3η5 η0þη1ð Þ�η23

ð2:251Þ

θ0 ¼ η0G0

; θ1 ¼ G5 η1G0�η0G1ð ÞþG3 η0G3�η3G0ð ÞG0 G5 G0þG1ð Þ�G2

3½ � ; θ2 ¼ 32 � η1þη2ð ÞG0�η0 G1þG2ð Þ

G0 3=4G0þG1þG2ð Þ

θ3 ¼ η3 G0þG1ð Þ�G3 η0þη1ð ÞG5 G0þG1ð Þ�G2

3; θ4 ¼ η5G3�η3G5

G5 G0þG1ð Þ�G23; θ5 ¼ η5 G0þG1ð Þ�η3G3

G5 G0þG1ð Þ�G23

ð2:252Þ

(2.211,2)

(2.221,2)

(2.231)

(2.232)

(2.241,2)

(2.251)

(2.252)

106 A.I. Leonov

Note that E nð Þ ¼ ) nð Þ �G�1 nð Þ and s nð Þ ¼ G nð Þ & )�1 nð Þ ¼ E�1 nð Þ are not ON-operators.5. Evolution equation for elastic (transient) strain " and elastic rotation Ω

e,

obtained upon substituting (2.241,2) in (2.4), is:

�*eþ s nð Þ & *

e¼ g , �"þ

X2k¼0

skak nð Þ & "þ s3a3 nð Þ & 4e

¼ e;�4

eþ s4a4 nð Þ & "þ s5a5 nð Þ & 4

e¼ ω ð2:261Þ

Equations (2.261) written in the common tensor form are presented as:

�"þ s0"þ s1 n n � "þ " � n n� 2nn " : n n� �h i

þ s2 n n� δ.3

� �" : n n� �

þs3ðn n � 4e� 4

e� n nÞ ¼ e

�4

eþ s4 n n � "� " � n n

� �þ s5 n n � 4

eþ 4

e� n n

� �¼ 4

ð2:262ÞHere the basis scalars of N-operator of relaxation frequency sðnÞ are presented in(2.251).

6. Maxwell-like nematodynamic equations have the following equivalent forms:

J nð Þ& �σþ 6 nð Þ&σ ¼ g ; �σþ s nð Þ&σ ¼ G nð Þ&g ;

E nð Þ& �σþ σ ¼ ) nð Þ&gð2:27123Þ

The basis scalar parameters Jk ; 8 k and sk ; qk are expressed via the given modelparameters Gk and hk in 2.231, 2.232 and 2.241,2), respectively. The example of“split” expression for CE 2.221,2) is:

�σs þ P2k¼0

ak nð Þ& skσs � Gke� �

þ a3 nð Þ& s3σa � G3w� �

¼ 0

�σa þ a4 nð Þ& s4σs � G3e� �

þ a5 nð Þ& s5σa � G5w� �

¼ 0ð2:27 � 2Þ

7. The eigenvalues of N-operator of relaxation frequency sðnÞ due to (1.18) are:

ν1 ¼ s0; ν2 ¼ s0 þ 2=3s2; ν3;4 ¼ 1=2 s0 þ s1 þ s5 dð Þ;

d2 ¼ s0 þ s1 � s5ð Þ2�4s3s4:ð2:28Þ

Here sk are the basis parameters (2.251) of the N-operator of relaxationfrequency sðnÞ, Due to “Appendix,” all nk in (2.28) are positive, while the N-operator sðnÞ is not necessarily positive.

8. Basic representation theorem of non-symmetric linear nematic viscoelasticity:

Maxwell-like nematodynamic (2.271,2,3) are always presented in the equivalentforms of LEP CE’s (2.111, 2.112 or 2.161, 2.162), where the parameters are changedfor linear viscoelastic functionals.

(2.261)

(2.262)

(2.271,2,3)

(2.27*2)


The N-operator formulation is:

σ ¼X5k¼0

ak nð Þ& φk tð Þ*γ tð Þn o

φk tð Þ*γ tð Þ �Z t

�1φk t � t1ð Þγ t1ð Þdt1

0@ 1Aσs ¼

X2k¼0

ak nð Þ& φk*e� �

þ a3 nð Þ& φ3*ω� �

; σa ¼ a4 nð Þ& φ4*e� �

þ a5 nð Þ& φ5*ω� � ð2:291Þ

The common tensor formulation is:

σs ¼ φ0*eþ φ1* �n�n � eþ e � �n�n� 2�n�n �n�n : e� �h i

þ 2 φ1 þ φ2ð Þ* �n�n� δ.3

� ��n�n : e� �

þφ3* �n�n � w� w � �n�n� �

;σa ¼ �φ4* �n�n � e� e � �n�n� �

þ φ5* �n�n � wþ w � �n�n� �

ð2:292ÞHere φk tð Þare:φ0 tð Þ ¼ G0e

�ν1t; φ1 tð Þ ¼ �G0e�ν1t þ G0 þ G1ð Þ κ2e

�ν3t � κ1e�ν4tð Þ;

φ2 tð Þ ¼ �3=2G0e�ν1t þ 2 3=4G0 þ G1 þ G2ð Þe�ν2t

φ3 tð Þ ¼ � s3=dð Þ G0 þ G1ð Þe�ν1t þ s3=dð Þ G0 þ G1ð Þ�κ1G3s4=d½ �e�ν3t þ κ2G3e�ν4t

φ4 tð Þ ¼ � κ2G3 þ G5s4=dð Þe�ν3t þ κ1G3 þ G5s4=dð Þe�ν4t

φ5 tð Þ ¼ �G3 s3=dð Þe�ν1t þ G3 s3=dð Þ½ � � κ1G5s4=d�e�ν3t þ κ2G5e�ν4t

κ1 ¼ s0 þ s1 � ν3ð Þ=d; κ2 ¼ s0 þ s1 � ν4ð Þ=df gð2:30Þ

The derivation of the formulae (2.291, 2.292 and 2.30), which proves the theorem,is made in several steps.

1. Consider the first equation in (2.261),�*eþ s nð Þ�*

e¼ g . Searching for a

solution of the homogeneous equation in the form, Γhetð Þ ¼ bΓe�nt, reduces

solution of this equation to the eigenvalue problem (1.172), whereNr n; nð Þ ¼ s nð Þ � nI nð Þand eigenvalues nk are exposed in (2.28). Theeigenoperators Q nk ; nð Þare presented in (1.20), with substitutions rk ! sk .

2. Utilizing additionally the condition (1.21) yields the following solution of

initial homogeneous problem: Γhetð Þ ¼ P4

k¼1e�nk tQ nk ; nð Þ � Γh

e0ð Þ. Employing

then the standard technique yields the solution of the evolution (2.211,2)presented in N-operator form as a linear memory functional: Γ

etð Þ ¼P4

k¼1Q νk ; nð Þ � Rt

�1e�νk t�t1ð Þγ t1ð Þdt1. Utilizing here the relations (1.19 and

1.21) with substitutions rk ! sk yields:

Γetð Þ ¼

X5k¼0

ak nð Þ& χk tð Þ*γ tð Þn o

ð2:31ÞThe following notations have been used in (2.31):

χ0 tð Þ ¼ e�ν1t;χ1 tð Þ ¼ �e�ν1t þ κ2=dð Þe�ν3t � κ1=dð Þe�ν4t;χ2 tð Þ ¼ 3=2 e�ν2t � e�ν1tð Þ;χ3 tð Þ ¼ s3=dð Þ e�ν3t � e�ν1tð Þ; χ4 tð Þ ¼ s4=dð Þ e�ν4t � e�ν3tð Þ;χ5 tð Þ ¼ �κ1e

�ν3t þ κ2e�ν4t

ð2:32Þ

(2.291)

(2.292)

108 A.I. Leonov

3. Using (2.31) in the first relation of (2.161) yields (2.291, 2.292 and 2.30).Formulae (2.291, 2.292/2.30 and 2.31/2.32) correctly describe the two limiting

cases:

1. The initial elastic “jump”, i.e. Γeþ0ð Þ ¼ Γ

0and σ ¼ G nð Þ�Γ

0, when

g tð Þ ¼ Γ0d tð Þ, and

2. The case: Γe1ð Þ ¼ θ nð Þ& γ

0;σ 1ð Þ ¼ ) nð Þ&γ

0, when g tð Þ ¼ H tð Þg

0�g0¼ constÞ.

Here δ (t) and H (t) are Dirac delta and Heaviside functions, and G nð Þ; E nð Þ, and) nð Þare the N-operators of moduli, relaxation time, and viscosity, respectively.

3.3.2 TI-operator Presentation of Symmetric Theory

1. The TI-operator presentations of “elastic” and “viscous” CE’s (2.161 and2.161) are:

σ ¼ G nð Þ&" ¼ ) nð Þ&ep; G nð Þ ¼

X2k¼0

Gkak nð Þ;

) nð Þ ¼X2k¼0

ηkak nð Þ

ð2:3312Þ

Here G nð Þ � SG nð Þ and) nð Þ � Sη nð Þ are the TI operators of moduli andviscosity. Also, Gk and bhk are defined in (2.191 and 2.192) for k=0, 1, 2 with no-degenerating conditions Gk 6¼ 0, hk 6¼ 0. It is also worth reminding of simplifyingnotations, Gr

k $ Gk , hrk $ hk k ¼ 1; 2ð Þ, accepted in “Section 3.2.2” after renorm-alization procedure.2. The TI-operator presentations of free energy and dissipation are:

f ¼ 1=2"&G nð Þ&"; D ¼ ep&) nð Þ� e

p: ð2:3412Þ

3. The inverse relations expressing kinematical variables via stresses are:

" ¼ G�1 nð Þ&σ;G�1 nð Þ � J nð Þ ¼P3i¼0

Jkak nð Þ; " ¼ )�1 nð Þ&σ;)�1 nð Þ

� 6 nð Þ ¼P3i¼0

8kak nð Þ

ð2:3512ÞHere J nð Þand F nð Þare the compliance and fluidity TI-operators, respectively.Due to (1.29) the expressions for their respective basis scalars are:

J0 ¼ 1=G0; J1 ¼ � G1=G0

G0 þ G1; J2 ¼ � 3=2 G1 þ G2ð Þ=G0

3=4G0 þ G1 þ G2ð2:361Þ

80 ¼ 1=η0; 81 ¼ � η1=η0η0 þ η1

; 82 ¼ � 3=2 η1 þ η2ð Þ=η03=4η0 þ η1 þ η2

ð2:362Þ

(2.331,2)

(2.341,2)

(2.351,2)

(2.361)

(2.362)


4. The expressions epvia "

eor vice versa using dual equations (2.331,2) are:

ep¼ EðnÞ&"; " ¼ sðnÞ&e

p� ð2:371:2Þ

Here sðnÞ =)�1ðnÞ&GðnÞ ¼ 6ðnÞ&GðnÞ and EðnÞ ¼ G�1ðnÞ&)ðnÞ ¼ JðnÞ � )ðnÞare the TI-operators of relaxation frequencies and relaxation times, respectively.Using (2.361 and 2.362), their basis scalar parameters sk and θk are calculated as:

θ0 ¼ η0G0

; θ1 ¼ η1G0 � G1η0G0 G0 þ G1ð Þ ; θ2 ¼

3

2� G0 η1 þ η2ð Þ � η0 G1 þ G2ð Þ

G0 3=4G0 þ G1 þ G2ð Þ ð2:381Þ

s0 ¼ G0

η0; s1 ¼ G1η0 � η0G0

η0 η0 þ η1ð Þ ; s2 ¼ 3

2� η0 G1 þ G2ð Þ � G0 η1 þ η2ð Þ

η0 3=4η0 þ η1 þ η2ð Þ : ð2:382Þ

5. Evolution equation for elastic (transient) strain ", obtained upon substituting(2.371.2) in (2.4):

��:"þ s �n

� �� " ¼ e , ��:"þ

X2k¼0

skak nð Þ� " ¼ e: ð2:391Þ

It is written in the common tensor form as:

��:"þ s0"þ s1 �n�n � "þ " � �n�n� 2nn " : �n�n

� �h iþ s2 �n�n� δ

.3

� �" : �n�n� �

¼ e: ð2:392Þ

The basis scalars of TI-operator of relaxation frequency sðnÞ are presented in(2.382).6. Maxwell-like nematodynamic equations have the following equivalent forms:

J nð Þ� ��:σþF nð Þ�σ ¼ e; ��

:σþ s nð Þ� σ ¼ G nð Þ� e; E nð Þ� ��

:σþ σ

¼ ) nð Þ� e ð2:401; 2; 3Þ

The basis scalar parameters Jk,8k and sk,Ek are expressed via the given modelparameters Gk and ηk in (2.361, 2.362 and 2.381, 2.382), respectively.7. The eigenvalues of TI-operator of relaxation frequency sðnÞ due to (1.39) are:

ν1 ¼ s0 ¼ G0

η0; ν2 ¼ s0 þ s1 ¼ G0 þ G1

η0 þ η1; ν3 ¼ s0 þ 2

3s2 ¼ 3=4G0 þ G1 þ G2

3=4η0 þ η1 þ η2:

ð2:41Þ

Due to the stability conditions (2.181 and 2.182) all the eigenvalues vk are positiveand describe the relaxation frequencies, with the respective relaxation timesθk ¼ 1=νk .

(2.371,2)

(2.381)

(2.382)

(2.391)

(2.392)

(2.401,2,3)

110 A.I. Leonov

8. Basic representation theorem of symmetric linear nematic viscoelasticity:

Maxwell-like nematodynamic equations (2.401,2,3) are always presented in theequivalent forms of Ericksen CE’s (2.162), where parameters are changed for linearviscoelastic functionals, as:

σ ¼X2k¼0

ak nð Þ& φk tð Þ*e tð Þn o

φk tð Þ*e tð Þ �Z t

�1φk t � t1ð Þe t1ð Þdt1

0@ 1A ð2:421Þ

or:

σ ¼ φ0*eþ φ1* n n � eþ e � n n� 2n n n n : e� �h i

þ 2 φ1 þ φ2ð Þ* n n� δ.3

� �n n : e� �ð2:422Þ

With eigenvalues vk given in (2.41), φk (t) are given as:

φ0 tð Þ ¼ G0e�ν1t;φ1 tð Þ ¼ �G0e

�ν1t þ G0 þ G1ð Þe�ν2t;

φ2 tð Þ ¼ �3=2G0e�ν1t þ 2 3=4G0 þ G1 þ G2ð Þe�ν3t

ð2:43Þ

The proof is the same as in the non-symmetric case, and based on the solution ofevolution (2.391 and 2.392) for elastic (transient) strain,

" ¼X2k¼0

ak nð Þ� χk tð Þ*e tð Þn o

χ0 tð Þ ¼ e�ν1t;χ1 tð Þ ¼ e�ν2t � e�ν1t;χ2 tð Þ ¼ 3

2e�ν3t � e�ν1tð Þ

� �ð2:44Þ

The same limiting cases as in non-symmetric linear nematic viscoelasticity are validhere.

3.3.3 Soft Modes in Linear Nematic Viscoelasticity

Consider now non-generating TI operators, when the values of material parametersbelong to the marginal stability boundaries in (2.181 and 2.182). There are fourindependent marginal stability conditions:

G0 þ Gr1 ¼ G0 þ G1�G2

3

G5 ¼ 0; η0 þ ηr1 ¼ η0 þ η1�η23

η5 ¼ 0 ð2:451Þ

3=4G0 þ Gr1 þ Gr

2 ¼ 3=4G0 þ G1 þ G2 ¼ 0; 3=4η0 þ ηr1 þ ηr2 ¼ 3=4η0 þ η1 þ η2 ¼ 0:

ð2:452ÞThe nearly marginal, still stable situations happen when instead (2.451 and 2.452)

the four independent conditions are satisfied:

G0 þ G1 ¼ dGG0; h0 þ h1 ¼ dhh0 0 << dG; dh << 1ð Þ ð2:461Þ

3=4G0 þ G1 þ G2 ¼ 3=2kGG0; 3=4η0 þ η1 þ η2 ¼ 3=2kηη0 0 << kG; kη << 1� � ð2:462Þ

If one of (or both) the conditions (2.461) occurs, the behavior of viscoelastic (as the

(2.421)

(2.422)

(2.451)

(2.452)

(2.461)

(2.462)


viscous or weakly elastic) nematics is sensitive to magnetic field, with great effectsexpected when the field is applied. On the contrary, the conditions (2.462) seem to beinsensitive to magnetic field.

When the magnetic field is absent, one could a use the marginal conditions (2.451and 2.452) and employ the results of “Section 2.5.4”. We consider here for examplethe extreme case when both the elastic and viscous TI operators are completely soft.Then CE’s (2.331,2) take the form:

σ ¼ G0a nð Þ� " ¼ η0a nð Þ� ep; a nð Þ ¼ a0 nð Þ � a1 nð Þ � 3=2a2 nð Þ: ð2:47Þ

In this case,

G1 ¼ �G0;G2 ¼ G0=4; η1 ¼ �η0; η2 ¼ η0=4; ð2:48Þand free energy and dissipation are represented as:

f =G0 ¼ 1=2"&a nð Þ&" ¼ 1=2 "�� 2�n n:"2þ1=4 n n:"

� �2� 0

2D=η0 ¼ 1=2ep&a nð Þ&e

p¼ 1=2 e

p

�� 2�n n:e2pþ1=4 n n:e

p

� �2� 0:

ð2:49Þ

The TI operator a nð Þ is singular, i.e. neither a�1 nð Þ exists nor the formulae(2.361, 2.362, 2.371,2 and generally, 2.381, 2.382).

The evolution equation (2.391) for elastic (transient) strain ", for super-soft casedue to limit in formulae (1.421, 1.422 and 1.423) is represented as:

�eþ s0a nð Þ&e ¼ e , �eþ s0 "� �n�n�"� "��n�nþ 2nn ":n n� �

�3=2 n n� δ.3

� �":n n� �

� ¼ eh

ð2:50Þ

Here the only parameter, s0 ¼ G0=η0, is the relaxation frequency.

4. The eigenvalues of super-soft TI-operator of relaxation frequency s nð Þ ¼ s0a nð Þdue to (1.35) are:

ν1 ¼ s0 ¼ G0=η0; ν2 ¼ ν3 ¼ 0: ð2:51Þ

5. The basic representation theorem of symmetric linear nematic viscoelasticity inthe super-soft case is presented by the limit singular case of (2.421) as:

σ ¼ G0a n� �

&E tð Þ;E tð Þ ¼Z t

�1e t1ð Þe�s0 t�t1ð Þdt1

σ ¼ G0 E � �n�n�E � E � �n�nþ 2nn E:�n�n� �

� 3=2 �n�n� δ.3

� �E:�n�n� �h i ð2:52Þ

The derivation is based on direct use of (2.51) and the marginal stabilityconditions (2.451 and 2.452).

112 A.I. Leonov

In this case, the form of convolution formula (2.44) for elastic (transient) strains "remains the same, but the functions χk (t) there change for:

# 0 tð Þ ¼ e�s0t; # 1 tð Þ ¼ 1� e�s0t; # 2 tð Þ ¼ 3=2 1� e�s0tð Þ: ð2:53Þ

It means that the elastic strain tensor " can unrestrictedly grow in time. This is theasymptotic effect of the super-soft nematic viscoelastic behavior. Nevertheless, oneshould recall that the linear nematic viscoelasticity in the soft cases can only holdeither for a restricted time with a given constant value of strain rate e, or for a smallamplitude oscillatory flow.

4 Results and Discussions

This paper consists of two parts. The first part analyzes the mathematical propertiesof linear nematic (N-) operators, which appeared in nematic viscous and elastictheories. The N-operators were first introduced and described in our preprint [22]. Itis shown that N-operators are represented by a specific set of basis fourth ranktensors in both the general no-symmetric and particular symmetric cases. Theanalysis includes the group and spectral properties of N-operators in both non-symmetric and symmetric cases, as well as their singular properties in symmetriccase, with singular, semi-positively definite symmetric TI-operators. The mainobjective of this part is to introduce and develop a new theoretical tool for analysesof nematic theories, which highly simplifies theoretical description.

The second part of the paper, developed a theory of linear viscoelasticnematodynamics applicable to LCP in the mono-domain limit, when the Frankelasticity effects are absent. The new theory of N-operators developed in the firstpart of paper, was applied to general analysis of the linear viscoelasticnematodynamics of Maxwell-type. Continuum approach employed in the paper isbased on well-elaborated specific viscoelastic kinematics and non-equilibriumthermodynamics. The action of external fields is illustrated on example of magneticfield. Additionally, the concept of marginal stability introduced first in papers [11,12] and mathematically developed for singular TI operators in the “Section 2.5.4”, isemployed for analyses of soft viscoelastic modes in the “Section 3.3.3”. The mainresults are the reduction of the differential formulation to the LEP-like equationswhere the coefficients are changed for linear memory functionals whose parametersare calculated from original viscosities and moduli.

Similar formulae have been obtained in paper [13] using statistical derivation.Keeping aside discussion about the derivation and concepts in [13], even a quickcomparison of formulae for integral presentation of the theory in this paper withsimilar ones obtained in [13] (in the single relaxation mode Maxwell-type, ordumbbell limit) shows their qualitative difference, e.g. in the non-symmetric currentapproach, the anisotropic relaxation functions are presented via four independentexponential terms, whereas in [13] via three terms. In the symmetric case only twoindependent exponential terms describe the anisotropic relaxation functions in [13]instead of three terms in the present paper.


It is remarkable that the complete soft shearing and soft elongation case in thispaper is described when holding the complete anisotropy by formulae (2.52) withonly one relaxation time as in the isotropic case. When comparing these results withreal linear dynamic behavior of LCP’s one should involve the semi-soft descriptionof relaxations, which are grouping near the basic soft relaxation frequency s0 in(2.52).

The results of the present analysis could be directly applied to the description oflinear dynamic behavior of rigid enough industrial thermotropic LCP’s, where theproposed Maxwell-like phenomenology seems to explain the major effect, the exactbalance of the anisotropic viscous and elastic forces (including the magnetic torque),which cause extension, bending and twisting rigid enough macromolecules. For themore flexible, model types of LCP’s where the conformation effects should also betaken into account, the proposed approach could serve at least as the firstapproximation.

Appendix

A prove of positive definiteness of eigenvalues for a dual operator Np nð Þ in (1.14)represented via two positive Onsager operators

According to (1.14) Np nð Þ ¼ No�1q nð Þ&No

r nð Þ; where Nor nð Þ andNo

q nð Þ are positive,non- degenerating ON operators. The eigenvalues of operator Np nð Þ are expressed in(1.18) via its basis parameters p with substitution r→p(r,q) and the use of (1.15),where the basis parameters rk and qk under condition r4=−r3 satisfy inequalities (1.3).It is proved below that due to these inequalities, all eigenvalues vk of operator Np nð Þare positive.

The first two eigenvalues, v1 and v2 defined in (1.18) with the substitution r→p(r,q) and (1.15), are positive because

ν1 ¼ p0 ¼ r0=q0 > 0; ν2 ¼ p0 þ 2=3p2 ¼ r0 þ 2=3r2ð Þ= q0 þ 2=3q2ð Þ > 0:

Other two eigenvalues v3,4 due to (1.18) are presented as:

ν3;4 ¼ 1=2 p0 þ p1 þ p5 dð Þ; d2 ¼ p0 þ p1 þ p5ð Þ2�4 p5 p0 þ p1ð Þ þ p3p4½ �: ð3ÞHere p=p(r,q) is defined in (1.15). The proof that v3 and v4 are positive is givenbelow in three steps.

1. We firstly show that p0 þ p1 þ p5 > 0. Due to (1.15) this sum is represented as:

p0 þ p1 þ p5 ¼ r0 þ r1ð Þq5 þ q0 þ q1ð Þr5 � 2r3q3q5 q0 þ q1ð Þ � q23

: ð4Þ

The denominator in (4) is positive due to inequalities (1.3), and the numerator ispositive, because

r0 þ r1ð Þq5 þ q0 þ q1ð Þr5 � 2q3r3 � 2 r0 þ r1ð Þr5 q0 þ q1ð Þq5½ �1=2�2q3r3 > 2 q3r3j j � q3r3ð Þ > 0:

2. We secondly show that d2>0. Presenting d2 ¼ r0þr1ð Þq5� q0þq1ð Þr5½ �2þ4F

q5 q0þq1ð Þ�q23½ �2 , where F ¼r5 r0 þ r1ð Þr23g xð Þ; g xð Þ ¼ x� αð Þ x� βð Þ x � q3=r3ð Þ, α ¼ q0 þ q1ð Þ= r0 þ r1ð Þ;

114 A.I. Leonov

and β ¼ q5=r5, and establishing that min g xð Þ ¼ g xmð Þ ¼ g 1=2 αþ βð Þf g ¼�1=4 α� βð Þ2, yields:

d2 � r0 þ rð Þ1q5 � q0 þ q1ð Þr5� �2þ4 min F

q5 q0 þ q1ð Þ � q23� �2

¼ r0 þ r1ð Þq5 � q0 þ q1ð Þr5½ �2 q5 q0 þ q1ð Þ � q23� �

r5 r0 þ r1ð Þ q5 q0 þ q1ð Þ � q23� �2 > 0:

Using now (1.15), it is shown that

p5 p0 þ p1ð Þ þ p3p4 ¼ r5 r0 þ r1ð Þ � r23q5 q0 þ q1ð Þ � q23

> 0: ð5Þ

3. Finally, due to (3 and 4) it is seen that ν3 ¼ 1=2 p0 þ p1 þ p5 þ dð Þ > 0,and due to (3 and 5) it is clear that ν4 ¼ 1=2 p0 þ p1 þ p5 � dð Þ ¼p5 p0 þ p1ð Þ þ p3p4½ �=ν3 > 0.

References

1. Beris, A.N., Edwards, B.J.: Thermodynamics of flowing systems. Oxford University Press, Oxford(1999)

2. de Gennes, P.G.: Weak nematic gels. In: Helfrich, W., Kleppke, G. (eds.) Liquid crystals in one andtwo dimensional order, pp. 231–237. Springer, Berlin (1980)

3. de Gennes, P.G., Prost, J.: The physics of liquid crystals, 2nd edn. Clarendon, Oxford (1993)4. Doi, M., Edwards, S.F.: The theory of polymer dynamics. Clarendon, Oxford (1986)5. Einaga, Y., Berry, G.C., Chu, S.-G.: Rheological properties of rod-like polymers in solution. 3.

Transient and steady-state studies on nematic solutions. Polymer (Japan) 17, 239 (1985)6. Feng, J.J., Sgalari, G., Leal, L.G.: A theory for flowing nematic polymers with orientational distortion.

J. Rheol. 44, 1085–1101 (2000)7. Golubovich, L., Lubensky, T.C.: Nonlinear elasticity of amorphous solids. Phys. Rev. Lett. 63, 1082–

1085 (1989)8. Han, C.D., Ugaz, V.M., Burghardt, W.R.: Shear stress overshoots in flow inception of semi-flexible

thermotropic liquid crystalline polymers: experimental test of a parameter-free model prediction.Macromolecules 34, 3642–3645 (2001)

9. Kleman, M.: Points, lines and walls. Wiley, New York (1983)10. Larson, R.G.: The structure and rheology of complex fluids. Oxford Press, New York (1999)11. Leonov, A.I., Volkov, V.S.: General analysis of linear nematic elasticity. J. Eng. Phys. Thermophys.

77, 717–726 (2004)12. Leonov, A.I., Volkov, V.S.: Dissipative soft modes in viscous nematodynamics. Rheol. Acta 44, 331–

341 (2005)13. Long, D., Morse, D.C.: A Rouse-like model of liquid crystalline polymer melts: director dynamics and

linear viscoelasticity. J. Rheol. 46, 49–92 (2002)14. Lubensky, T.C., Mukhopadya, R.: Symmetries and elasticity of nematic gels. Phys. Rev. E 66, 011702

(2002)15. Marrucci, G., Greco, F.: Flow behavior of liquid crystalline polymers. Adv. Chem. Phys. 86, 331–404

(1993)16. Odell, P.A., Unger, G., Feijo, J.L.: A rheological, optical and X-ray study of the relaxation and

orientation of nematic PBZT. J. Polym. Sci.: Polym. Phys. 31, 141 (1993)17. Pleiner, H., Brand, H.R.: Macroscopic dynamic equations for nematic liquid crystalline side-chain

polymers. Mol. Cryst. Liq. Cryst. 199, 407–418 (1991)


18. Pleiner, H., Brand, H.R.: Local rotational degrees of freedom in nematic liquid-crystalline side-chainpolymers. Macromolecules 25, 895–901 (1992)

19. Ugaz, V.M., Burghardt, W.R.: In situ X-ray scattering study of a model thermotropic copolyesterunder shear: evidence and consequences of flow-aligning behavior. Macromolecules 31, 8474–8484(1998)

20. Volkov, V.S., Kulichikhin, V.G.: Macromolecular dynamics in anisotropic viscoelastic liquids.Macromolec. Symposia 81, 45–53 (1994)

21. Warner, M., Terentjev, E.M.: Liquid crystal elastomers. Clarendon Press, Oxford (2003)22. Leonov A.I.: Algebraic theory of linear viscoelastic nematodynamics. http://arxiv.org/e-print/cond-mat/

0409274, http://arxiv.org/e-print/cond-mat/0409275

116 A.I. Leonov

http://arxiv.org/e-print/cond-mat/0409274



Math Phys Anal Geom (2008) 11:117–129DOI 10.1007/s11040-008-9043-x

A Wegner-type Estimate for Correlated Potentials

Victor Chulaevsky

Received: 19 February 2008 / Accepted: 5 May 2008 /Published online: 5 July 2008© Springer Science + Business Media B.V. 2008

Abstract We propose a fairly simple and natural extension of Stollmann’slemma to correlated random variables. This extension allows to obtainWegner-type estimates even in various problems of spectral analysis of randomoperators where the original Wegner’s lemma is inapplicable, e.g., for cor-related random potentials with singular marginal distributions and for multi-particle Hamiltonians.

Keywords Wegner estimate · Stollmann’s lemma · Multi-scale analysis

Mathematics Subject Classifications (2000) Primary 35P20 · Secondary 47F05

1 Introduction

The regularity problem for the limiting distribution of eigenvalues of infinitedimensional self-adjoint operators appears in many problems of mathematicalphysics. Specifically, consider a lattice Schrödinger operator (LSO, for short)H: �2(Zd) → �2(Zd) given by

(Hψ)(x) =∑

y: |y−x|=1

ψ(y) + V(x)ψ(x); x, y ∈ Zd.

V. Chulaevsky (B)Département de Mathématiques, Université de Reims, Moulin de la Housse,B.P. 1039 51687 Reims, Francee-mail: [email protected]

118 V. Chulaevsky

For each finite subset � ⊂ Zd, let E�

j , j = 1, . . . , |�|, be eigenvalues of H withDirichlet b.c. in �. Consider the family of finite sets �L = [−L, L]d ∩ Z

d anddefine the following quantity (which does not necessarily exist for an arbitraryLSO):

k(E) = limL→∞

1

(2L + 1)dcard

{j: E�L

j � E}

.

If the above limit exists, k(E) is called the limiting distribution function (LDF)of e.v. of H. It is not difficult to construct various examples of a function V:Z

d → R (called the potential of the operator H) for which the LDF does notexist. One can prove the existence of LDF for periodic potentials V, but evenin this, relatively simple situation the existence of k(E) is not a trivial fact.

However, the existence of k(E) can be proved for a large class ofergodic random potentials. Namely, consider an ergodic dynamical system(�,F, P, {Tx, x ∈ Z

d}) with discrete time Zd and a measurable function (some-

times called a hull) v: � → R. Then we can introduce a family of samplepotentials

V(x, ω) = v(Txω), x ∈ Zd,

labeled by ω ∈ �. Under the assumption of ergodicity of {Tx} (and, forexample, boundedness of function v), the quantity

k(E, ω) = limL→∞

1

(2L + 1)dcard

{j: E�L

j (ω) � E}

is well-defined P-a.s. Moreover, k(E, ω) is P-a.s. independent of ω, so its valuefor a.e. ω is natural to take as k(E). In such a context, k(E) is usually called theintegrated density of states (IDS, for short). It admits an equivalent definition:

k(E) = E[( f, �(−∞,E](H(ω)) f )

],

where f ∈ �2(Zd) is any vector of unit norm, and �(−∞,E](H(ω)) is the spectralprojection of H(ω) on (−∞, E]. The reader can find a detailed discussionof the existence problem of IDS in excellent monographs by Carmona andLacroix [5] and by Pastur and Figotin [16]. See also articles [3, 4, 6, 10, 15].

It is not difficult to see that k(E) can be considered as the distribution func-tion of a normalized measure, i.e. probability measure, on R. If this measuredk(E), called the measure of states, is absolutely continuous with respect to theLebesgue measure dE, its density (or Radon–Nikodim derivative) dk(E)/dEis called the density of states (DoS). In physical literature, it is customary toneglect the problem of existence of such density, for if dk(E)/dE is not afunction, then “it is simply a generalized function”. However, the real problemis not terminological. The actual, explicit estimates of the probabilities of theform

P

{∃ eigenvalue E�L

j ∈ (a, a + ε)}

for an LSO H�L in a finite cube �L of size L, for small ε, often depend essen-tially upon the existence and the regularity properties of the DoS dk(E)/dE.

A Wegner-type estimate for correlated potentials 119

Apparently, the first fairly general result relative to the existence andboundedness of the DoS is due to Wegner [21].

Lemma 1.1 (Wegner) Assume that {V(x, ω), x ∈ Zd} are i.i.d. r.v. with a

bounded density pV(u) of their common probability distribution: ‖pV‖∞ = C <

∞. Then the DoS dk(E)/dE exists and is bounded by the same constant C.

The proof can be found, for example, in the monographs [5] and [16].This estimate and some of its generalizations have been used in the multi-

scale analysis (MSA) developed in the works by Fröhlich and Spencer [13],Fröhlich et al. [12], von Dreifus and Klein [19, 20], and in a number ofmore recent works where the so-called Anderson Localization phenomenonhas been observed. Namely, it has been proven that all eigenfunctions ofrandom LSOs decay exponentially at infinity with probability one (for P-a.e. sample of random potential V(ω)). Von Dreifus and Klein [20] provedan analog of Wegner estimate and used it in their proof of localization forGaussian and some other correlated (but non-deterministic) potentials. Theauthor of this paper recently proved, in a joint work with Suhov [8], ananalog of Wegner estimate for a system of two or more interacting quantumparticles on the lattice under the assumption of analyticity of the probabilitydensity pV(u), using a rigorous path integral formula by Molchanov (see adetailed discussion of this formula in the monograph [5]). In order to relaxthe analyticity assumption in a multi-particle context, Chulaevsky and Suhov[9] used later a more general and flexible result: a lemma proved by Stollmann(cf. [17] and [18]) which we discuss below.

In the present work, we propose a fairly simple and natural extension ofStollmann’s lemma to correlated, but still non-deterministic random fieldsgenerating random potentials. Our main motivation here is to lay out a wayto interesting applications to localization problems for multi-particle systems.

2 Stollmann’s Lemma for Product Measures

Recall the Stollmann’s lemma and its proof for independent random variables.Let m � 1 be a positive integer, and J an abstract finite set with |J|(= cardJ) =m. Consider the Euclidean space R

J ∼= Rm with the standard basis (e1, . . . , em),

and its positive orthant

RJ+ = {

q ∈ RJ: q j � 0, j = 1, 2, . . . , m

}.

Definition 2.1 Let J be a finite set with |J| = m. Consider a function : RJ →

R. It is called diagonally monotone (DM, for short) if it satisfies the followingconditions:

(1) for any r ∈ RJ+ and any q ∈ R

J ,

(q + r) � (q); (1)

120 V. Chulaevsky

(2) moreover, for e = e1 + · · · + em ∈ RJ , for any q ∈ R

J and for any t > 0

(q + t · e) − (q) � t. (2)

It is convenient to introduce the notion of DM operators considered asquadratic forms. In the following definition, we use the same notations asabove.

Definition 2.2 Let H be a Hilbert space. A family of self-adjoint operatorsB(q): H → H, q ∈ R

J , is called DM if,

∀ q ∈ RJ ∀ r ∈ R

J+ B(q + r) � B(q),

in the sense of quadratic forms, and for any vector f ∈ H with ‖ f‖ = 1, thefunction f : R

J → R defined by

f (q) = (B(q) f, f )

is DM.

In other words, the quadratic form QB(q)( f ):= (B(q) f, f ) as a function ofq ∈ R

J is non-decreasing in any q j, j = 1, . . . , |J|, and

(B(q + t · e) f, f ) − (B(q) f, f ) � t · ‖ f‖2.

Remark 2.3 By virtue of the min-max principle for self-adjoint operators, if anoperator family H(q) in a finite-dimensional Hilbert space H is DM, then eacheigenvalue EB(q)

k of B(q) is a DM function.

Remark 2.4 If H(q), q ∈ RJ , is a DM operator family in a Hilbert space H, and

H0 : H → H is an arbitrary self-adjoint operator, then the family H0 + H(q) isalso DM.

This explains why the notion of diagonal monotonicity is relevant to thespectral theory of random operators. Note also, that this property applies tophysically interesting examples where dim H = +∞, but H(q) have, e.g., acompact resolvent, as in the case of Schrödinger operators in a finite cube withDirichlet b.c. and with a bounded potential, so the respective spectrum is purepoint, and even discrete.

For any measure μ on R, we will denote by μJ the product measure μ ×· · · × μ on R

J . Furthermore, for any probability measure μ and for any ε > 0,define the following quantity:

s(μ, ε) = supa∈R

μ([a, a + ε])

We will denote by μm−1j the marginal probability distribution induced by μJ on

q′�= j = (q1, . . . , q j−1, q j+1, . . . , qm).

Lemma 2.5 (Stollmann [17]) Let J be a finite index set, |J| = m, μ be aprobability measure on R, and μJ be the product measure on R

J with marginal


measures μ. If the function : RJ → R is DM, then for any open interval I ⊂ R

we have

μJ{ q: (q) ∈ I } � m · s(μ, |I|).

We provide below a proof of Stollmann’s lemma; this will allow to extend itto the case of correlated potentials.

Proof Let I = (a, b), b − a = ε > 0, and consider the set

A = { q: (q) � a }.Furthermore, define recursively sets Aε

j , j = 0, . . . , m, by setting

Aε0 = A, Aε

j = Aεj−1 + [0, ε]e j :=

{q + te j : q ∈ Aε

j−1, t ∈ [0, ε]}.

Obviously, the sequence of sets Aεj , j = 1, 2, ..., is increasing with j. The DM

property implies

{ q: (q) < b } ⊂ Aεm.

Indeed, if (q) < b , then for the vector r := q − ε · e we have by (2):

(r) � (r + ε · e) − ε = (q) − ε � b − ε � a,

meaning that r ∈ { � a } = A and, therefore,

q = r + ε · e ∈ Aεm.

Now, we conclude that

{ q: (q) ∈ I } = { q: (q) ∈ (a, b) }= { q: (q) < b } \ { q: (q) � a } ⊂ Aε

m \ A.

Furthermore,

μm{ q: (q) ∈ I } � μm (Aε

m \ A)

= μm

⎛

⎝m⋃

j=1

(Aε

j \ Aεj−1

)⎞

⎠ �m∑

j=1

μm(

Aεj \ Aε

j−1

).

For q′�=1 = (q2, . . . , qm) ∈ R

m−1, set

I1(q′�=1) = {

q1 ∈ R : (q1, q′�=1) ∈ Aε

1 \ A}.

By definition of the set Aε1, this is an interval of length not bigger than ε. Since

μJ is a product measure, we have

μm(Aε1 \ A) =

∫dμm−1(q′

�=1)

∫

I1

dμ(q1) � s(μ, ε). (3)

Similarly, we obtain for j = 2, . . . , m

μm(Aεj \ Aε

j−1) � s(μ, ε),

122 V. Chulaevsky

yielding

μm{ q: (q) ∈ I } �m∑

j=1

μm(Aεj \ Aε

j−1) � m · c(μ, ε). �

Now, taking into account the above Remark 2.4, Lemma 2.5 yields immedi-ately the following estimate.

Lemma 2.6 Let H� be an LSO with a random potential V(x; ω) in a finitebox � ⊂ Z

d with Dirichlet b.c., and (H�) its spectrum, i.e. the collection ofits eigenvalues E(�)

j , j = 1, . . . , |�|. Assume that r.v. V(x; ·) are i.i.d. with themarginal distribution μV. Then

P { dist((H�(ω), E) � ε } � |�|2s(μV, 2ε).

This is an analogue of Wegner bound. One visible distinction is the formof its volume dependence: the factor |�|2 instead of |�| in the conventionalWegner bound. One has to keep in mind, however, that

• Stollmann’s lemma on monotone functions is sharp (cf. [17]); eigenvalues,however, are particular monotone functions, which explains why conven-tional Wegner bound has the factor of |�|1;

• while Wegner’s method requires the ensemble of random variables gener-ating (or controlling) potential in a volume of cardinality |�| to have |�|degrees of freedom, correlated or not, the above approach works fine evenfor ensembles with one degree of freedom (the parameter m above mayequal 1);

• in applications to the MSA, any upper bound of the form Const |�|Nε, oreven e|�|β ε, with β ∈ (0, 1) would be sufficient to make the MSA inductivescheme work;

• although the above version of Wegner bound does not allow to establishthe existence of DoS even in models where the latter does exist (as canbe shown with Wegner bound), the existence of the (limiting) DoS is notquite helpful per se for the finite-volume MSA, where upper bounds forthe probability of high concentration of eigenvalues are vital;

• in applications to multi-particle localization problems with a short-range(or decaying) interaction between quantum particles, the existence of DoSfor external potentials with regular marginal distributions can be proved bydifferent methods. For example, Klopp and Zenk (2003, preprint) provedthat the IDS for a multi-particle quantum system in R

d with a decayingparticle interaction is the same as for the model without interaction. Thisresult, quite natural from a physical point of view, was proved with thehelp of Helffer–Sjöstrand formula for almost analytic extensions. A similarresult can be proved in a simpler way for lattice systems.


3 Extension to Multi-particle Systems

Results of this section have been obtained by the author and Suhov [9].Let N > 1 and d � 1 be two positive integers and consider a random LSO

H = H(ω) which can be used, in the framework of tight-binding approxima-tion, as the Hamiltonian of a system of N quantum particles in Z

d with arandom external potential V and an interaction potential U . Specifically, letx1, . . . , xN ∈ Z

d be the positions of quantum particles in the lattice Zd, and

x = (x1, . . . , xN). Let {V(x; ω), x ∈ Zd} be a random field on Z

d describingthe external potential acting on all particles, and U: (x1, . . . , xN) �→ R be theinteraction energy of the particles. In physics, U is usually to be a symmetricfunction of its N arguments x1, . . . , xN ∈ Z

d. We will assume in this sectionthat the system in question obeys either Fermi or Bose quantum statistics, soit is convenient to assume U to be symmetric. Note, however, that the resultsof this section can be extended, with natural modifications, to more generalinteractions U . Further, U is assumed to be a finite-range interaction:

supp U ⊂ {x : max(|x j − xk| � r0)}, r0 < ∞.

Such an assumption is required in the proof of Anderson localization for multi-particle systems. However, it is irrelevant to the Wegner-type estimate we aregoing to discuss below.

Now, let H be as follows:

(H(ω) f )(x) =N∑

j=1

(�( j ) + V(x j; ω)

) + U(x),

where �( j ) is the lattice Laplacian acting on the j-th particle, i.e.

�( j ) = 11

⊗ . . . ⊗ �j

⊗ . . . ⊗ 1N

acting in the Hilbert space �2(ZNd). For any finite “box”

� = �(1) × . . . × �(N) ⊂ ZNd

one can consider the restriction, H�(ω), of H(ω) on � with Dirichlet b.c. It iseasy to see that the potential

W(x) =N∑

j=1

V(x j; ω) + U(x)

is no longer an i.i.d. random field on ZNd, even if V is i.i.d. Therefore, neither

version of the Wegner bound applies directly. But, in fact, Stollmann’s lemmadoes apply to multi-particle systems, virtually in the same way as to single-particle ones.

Lemma 3.1 Assume that r.v. V(x; ·) are i.i.d. with marginal distribution μV.Then

P { dist((H�(ω), E) � ε } � |�| · M(�) · s(μV, 2ε),

124 V. Chulaevsky

with

M(�) =N∑

j=1

card �( j ).

A reader familiar with the MSA method may notice that, in fact, the latterrequires two different kinds of Wegner-type bounds: for individual finite vol-umes � and for couples of disjoint (or, more generally, distant) finite volumes�, �′. In the conventional, single-particle MSA the two-volume bound canbe deduced (under certain conditions) from its single-volume counterpart.This is far from obvious for multi-particle (even two-particle) systems withan interaction. A detailed discussion of the multi-particle MSA scheme isbeyond the scope of this short note (for details, see [9]). Recently, Kirsch [14]proved an analog of Wegner bound for single volumes (but not for couples ofvolumes) under a more restrictive assumption of existence and boundedness ofthe marginal probability distribution of the potential of a multi-particle latticeAnderson model with interaction.

4 Extension to Correlated Random Variables

Fix a positive integer m � 1 and consider the Euclidean space Rm with

coordinates q = (q1, . . . , qm). For a given point q ∈ Rm, set q′

�= j := (q1, . . . ,

q j−1, q j+1, . . . , qm). Let μm be a probability measure on Rm with marginal dis-

tributions μm−1j (q′

�= j) of order m − 1, and conditional distributions μ1j(q j | q′

�= j)

of order 1; here j = 1, . . . , m. In the case where the measure μ1j(q j | q′

�= j) isabsolutely continuous, we denote by p(q j|q′

�= j) its density with respect to theLebesgue measure dq j. The measure μm (unlike the measure μJ in Section 2)is no longer assumed to be a product measure (we emphasize this fact bychanging notation). For every ε > 0, define the following quantities, measuringin different ways continuity properties of μm:

C1(μm, ε) = max

jsupa∈R

∫dμm−1(q′

�= j)

∫ a+ε

adμ(q j|q′

�= j), (4)

C2(μm, ε) = max

jess sup

q′�= j∈Rm−1

supa∈R

∫ a+ε

adμ(q j|q′

�= j), (5)

and

C3(μm) = ess sup

q′�= j∈Rm−1, q j∈R

p(q j|q′�= j). (6)

Since μm is a finite (even probability) measure, the quantities C1(μm, ε) and

C2(μm, ε) are always finite, and bounded by 1, while C3(μ

m, ε) may be infinite(in which case, naturally, it is useless). If the density p(q j|q′

�= j) exists, we have

dμ(q j|q′�= j) = p(q j|q′

�= j)dq j


and if C3(μm, ε) < ∞, we can write

C2(μm, ε) = max

jess sup

q′�= j∈Rm−1

supa∈R

∫ a+ε

ap(q j|q′

�= j)dq j

� maxj

ess supq′

�= j∈Rm−1

supa∈R

∫ a+ε

aC3(μ

m) dq j � C3(μm) ε.

Also, it is easy to see that

C1(μm, ε) = max

jsupa∈R

∫dμm−1(q′

�= j)

∫ a+ε

adμ(q j|q′

�= j)

�∫

C2(μm, ε) dμm−1(q′

�= j) = C2(μm, ε),

since μm−1 is a probability measure. Therefore,

C1(μm, ε) � C2(μ

m, ε) � C3(μm, ε) ε. (7)

Remark 4.1 In applications to localization problems, the aforementioned con-tinuity moduli C1(μ

m, ε), C2(μm, ε) need to decay not too slowly as ε → 0.

A power decay of order O(εβ) with β > 0 is certainly sufficient, but it can beessentially relaxed. For example, it suffices to have an upper bound of the form

C1

(μm, e−Lβ

)� Const · L−B,

uniformly for all sufficiently large L > 0 with some (arbitrarily small) β > 0and with B > 0 which should sufficiently big, depending on the specific spectralproblem.

Using notations of the previous section, one can formulate the followinggeneralization of Stollmann’s lemma.

Lemma 4.1 Let J be a finite set with |J| = m, so that we can identify RJ with R

m.Let : R

J → R be a DM function and μm a probability measure on Rm ∼ R

J

with C1(μm, ε) < ∞. Then for any interval I ⊂ R of length |I| = ε > 0, we have

μm{ q: (q) ∈ I } � m · C1(μ, ε).

Proof We proceed as in the proof of Stollmann’s lemma and introduce in Rm

the sets A = { q: (q) � a } and Aεj , j = 0, . . . , m. Here, again, we have

{ q: (q) ∈ I } =⊂ Aεm \ A

and

μm{ q: (q) ∈ I } �m∑

j=1

μm(

Aεj \ Aε

j−1

).

126 V. Chulaevsky

For q′�=1 ∈ R

m−1, we set

I1(q′�=1) = {

q1 ∈ R : (q1, q′�=1) ∈ Aε

1 \ A}.

Furthermore, we come to the following upper bound which generalizes (3):

μm (Aε

1 \ A) =

∫dμm−1(q′)

∫

I1

dμ(q1|q′) � C1(μ, ε). (8)

Similarly, we obtain for j = 2, . . . , m

μm(

Aεj \ Aε

j−1

)� C1(μ, ε),

yielding

μm{ q: (q) ∈ I } �m∑

j=1

μm(

Aεj \ Aε

j−1

)� m · C1(μ, ε). �

5 Application to Gibbs Fields with Continuous Spin

There exists a large variety of correlated random lattice fields for which thehypothesis of Lemma 4.2 can be easily verified. For example, conditionaldistributions of Gibbs fields are given explicitly in terms of their respectiveinteraction potentials.

Gaussian fields can also be considered as a particular class of Gibbsian fields.The reader can find in the article by von Dreifus and Klein [20] a detaileddiscussion of such models and a proof of Wegner estimate for homogeneousnon-deterministic Gaussian potentials. It suffices to notice, actually, that forsuch potentials the conditional density of a single-site value V(x0; ·) given allother values {V(y; ·), y �= x0} exists and is bounded. Therefore, Lemma 4.2applies, but so does the traditional Wegner’s method.

Anderson localization for Gibbsian potentials on the lattice was provedby von Dreifus and Klein [20] under a rather strong assumption of completeanalyticity in the sense of Dobrushin and Shlosman (see, e.g., [11]) of theGibbsian field V(x; ω), with continuous spins, generating the potential of therespective LSO. We show in this section how Lemma 4.2 allows to relaxthe complete analyticity hypothesis to a quite general, single-site conditionon the Hamiltonian generating the respective Gibbs state for a model ofthe classical statistical mechanics with continuous spins. Indeed, originalDobrushin–Shlosman techniques are adapted to models with a finite number ofspin values. Bourgain and Kenig [2] considered continuous Anderson modelswhere amplitudes determining the random potential take two values. Recently,Aizenman et al. [1] extended this result to a quite general case, includingrandom variables taking with positive probability any finite number n > 1 ofvalues. Unfortunately, no analog of such techniques is known so far for thelattice models.


It is worth mentioning that the condition described below can hold insome models where the marginal density does not exist, in which case moretraditional methods do not apply.

Consider a lattice Gibbs field Sx(ω) with bounded continuous spins,

S: � × Zd → S = [a, b ] ⊂ R

generated by a short-range, bounded, two-body interaction potential u•(·, ·).The spin space is assumed to be equipped with a measure dS which may,in principle, be singular with respect to Lebesgue measure on [a, b ] (moregeneral spin spaces S and measures dS can also be considered). In other words,consider the formal Hamiltonian

H(S) =∑

x∈Zd

h(Sx) +∑

x∈Zd

∑

|y−x|�R

u|x−y|(Sx, Sy),

where h: S → R is the self-energy of a given spin. Assume that the interactionpotentials u|x−y|(Sx, Sy) vanish for |x − y| > R and are uniformly bounded:

maxl�R

supS,S′∈S

|ul(S, S′)| < ∞.

Then for any lattice point x and any configuration S′ = S′�=x of spins outside {x},

the single-site conditional distribution of Sx given the external configuration S′admits a bounded density with respect to measure dS, namely,

p(Sx | S′�=x) = e−βU(Sx|S′

)

(β, S′)= e−βU(Sx|S′

)∫S e−βU(S′′|S′

) dS′′

with

U(Sx|S′) :=∑

y: |y−x|�R

u|x−y|(Sx, S′y)

satisfying the upper bound

|U(Sx|S′)| � (2R + 1)d supS,S′′∈S

|ul(S, S′′)| < ∞.

A similar property is valid for sufficiently rapidly decaying long-range interac-tion potentials, for example, under the condition

supS,S′′∈S

|u|y|(S, S′′)| � Const|y|d+1+δ

, δ > 0. (9)

as well as for more general, but still uniformly summable many-body interac-tions. Below we give one simple example of application of Wegner–Stollmann-type bound to such random potentials.

Lemma 5.1 Let � ⊂ Zd be a finite subset of the lattice, �′ ⊂ Z

d \ � any subsetdisjoint with � (�′ may be empty), and let Sx(ω) be a Gibbs field in � withcontinuous spins S ∈ S = [a, b ] generated by a two-body interaction potentialul(S, S′′) satisfying condition (9), with any b.c. on Z

d \ �. Consider a LSO

128 V. Chulaevsky

H� with the random potential V(x, ω) = Sx(ω). Then for any interval I ⊂ R

of length ε > 0, we have

P{(H�) ∩ I �= ∅ | V(y, ·), y ∈ �′ } � C(V) |�|2 ε, C(V) < ∞.

In the case of unbounded spins and/or interaction potentials, the uniformboundedness of conditional single-spin distributions does not necessarily hold,since the energy of interaction of a given spin S0 with the external configurationS′ may be arbitrarily large (depending on a particular form of interaction) andeven infinite, if S′

y → ∞ too fast. In such situations, our general condition(4) might still apply, provided that rapidly growing configurations S′ havesufficiently small probability, so that the outer integral in the r.h.s. of (4)converges.

6 Conclusion

Wegner-type bounds of the IDS in finite volumes are a key ingredient ofthe MSA of spectra of random Schrödinger (and some other) operators. Theproposed simple extension of Stollmann’s lemma shows that a very generalassumption on correlated random fields generating the potential rules out anabnormal accumulation of eigenvalues in finite volumes. This extension appliesalso to multi-particle systems with interaction.

Acknowledgements I would like to thank Senya Shlosman for numerous and very fruitfuldiscussions of Dobrushin–Shlosman techniques. I also thank Lana Jitomirskaya and Abel Kleinand the University of California at Irvine for their warm hospitality during my stay at UCI andstimulating discussions. I thank Anne Boutet de Monvel and Peter Stollmann for many fruitfuldiscussions of Wegner-type estimates.

References

1. Aizenman, A., Germinet, F., Klein, A., Warzel, S.: On Bernoulli decompositions for randomvariables, conncenration bounds, and spectral localization. arXiv:0707.0095 (2007)

2. Bourgain, J., Kenig, C.: On localization in continuous Anderson–Bernoulli model in highredimensions. Invent. Math 161, 389–426 (2005)

3. Bovier, A., Campanino, M., Klein, A., Perez, F.: Smoothness of the density of states in theAnderson model at high disorder. Comm. Math. Phys. 114, 439–461 (1988)

4. Campanino, M., Klein, A.: A supersymmetric transfer matrix and differentiability of thedensity of states in the one-dimensional Anderson model. Comm. Math. Phys. 104, 227–241(1986).

5. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators. Birkhäuser,Boston (1990)

6. Craig, W., Simon, B.: Log Hölder continuity of the integrated density of states for stochasticJacobi matrices. Comm. Math. Phys. 90, 207–218 (1983)

7. Chulaevsky, V.: A simple extension of Stollmann’s lemma to correlated potentials. Universitéde Reims. arXiv:0705:2873 May (2007)

8. Chulaevsky, V., Suhov, Y.: Anderson localisation for an interacting two-particle quantumsystem on Z. Université de Reims. arXiv:0705.0657 May (2007)

9. Chulaevsky, V., Suhov, Y.: Wegner bounds for a two-particle tight binding model.arXiv:0708:2056. Comm. Math. Phys. (2008, in press)


10. Constantinescu, F., Fröhlich, J., Spencer, T.: Analyticity of the density of states and replicamethod for random Schrödinger operators on a lattice. J. Stat. Phys. 34, 571–596 (1983)

11. Dobrushin, R.L., Shlosman, S.: Completely analytical interactions: constructive description.J. Statist. Phys. 46, 983–1014 (1987)

12. Fröhlich, J., Martinelli, F., Scoppola, E., Spencer, T.: A constructive proof of localization inAnderson tight binding model. Comm. Math. Phys. 101, 21–46 (1985)

13. Fröhlich, J., Spencer, T.: Absence of diffusion in the Anderson tight binding model for largedisorder or low energy. Comm. Math. Phys. 88, 151–184 (1983)

14. Kirsch, W.: A Wegner estimate for multi-particle random Hamiltonians. arXiv:0704:2664April (2007)

15. Pastur, L.A.: Spectral properties of disordered systems in one-body approximation. Comm.Math. Phys. 75, 179 (1980)

16. Pastur, L.A., Figotin, A.L.: Spectra of Random and Almost Periodic Operators. Springer,Berlin (1992)

17. Stollmann, P.: Wegner estimates and localization for continuous Anderson models with somesingular distributions. Arch. Math. 75, 307–311 (2000)

18. Stollmann, P.: Caught by Disorder. Birkhäuser, Boston (2001)19. Von Dreifus, H., Klein, A.: A new proof of localization in the Anderson tight binding model.

Comm. Math. Phys. 124, 285–299 (1989)20. Von Dreifus, H., Klein, A.: Localization for Schrödinger operators with correlated potentials.

Comm. Math. Phys. 140, 133–147 (1991)21. Wegner, F.: Bounds on the density of states in disordered systems. Z. Phys. B. Condens. Matter

44, 9–15 (1981)


The Two-Spectra Inverse Problem for Semi-infiniteJacobi Matrices in The Limit-Circle Case

Luis O. Silva · Ricardo Weder

Received: 2 August 2007 / Accepted: 28 May 2008 /Published online: 12 July 2008© Springer Science + Business Media B.V. 2008

Abstract We present a technique for reconstructing a semi-infinite Jacobioperator in the limit circle case from the spectra of two different self-adjointextensions. Moreover, we give necessary and sufficient conditions for two realsequences to be the spectra of two different self-adjoint extensions of a Jacobioperator in the limit circle case.

Keywords Jacobi matrices · Two-spectra inverse problem · Limit circle case

Mathematics Subject Classifications (2000) 47B36 · 49N45 · 81Q10 · 47A75 ·47B37 · 47B39

Research partially supported by CONACYT under Project P42553F.

L. O. Silva partially supported by PAPIIT-UNAM through grant IN-111906.

Ricardo Weder is a fellow of Sistema Nacional de Investigadores.

L. O. Silva · R. Weder (B)Departamento de Métodos Matemáticos y Numéricos,Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas,Universidad Nacional Autónoma de México, México, D. F. C.P. 04510, Mexicoe-mail: [email protected]

L. O. Silvae-mail: [email protected]

132 L.O. Silva, R. Weder

1 Introduction

In the Hilbert space l2(N), consider the operator J whose matrix representationwith respect to the canonical basis in l2(N) is the semi-infinite Jacobi matrix

⎛⎜⎜⎜⎜⎜⎜⎜⎝

q1 b1 0 0 · · ·b1 q2 b2 0 · · ·0 b2 q3 b3

0 0 b3 q4. . .

......

. . .. . .

⎞⎟⎟⎟⎟⎟⎟⎟⎠

, (1.1)

where bn > 0 and qn ∈ R for n ∈ N. This operator is densely defined in l2(N)

and J ⊂ J∗ (see Section 2 for details on how J is defined).It is well known that J can have either (1, 1) or (0, 0) as its deficiency indices

[2, Sec. 1.2 Chap. 4], [31, Cor. 2.9]. By our definition (see Section 2), J is closed,so the case (0, 0) corresponds to J = J∗, while (1, 1) implies that J is a non-trivial restriction of J∗. The latter operator is always defined on the maximaldomain in which the action of the matrix (1.1) makes sense [3, Sec. 47].

Throughout this work we assume that J has deficiency indices (1, 1). Jacobioperators of this kind are referred as being in the limit circle case and themoment problem associated with the corresponding Jacobi matrix is said tobe indeterminate [2, 31]. In the limit circle case, all self-adjoint extensions ofa Jacobi operator have discrete spectrum [31, Thm. 4.11]. The set of all self-adjoint extensions of a Jacobi operator can be characterized as a one parameterfamily of operators (see Section 2).

The main results of the present work are Theorem 1 in Section 3 andTheorem 2 in Section 4. In Theorem 1 we show that a Jacobi matrix can berecovered uniquely from the spectra of two different self-adjoint extensions ofthe Jacobi operator J corresponding to that matrix. Moreover, these spectraalso determine the parameters that define the self-adjoint extensions of J forwhich they are the spectra. The proof of Theorem 1 is constructive and it givesa method for the unique reconstruction. The uniqueness of this reconstructionin a more restricted setting has been announced in [13] without proof. InTheorem 2 we give necessary and sufficient conditions for two sequences tobe the spectra of two self-adjoint extensions of a Jacobi operator in the limitcircle case. This is a complete characterization of the spectral data for the two-spectra inverse problem of a Jacobi operator in the limit circle case.

In two spectra inverse problems, one may reconstruct a certain self-adjointoperator from the spectra of two different rank-one self-adjoint perturbationsof the operator to be reconstructed. This is the case of recovering the potentialof a Schrödinger differential expression in L2(0, ∞), being regular at theorigin and limit point at ∞, from the spectra of two operators defined by thedifferential expression with two different self-adjoint boundary conditions atthe origin [4, 5, 9, 15, 17, 21, 24, 27, 28]. Necessary and sufficient conditions forthis inverse problem are found in [27]. Characterization of spectral data of a

The two-spectra inverse problem 133

related inverse problem was obtained in [10]. The inverse problem consistingin recovering a Jacobi matrix from the spectra of two rank-one self-adjointperturbations, was studied in [8, 12, 19, 20, 32, 34]. A complete characterizationof the spectral data for this two-spectra inverse problem is given in [29].

In the formulation of the inverse problem studied in the present work, theaim is to recover a symmetric non-self-adjoint operator from the spectra ofits self-adjoint extensions, as well as the parameters that characterize the self-adjoint extensions. There are results for this setting of the two spectra inverseproblem, for instance in [14, 23] for Sturm–Liouville operators, and in [13] forJacobi matrices.

It is well known that self-adjoint extensions of symmetric operators withdeficiency indices (1, 1) can be treated within the rank-one perturbation theory(cf. [6, Sec. 1.1–1.3] and, in particular, [6, Thm. 1.3.3]). Thus, both settingsmay be regarded as particular cases of a general two-spectra inverse problem.A consideration similar to this is behind the treatment of inverse problemsin [11]. For Jacobi operators, however, the type of rank-one perturbations inthe referred formulations of the inverse spectral problem are different [6].Indeed, in the setting studied in [29], one has the so-called bounded rank-one perturbations [6, Sec. 1.1]. This means that all the family of rank-oneperturbations share the same domain. In contrast the present work dealswith singular rank-one perturbations [6, Sec. 1.3], meaning that every elementof the family of rank-one perturbations has different domain. Note that fordifferential operators both settings involve a family of singular rank-oneperturbations.

The paper is organized as follows. In Section 2, we introduce Jacobi oper-ators, in particular the class whose corresponding Jacobi matrix is in the limitcircle case. Here we also present some preliminary results and lay down somenotation used throughout the text. Section 3 contains the uniqueness resulton the determination of a Jacobi matrix by the spectra of two self-adjointextensions. The proof of this assertion yields a reconstruction algorithm.Finally in Section 4, we give a complete characterization of the spectral datafor the two spectra inverse problem studied here.

2 Preliminaries

Let l f in(N) be the linear space of sequences with a finite number of non-zeroelements. In the Hilbert space l2(N), consider the operator J defined for everyf = { fk}∞k=1 in l f in(N) by means of the recurrence relation

(J f )k := bk−1 fk−1 + qk fk + bk fk+1 , k ∈ N \ {1} , (2.1)

(J f )1 := q1 f1 + b1 f2 , (2.2)

where, for n ∈ N, bn is positive and qn is real. Clearly, J is symmetric since itis densely defined and Hermitian due to (2.1) and (2.2). Thus J is closable andhenceforth we shall consider the closure of J and denote it by the same letter.


We have defined the operator J so that the semi-infinite Jacobi matrix(1.1) is its matrix representation with respect to the canonical basis {en}∞n=1in l2(N) (see [3, Sec. 47] for the definition of the matrix representation of anunbounded symmetric operator). Indeed, J is the minimal closed symmetricoperator satisfying

(Jen, en) = qn, (Jen, en+1) = (Jen+1, en) = bn ,

(Jen, en+k) = (Jen+k, en) = 0 ,n ∈ N, k ∈ N \ {1} .

We shall refer to J as the Jacobi operator and to (1.1) as its associated matrix.The spectral analysis of J may be carried out by studying the following

second order difference system

bn−1 fn−1 + qn fn + bn fn+1 = ζ fn , n > 1 , ζ ∈ C , (2.3)

with the “boundary condition”

q1 f1 + b1 f2 = ζ f1 . (2.4)

If one sets f1 = 1, then f2 is completely determined by (2.4). Having f1 and f2,equation (2.3) gives all the other elements of a sequence { fn}∞n=1 that formallysatisfies (2.3) and (2.4). Clearly, fn is a polynomial of ζ of degree n − 1, so wedenote fn =: Pn−1(ζ ). The polynomials Pn(ζ ), n = 0, 1, 2, . . . , are referred toas the polynomials of the first kind associated with the matrix (1.1) [2, Sec. 2.1Chap. 1].

The sequence P(ζ ) := {Pk−1(ζ )}∞k=1 is not in l f in(N), but it may happen that

∞∑k=0

|Pk(ζ )|2 < ∞ , (2.5)

in which case P(ζ ) ∈ Ker(J∗ − ζ I ).The polynomials of the second kind Q(ζ ) := {Qk−1(ζ )}∞k=1 associated with

the matrix (1.1) are defined as the solutions of

bn−1 fn−1 + qn fn + bn fn+1 = ζ fn , n ∈ N \ {1} ,

under the assumption that f1 = 0 and f2 = b1−1. Then

Qn−1(ζ ) := fn , ∀n ∈ N .

Qn(ζ ) is a polynomial of degree n − 1.As pointed out in the introduction, J has either deficiency indices (1, 1)

or (0, 0) [2, Sec. 1.2 Chap. 4] and [31, Cor. 2.9]. These cases correspond to thelimit circle and limit point case, respectively. In terms of the polynomials of thefirst kind, J has deficiency indices (0, 0) if for one ζ ∈ C \ R the series in (2.5)diverges. In the limit circle case (2.5) holds for every ζ ∈ C [2, Thm. 1.3.2], [31,Thm. 3] and, therefore, P(ζ ) is always in Ker(J∗ − ζ I ). Another peculiarityof the limit circle case is that every self-adjoint extension of J has purelydiscrete spectrum [31, Thm. 4.11]. Moreover, the resolvent of every self-adjointextension is a Hilbert-Schmidt operator [33, Lem. 2.19].


In what follows we always consider J to have deficiency indices (1, 1). Thebehavior of the polynomials of the first kind determines this class. Thereare various criteria for establishing whether a Jacobi operator is symmetricbut non-self-adjoint. These criteria may be given in terms of the momentsassociated with the matrix, for instance the criterion [31, Prop. 1.7] due toKrein. A criterion in terms of the matrix entries is the following result whichbelongs to Berezans′kiı [2, Chap. 1], [7, Thm. 1.5 Chap. 7].

Proposition 1 Suppose that supn∈N|qn| < ∞ and that

∑∞n=1

1b n

< ∞. If there isN ∈ N such that for n > N

bn−1bn+1 ≤ bn2 ,

then the Jacobi operator whose associated matrix is (1.1) is in the limitcircle case.

Jacobi operators in the limit circle case may be used to model physicalprocesses. For instance Krein’s mechanical interpretation of Stieltjes contin-ued fractions [22], in which one has a string carrying point masses with a certaindistribution along the string, is modeled by an eigenvalue equation of a Jacobioperator [2, Appendix]. There are criteria in terms of the point masses andtheir distribution [2, Thm. 0.4 Thm. 0.5 Appendix] for the corresponding Jacobioperator to be in the limit circle case.

In this work, all self-adjoint extensions of J are assumed to be restrictionsof J∗. When dealing with all self-adjoint extensions of J, including those whichimply an extension of the original Hilbert space, the self-adjoint restrictions ofJ∗ are called von Neumann self-adjoint extensions of J (cf. [3, Appendix I],[31, Sec. 6]).

There is also a well known result for J in the limit circle case, namely, thatJ is simple [2, Thm. 4.2.4]. In its turn this imply that the eigenvalues of anyself-adjoint extension of J have multiplicity one [3, Thm.3 Sec. 81].

Let us now introduce a convenient way of parametrizing the self-adjointextensions of J in the symmetric non-self-adjoint case. We first define theWronskian associated with J for any pair of sequences ϕ = {ϕk}∞k=1 and ψ ={ψk}∞k=1 in l2(N) as follows

Wk(ϕ, ψ) := bk(ϕkψk+1 − ψkϕk+1) , k ∈ N .

Now, consider the sequences v(τ) = {vk(τ )}∞k=1 such that, for k ∈ N,

vk(τ ) := Pk−1(0) + τ Qk−1(0) , τ ∈ R , (2.6)

andvk(∞) := Qk−1(0) . (2.7)

All the self-adjoint extensions J(τ ) of the symmetric non-self-adjointoperator J are restrictions of J∗ to the set [33, Lem. 2.20]

Dτ :={

f ={ fk}∞k=1 ∈ dom(J∗) : limn→∞ Wn

(v(τ), f

) = 0}

, τ ∈R∪{∞} . (2.8)


Different values of τ imply different self-adjoint extensions, so J(τ ) is a self-adjoint extension of J uniquely determined by τ [33, Lem. 2.20]. Observe thatthe domains Dτ are defined by a boundary condition at infinity given by τ . Wealso remark that given two sequences ϕ and ψ in dom(J∗) the following limitalways exists [33, Sec. 2.6]

limn→∞ Wn(ϕ, ψ) =: W∞(ϕ, ψ) .

It follows from [31, Thm. 3] that, in the limit circle case, P(ζ ) and Q(ζ ) are indom(J∗) for every ζ ∈ C.

From what has just been said, one can consider the functions (see also[2, Sec. 2.4 Chap. 1, Sec. 4.2 Chap. 2])

W∞(P(0), P(ζ )) =: D(ζ ) ,

W∞(Q(0), P(ζ )) =: B(ζ ) . (2.9)

The notation for these limits has not been chosen arbitrarily; they are theelements of the second row of the Nevanlinna matrix associated with thematrix (1.1) and they are usually denoted by these letters [2, Sec. 4.2 Chap. 2],[31, Eq. 4.17].

It is well known that the functions D(ζ ) and B(ζ ) are entire of at mostminimal type of order one [2, Thm. 2.4.3], [31, Thm. 4.8], that is, for each ε > 0there exist constants C1(ε), C2(ε) such that

|D(ζ )| � C1(ε)eε|ζ | , |B(ζ )| � C2(ε)eε|ζ | .

If P(ζ ) is in Dτ the following holds

0 = W∞(v(τ ), P(ζ )) ={

D(ζ ) + τ B(ζ ) if τ ∈ R

B(ζ ) if τ = ∞ .

Thus, the zeros of the function

Rτ (ζ ) :={

D(ζ ) + τ B(ζ ) if τ ∈ R

B(ζ ) if τ = ∞ (2.10)

constitute the spectrum of the self-adjoint extension J(τ ) of J.A Jacobi matrix of the form (1.1) determines, in a unique way, the se-

quence P(t) = {Pn−1(t)}∞n=1, t ∈ R. This sequence is orthonormal in any spaceL2(R, dρ), where ρ is a solution of the moment problem associated withthe Jacobi matrix (1.1) [2, Sec. 2.1 Chap. 2]. The elements of the sequence{Pn−1(t)}∞n=1 form a basis in L2(R, dρ) if ρ is an N-extremal solution of themoment problem [2, Def. 2.3.3] or, in other words, if ρ can be written as

ρ(t) = ⟨E(t)e1, e1

⟩, t ∈ R , (2.11)

where E(t) is the spectral resolution of the identity for some von Neumann self-adjoint extension of the Jacobi operator J associated with (1.1) [2, Thm. 2.3.3,Thm. 4.1.4].


Let ρ be given by (2.11), then we can consider the linear isometric operatorU which maps the canonical basis {en}∞n=1 in l2(N) into the orthonormal basis{Pn(t)}∞n=0 in L2(R, dρ) as follows

Uen = Pn−1 , n ∈ N . (2.12)

By linearity, one extends U to the span of {en}∞n=1 and by continuity, toall l2(N). Clearly, the range of U is all L2(R, dρ). The Jacobi operator J givenby the matrix (1.1) is transformed by U into the operator of multiplicationby the independent variable in L2(R, dρ) if J = J∗, and into a symmetric re-striction of the operator of multiplication if J �= J∗. Following the terminologyused in [11], we call the operator U JU−1 in L2(R, dρ) the canonicalrepresentation of J.

By virtue of the discreteness of σ(J(τ )) in the limit circle case (here and inthe sequel, σ(A) stands for the spectrum of operator A), the function ρτ givenby (2.11), with E(t) being the resolution of the identity of J(τ ), can be writtenas follows

ρτ (t) =∑λk�t

a(λk)−1 , λk ∈ σ(J(τ )) ,

where the positive constant a(λk) is the so-called normalizing constant of J(τ )

corresponding to λk. In the limit circle case it is easy to obtain the followingformula for the normalizing constants [2, Sec. 4.1 Chap 3], [31, Thm. 4.11]

a(λk) = ‖P(λk)‖2l2(N) , λk ∈ σ(J(τ )) . (2.13)

Formula (2.13), which gives the jump of the spectral function at λk, also holdstrue in the limit point case, when λk is an eigenvalue of J [7, Thm. 1.17 Chap. 7].

It turns out that the spectral function ρτ uniquely determines J(τ ). Indeed,there are two ways of recovering the matrix from the spectral function.One method, developed in [16] (see also [32]), makes use of the asymptoticbehaviour of the Weyl m-function

mτ (ζ ) :=∫

R

ρτ (t)t − ζ

and the Ricatti equation [16, Eq. 2.15], [32, Eq. 2.23],

b 2nm(n)

τ (ζ ) = qn − ζ − 1

m(n−1)τ (ζ )

, n ∈ N , (2.14)

where m(n)τ (ζ ) is the Weyl m-function of the Jacobi operator associated with

the matrix (1.1) with the first n columns and n rows removed.The other method for the reconstruction of the matrix is more straight-

forward (see [7, Sec. 1.5 Chap. 7 and, particularly, Thm. 1.11]). The starting


point is the sequence {tk}∞k=0, t ∈ R. From what we discussed above, all theelements of the sequence {tk}∞k=0 are in L2(R, dρτ ) and one can apply, inthis Hilbert space, the Gram-Schmidt procedure of orthonormalization to thesequence {tk}∞k=0. One, thus, obtains a sequence of polynomials {Pk(t)}∞k=0normalized and orthogonal in L2(R, dρτ ). These polynomials satisfy a threeterm recurrence equation [7, Sec. 1.5 Chap. 7], [31, Sec. 1]

tPk−1(t)=bk−1 Pk−2(t) + qk Pk−1(t) + bk Pk(t) , k ∈ N \ {1} , (2.15)

tP0(t)=q1 P0(t) + b1 P1(t) , (2.16)

where all the coefficients bk (k ∈ N) turn out to be positive and qk (k ∈ N) arereal numbers. The system (2.15) and (2.16) defines a matrix which is the matrixrepresentation of J.

After obtaining the matrix associated with J, if it turns out to be non-self-adjoint, one can easily obtain the boundary condition at infinity which definesthe domain of J(τ ). The recipe is based on the fact that the spectra of differentself-adjoint extensions are disjoint [2, Sec. 2.4 Chap. 4]. Take an eigenvalue,λ, of J(τ ), i. e., λ is a point of discontinuity of ρτ or a pole of mτ . Since thecorresponding eigenvector P(λ) = {Pk−1(λ)}∞k=1 is in dom(J(τ )), it must be that

W∞(v(τ), P(λ)

) = 0 .

This implies that either W∞(Q(0), P(λ)

) = 0, which means that τ = ∞, or

τ = − W∞(P(0), P(λ)

)

W∞(Q(0), P(λ)

) .

Notation We conclude this section with a remark on the notation. Theelements of the unbounded set σ(J(τ )), τ ∈ R ∪ ∞, may be enumerated indifferent ways. Let σ(J(τ )) = {λk}k∈K, where K is a countable set throughwhich the subscript k runs. If σ(J(τ )) is either bounded from above or below,one may take K = N. If σ(J(τ )) is unbounded below and above, one may setK = Z. Of course, other choices of K are possible. Since the particular choiceof K is not important in our formulae, we shall drop K from the notation andsimple write {λk}k. All our formulae will be written so that they are independentof the way the elements of a sequence are enumerated, so our convention fordenoting sequences should not lead to misunderstanding. Similarly, we write∑

k yk instead of∑

k∈K yk, and the convergence of the series to a number cmeans that for any sequence of sets {K j}∞j=1, with K j ⊂ K j+1 ⊂ K, such that⋃∞

j=1 K j = K, the sequence {∑k∈K jyk}∞j=1 tends to c whenever j → ∞.

3 Unique Reconstruction of the Matrix

In this section we show that, given the spectra of two different self-adjointextensions J(τ1), J(τ2) of the Jacobi operator J in the limit circle case, one can


always recover the matrix, being the matrix representation of J with respectto the canonical basis in l2(N), and the two parameters τ1, τ2 that define theself-adjoint extensions. It has already been announced [13, Thm. 1] that, whenτ1, τ2 ∈ R and τ1 �= τ2, the spectra σ(J(τ1)) and σ(J(τ2)) uniquely determinethe matrix of J and the numbers τ1 and τ2. A similar result, but in a moregeneral setting can be found in [11, Thm. 7].

Consider the following expression which follows from the Christoffel–Darboux formula [2, Eq. 1.17]:

n−1∑k=0

P2k(ζ ) = bn

(Pn−1(ζ )P′

n(ζ ) − Pn(ζ )P′n−1(ζ )

) = Wn(P(ζ ), P′(ζ )) .

It is easy to verify, taking into account the analogue of the Liouville–Ostrogradskii formula [2, Eq. 1.15], that

Wn(P(ζ ), P′(ζ )) = Wn(P(0), P(ζ ))Wn(Q(0), P′(ζ ))

−Wn(Q(0), P(ζ ))Wn(P(0), P′(ζ )) .

Thus,∞∑

k=0

P2k(ζ ) = W∞(P(ζ ), P′(ζ ))

= D(ζ )B′(ζ ) − B(ζ )D′(ζ ) .

Indeed, due to the uniform convergence of the limits in (2.9) [2, Sec. 4.2Chap. 2], the following is valid

B′(ζ ) = W∞(Q(0), P′(ζ ))

D′(ζ ) = W∞(P(0), P′(ζ )) .

Now, a straightforward computation yields (τ1, τ2 ∈ R, τ1 �= τ2)

Rτ1(ζ )R′τ2(ζ ) − R′

τ1(ζ )Rτ2(ζ ) = (τ2 − τ1)

[D(ζ )B′(ζ ) − B(ζ )D′(ζ )

].

On the other hand one clearly has

Rτ1(ζ )R′∞(ζ ) − R′

τ1(ζ )R∞(ζ ) = D(ζ )B′(ζ ) − B(ζ )D′(ζ ) , τ1 ∈ R .

Hence,

a(ζ ) :=∞∑

k=0

P2k(ζ )=

⎧⎪⎨⎪⎩

Rτ1(ζ )R′τ2(ζ ) − R′

τ1(ζ )Rτ2(ζ )

τ2 − τ1τ1 �= τ2 , τ1, τ2 ∈ R

Rτ1(ζ )R′∞(ζ ) − R′τ1(ζ )R∞(ζ ) τ1 ∈ R .

(3.1)

It follows from (2.13) that the values of the function a(ζ ) evaluated at thepoints of the spectrum of some self-adjoint extension of J are the correspond-ing normalizing constants of that extension.


The analogue of (3.1) with τ1 �= τ2 and τ1, τ2 ∈ R, for the Schrödingeroperator in L2(0, ∞) being in the limit circle case is [14, Eq. 1.20]. Formula [14,Eq. 1.20] plays a central rôle in proving the unique reconstruction theorem forthat operator [14, Thm. 1.1]. The discrete counterpart of [14, Thm. 1.1] is [13,Thm. 1]. It is worth mentioning that the reconstruction technique we presentbelow is also based on (3.1).

It is well known that the spectra of any two different self-adjoint extensionsof J are disjoint [2, Sec. 2.4 Chap. 4]. One can easily conclude this from (3.1).Moreover, the following assertion holds true.

Proposition 2 The eigenvalues of two different self-adjoint extensions of aJacobi operator interlace, that is, there is only one eigenvalue of a self-adjointextension between two eigenvalues of any other self-adjoint extension.

Remark 1 One may arrive at this assertion via rank-one perturbation theory,in particular by recurring to the Aronzajn–Krein formula [30, Eq. 1.13].Nonetheless, we provide below a simple proof to illustrate the use of (3.1). Theproof of this statement for regular simple symmetric operators can be found in[18, Prop. 3.4 Chap. 1].

Proof The proof of this assertion follows from the expression (3.1). It is similarto the proof of [2, Thm. 1.2.2].

Note that (2.10) implies that the entire function Rτ (ζ ), τ ∈ R ∪ {∞}, isreal, i. e., it takes real values when evaluated on the real line. Let λk < λk+1

be two neighboring eigenvalues of the self-adjoint extension J(τ2) of J, withτ2 ∈ R ∪ {∞}. So λk, λk+1 are zeros of Rτ2 and by (3.1) these zeros are simple.Since R′

τ2(λk) and R′

τ2(λk+1) have different signs, it follows from (3.1) that

Rτ1(λk) and Rτ1(λk+1) (τ1 ∈ R ∪ {∞}, τ1 �= τ2) have also opposite signs. Fromthe continuity of Rτ1 on the interval [λk, λk+1], there is at least one zero ofRτ1 in (λk, λk+1). Now, suppose that in this interval there is more than onezero of Rτ1 , so one can take two neighboring zeros of Rτ1 in (λk, λk+1). Byreproducing the argumentation above with τ1 and τ2 interchanged, one obtainsthat there is at least one zero of Rτ2 somewhere in (λk, λk+1). This contradictsthe assumption that λk and λk+1 are neighbors. �

The assertion of the following proposition is a well established fact (see,for instance [23, Thm. 1]). We, nevertheless, provide the proof for the reader’sconvenience and because we introduce in it notation for later use. Note thata non-constant entire function of at most minimal type of order one musthave zeros, otherwise, by Weierstrass theorem on the representation of entirefunctions by infinite products [25, Thm. 3 Chap. 1], it would be a function of atleast normal type.

Before stating the proposition we remind the definition of convergenceexponent of a sequence of complex numbers (see [25, Sec. 4 Chap. 1]). The


convergence exponent ρ1 of a sequence {νk}k of non-zero complex numbersaccumulating only at infinity is given by

ρ1 := inf

⎧⎨⎩γ ∈ R : lim

r→∞∑

|νk|�r

1

|νk|γ < ∞⎫⎬⎭ . (3.2)

We also remark that, as it is customary, whenever we say that an infiniteproduct is convergent we mean that at most a finite number of factors maybe zero and the partial product formed by the non-vanishing factors tends to anumber different from zero [1, Sec. 2.2 Chap. 5].

Proposition 3 Let f (ζ ) be an entire function of at most minimal type oforder one with an infinite number of zeros. Let the elements of the sequence{νk}k, which accumulate only at infinity, be the non-zero roots of f , where{νk}k contains as many elements for each zero as its multiplicity. Assume thatm ∈ N ∪ {0} is the order of the zero of f at the origin. Then there exists a complexconstant C such that

f (ζ ) = Cζ m limr→∞

∏|νk|�r

(1 − ζ

νk

), (3.3)

where the limit converges uniformly on compacts of C.

Proof The convergence exponent ρ1 of the zeros of an arbitrary entire functiondoes not exceed its order [25, Thm. 6 Chap. 1]. Then, for a function of at mostminimal type of order one, ρ1 � 1. According to Hadamard’s theorem [25,Thm. 13 Chap. 1], the expansion of f in an infinite product has either the form:

f (ζ ) = ζ meaζ+b limr→∞

∏|νk|�r

G(

ζ

νk; 0

), a, b ∈ C (3.4)

if the limit

limr→∞

∑|νk|�r

1

|νk| (3.5)

converges, or

f (ζ ) = ζ mecζ+d limr→∞

∏|νk|�r

G(

ζ

νk; 1

), c, d ∈ C (3.6)

if (3.5) diverges. We have used here the Weierstrass primary factors G (fordetails see [25, Sec. 3 Chap. 1]). Let us suppose that the order is one and (3.5)diverges, then, in view of the fact that f is of minimal type, by a theorem dueto Lindelöf [25, Thm. 15 a Chap. 1], we have in particular that

limr→∞

∑|νk|�r

ν−1k = −c .


This implies the uniform convergence of the series limr→∞∑

|νk|�rζ

νkon

compacts of C. In its turn, since ρ1 = 1, this yields that

limr→∞

∏|νk|�r

(1 − ζ

νk

)

is uniformly convergent on any compact of C. Therefore,

limr→∞

∏|νk|�r

G(

ζ

νk; 1

)= e−cζ lim

r→∞∏

|νk|�r

(1 − ζ

νk

).

Thus, (3.6) can be written as (3.3).Suppose now that the limit (3.5) converges. If the order of the function is

less than one, then, by [25, Thm. 13 Chap. 1], one may write (3.4) as (3.3). If theorder of the function is one, by [25, Thm. 12, Thm. 15 b Chap. 1], one concludesagain that (3.4) can be written as (3.3) (cf. Thm 15 in the Russian version of[25] or, alternatively, [26, Lect. 5]). �

Let {λn(τ )}n be the eigenvalues of J(τ ). In view of the fact that Rτ (ζ ) is anentire function of at most minimal type of order one, by Proposition 3, one canalways write

Rτ (ζ ) = Cτ ζδτ lim

r→∞∏

0<|λk(τ )|�r

(1 − ζ

λk(τ )

), τ ∈ R ∪ {∞} , (3.7)

where Cτ ∈ R \ {0} and δτ is the Kronecker delta, i. e., δτ = 1 if τ = 0, andδτ = 0 otherwise. The limits in (3.7) converge uniformly on compacts of C.Note that when τ = 0 we have naturally excluded λk(0) = 0 from the infiniteproduct.

When writing (3.7), we have taken into account, on the one hand, thatR0(0) = 0, which follows from (2.10) and the definition of the function D,and on the other, that different self-adjoint extensions have disjoint spectra(see Section 2).

Now, let us consider the following expressions derived from the Green’sformula [2, Eqs. 1.23, 2.28]

D(ζ ) = ζ

∞∑k=0

Pk(0)Pk(ζ ) , (3.8)

B(ζ ) = −1 + ζ

∞∑k=0

Qk(0)Pk(ζ ) . (3.9)

Again we verify from (3.8) that D(0) = 0, while from (3.9) we have B(0) =−1. Therefore Rτ (0) = −τ for every τ ∈ R, and R∞(0) = −1. Thus, Cτ = −τ

provided that τ ∈ R and τ �= 0, and C∞ = −1.


To simplify the writing of some of the formulae below, let us introduceRτ (ζ ) := Rτ (ζ )

Cτ, that is,

Rτ (ζ ) := ζ δτ limr→∞

∏0<|λk(τ )|�r

(1 − ζ

λk(τ )

), τ ∈ R ∪ {∞} , (3.10)

where δτ is defined as in (3.7).Due to the uniform convergence of the expression

ddζ

⎡⎣ ∏

0<|λk(τ )|�r

(1 − ζ

λk(τ )

)⎤⎦ , as r → ∞ ,

one has

R′τ (λ j(τ ))=

⎧⎪⎪⎨⎪⎪⎩

−[λ j(τ )]δτ −1 limr→∞

∏0 <

∣∣λk(τ )∣∣ � r

k �= j

(1 − λ j(τ )

λk(τ )

), λ j(τ ) �= 0

1 λ j(τ ) = 0

(3.11)

By (3.1) and (3.11), one obtains

C0 = a(0). (3.12)

Theorem 1 Let τ1, τ2 ∈ R ∪ {∞} with τ1 �= τ2. The spectra {λk(τ1)}k, {λk(τ2)}k

of two different self-adjoint extensions J(τ1), J(τ2) of a Jacobi operator J inthe limit circle case uniquely determine the matrix associated with J, and thenumbers τ1 and τ2.

Proof For definiteness assume that τ1 �= 0, in other words that the sequence{λk(τ1)}k does not contain any zero element.

By (3.1), we have

a(λk(τ1)) = MRτ2(λk(τ1))R′τ1(λk(τ1)) , (3.13)

where

M =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

τ1τ2τ1−τ2

if τ1, τ2 ∈ R \ {0}τ2 if τ1 = ∞ , τ2 �= 0

−τ1 if τ2 = ∞−C0 if τ2 = 0

(3.14)

Now, since {a(λk(τ1))}k are the normalizing constants of J(τ1) we must have

1 =∑

k

1

a(λk(τ1))= 1

M

∑k

1

Rτ2(λk(τ1))R′τ1(λk(τ1))

.

Therefore

M =∑

k

1

Rτ2(λk(τ1))R′τ1(λk(τ1))

. (3.15)


Thus, M is completely determined by the sequences {λk(τ2)}k and {λk(τ1)}k.Inserting the obtained value of M into (3.13) one obtains the normalizingconstants. Having the normalizing constants allows us to construct the spectralmeasure for J(τ1). Then, by standard methods (see Section 2), one reconstructsthe matrix associated with J and the boundary condition at infinity τ1. From thevalue of M and τ1 one obtains τ2, by using the first three cases in (3.14). When0 ∈ {λk(τ2)}k, one does not use (3.14), since it is already known that τ2 = 0. �

Remark 2 Note that the proof of Theorem 1 gives a reconstruction method ofthe Jacobi matrix. Although mentioned earlier, we also remark here that theassertion of Theorem 1, for the case of τ1, τ2 ∈ R, was announced without proofin [13, Thm. 1].

4 Necessary and Sufficient Conditions

In this section we give a complete characterization of our two-spectra inverseproblem. We remind the reader about the remark on the notation at the endof Section 2.

First we prove the following simple proposition related to the converse ofProposition 3.

Proposition 4 Let {νk}k be an infinite sequence of non-vanishing complexnumbers accumulating only at ∞, and whose convergence exponent ρ1 does notexceed one. Suppose that the infinite product

limr→∞

∏|νk|�r

(1 − ζ

νk

)(4.1)

converges uniformly on any compact of C. Then this product is an entirefunction of at most minimal type of order one if either (3.5) converges or if(3.5) diverges but the following holds

limr→∞

n(r)r

= 0 , (4.2)

where n(r) is the number of elements of {νk}k in the circle |ζ | < r.

Proof Clearly, by the conditions of the theorem, one can express (4.1) in termsof canonical products [25, Sec. 3 Chap. 1] either in the form

limr→∞

∏|νk|�r

(1 − ζ

νk

)= lim

r→∞∏

|νk|�r

G(

ζ

νk; 0

)(4.3)

whenever (3.5) converges, or in the form

limr→∞

∏|νk|�r

(1 − ζ

νk

)= ecζ lim

r→∞∏

|νk|�r

G(

ζ

νk; 1

)(4.4)


otherwise, where

limr→∞

∑|νk|�r

ν−1k = −c. (4.5)

In the case (4.3), in which the genus of the product is less than the convergenceexponent of {νk}k, it is clear that (4.1) does not grow faster than an entirefunction of minimal type of order one. Indeed, by [25, Thm. 7 Chap. 1],the order of a canonical product is equal to the convergence exponent, sowhen ρ1 < 1 the assertion is obvious. For ρ1 = 1 the statement follows from[25, Thm. 15 b Chap. 1].

If we have the representation (4.4), then ρ1 = 1. By [25, Thm. 7 Chap. 1],the canonical product has order one. Since the product of functions of the sameorder is of that same order, the order of (4.1) is one. Then, the assertion followsfrom [25, Thm. 15 a Chap. 1] due to (4.2) and (4.5). �

Before passing on to the main results of this section, we establish anauxiliary result which is related to part of the proof of Theorem 1 in theAddenda and Problems of [2, Chap. 4].

Lemma 1 Consider an infinite real sequence {κ j} j and a sequence {α j} j

of positive numbers such that

∑j

κ2mj

α j< ∞ for all m = 0, 1, . . .

Let F be an entire function of at most minimal type of order one whose zeros,{κ j} j, are simple, and such that

|F(it)| → ∞ as t → ±∞ , t ∈ R . (4.6)

If∑

j

α j(1 + κ2

j

) [F′(κ j)

]2 < ∞ ,

then

∑j

κmj

F′(κ j)(4.7)

is absolutely convergent for m = 0, 1, . . . , and the absolutely convergentexpansion

1

F(ζ )=∑

j

1

F′(κ j)(ζ − κ j)

holds true for all ζ ∈ C \ {κ j} j.


Proof The absolutely convergence of (4.7) follows from

∑j

∣∣∣∣κm

j

F′(κ j)

∣∣∣∣ =∑

j

∣∣∣∣∣∣

√α j√

1 + κ2j F

′(κ j)

∣∣∣∣∣∣

∣∣∣∣∣∣κm

j

√1 + κ2

j√α j

∣∣∣∣∣∣

�√√√√∑

j

α j(1 + κ2

j

)[F′(κ j)]2

√√√√∑j

κ2mj + κ2m+2

j

α j< ∞

Construct the function

h(ζ ) := 1 − F(ζ )∑

j

1

F′(κ j)(ζ − κ j),

where the series is absolutely convergent in compact subsets of C \ {κ j} j

because of (4.7). Clearly, h(κ j) = 0 for any j. Moreover, it turns out that his an entire function of at most minimal type of order one. To show this, firstconsider the case when

∑j

∣∣κ j∣∣−1

< ∞. Here, by what we have discussed inthe proof of Proposition 3, the function F(ζ )/(ζ − κ j) can be expressed by acanonical product of genus zero. It follows from [25, Lem. 3 Chap. 1] (see alsothe proof of [25, Thm. 4 Chap. 1]) that, on the one hand,

max|ζ |=r

∣∣∣∣F(ζ )

(ζ − κ j)

∣∣∣∣ < exp (C(α)rα) , ρ1 < α < 1 ,

for any r > 0 provided that ρ1 < 1. If, on the other hand, ρ1 = 1, then for anyε > 0, there exists R0 > 0 such that

max|ζ |=r

∣∣∣∣F(ζ )

(ζ − κ j)

∣∣∣∣ < exp (εr) (4.8)

for all r > R0. Hence, in any case, we have the uniform, with respect to j,asymptotic estimation (4.8) when

∑j

∣∣κ j∣∣−1

< ∞.

Suppose now that∑

j

∣∣κ j∣∣−1 = ∞. In this case, as was shown in the proof of

Proposition 3,

F(ζ )

(ζ − κ j)= − 1

κ jζ me(c+κ−1

j )ζ+d limr→∞

∏∣∣κk∣∣ � r

k �= j

G(

ζ

κk; 1

),

where

limr→∞

∑|κk|�r

κ−1k = −c . (4.9)


On the basis of the estimates found in the proof of [25, Thm. 15 Chap. 1] (see inparticular the inequality next to [25, Eq. 1.43]), one can find R1, independentof j, such that

max|ζ |=r

∣∣∣∣F(ζ )

(ζ − κ j)

∣∣∣∣ < exp

⎡⎣r

⎛⎝∣∣∣∣∣∣c +∑

|κk|�r

κ−1k

∣∣∣∣∣∣+ C

(lim sup

r→∞n(r)

r+ ε

)+ O

(1

r

)⎞⎠⎤⎦

for all r > R1 and ε > 0 (see the definition of n(r) in the statement ofProposition 4). Note that if

∣∣κ j∣∣ � r, then the above inequality follows directly

from the inequality next to [25, Eq. 1.43]. If∣∣κ j∣∣ > r, the same inequality holds

due to

∣∣∣∣∣∣c + κ−1

j +∑

|κk|�r

κ−1k

∣∣∣∣∣∣�

∣∣∣∣∣∣c +

∑|κk|�r

κ−1k

∣∣∣∣∣∣+ 1

r.

Since F does not grow faster than a function of minimal type of order one, by[25, Thm. 15 a Chap. 1], one again verifies that, for any ε > 0, (4.8) holds for allr greater than a certain R2 depending only on the velocity of convergence inthe limits (4.9) and (4.2).

Thus, one concludes that, for any ε > 0, there is R > 0 such that

max|ζ |=r

∣∣∣∣∣∣F(ζ )

∑j

1

F′(κ j)(ζ − κ j)

∣∣∣∣∣∣�∑

j

1∣∣F′(κ j)∣∣ max

|ζ |=r

∣∣∣∣F(ζ )

(ζ − κ j)

∣∣∣∣ < exp(εr)

for all r > R, which shows that h is an entire function of at most minimal typeof order one.

Now, the function h/F is also an entire function of at most minimal type oforder one [25, Cor. Sec. 9 Chap. 1]. By the hypothesis (4.6),

limt → ±∞

t ∈ R

h(it)F(it)

= 0 ,

which implies that h/F ≡ 0 (see Corollary of [25, Sec. 14 Chap. 1]). �

Theorem 2 Let {λk}k and {μk}k be two infinite sequences of real numberssuch that

a) {λk}k ∩ {μk}k = ∅. For definiteness we assume that 0 �∈ {λk}k

b) the sequences accumulate only at the point at infinity.c) λk �= λ j, μk �= μ j for k �= j.


Then there exist unique τ1, τ2 ∈ R ∪ {∞}, with τ1 �= 0, τ1 �= τ2, and a uniqueJacobi operator J �= J∗ such that {λk}k = σ(J(τ1)) and {μk}k = σ(J(τ2)) if andonly if the following conditions are satisfied.

1. The convergence exponents of the sequence {λk}k, and of the non-zeroelements of {μk}k do not exceed one. Additionally, if

limr→∞

∑|λk|�r

1

|λk| = ∞ , require that limr→∞

nλ(r)r

= 0 ,

and if

limr→∞

∑0<|μk|�r

1

|μk| = ∞ , require that limr→∞

nμ(r)r

= 0 ,

where nλ(r) and nμ(r) are the number of elements of {λk}k and {μk}k,respectively, in the circle |ζ | < r.

2. The limits

limr→∞

∏|λk|�r

(1 − ζ

λk

)lim

r→∞∏

0<|μk|�r

(1 − ζ

μk

),

converge uniformly on compact subsets of C, and they define the functions

Rλ(ζ ) := limr→∞

∏|λk|�r

(1 − ζ

λk

)(4.10)

Rμ(ζ ) := ζ δ limr→∞

∏0<|μk|�r

(1 − ζ

μk

), (4.11)

where δ = 1 if 0 ∈ {μk}k, and δ = 0 otherwise.3. All numbers Rμ(λ j)R′

λ(λ j) have the same sign for all j. The same is true forthe numbers Rλ(μ j)R′

μ(μ j).4. For every m = 0, 1, 2, . . . the series below are convergent and the following

equalities hold ∑j

λmj

Rμ(λ j)R′λ(λ j)

= −∑

j

μmj

Rλ(μ j)R′μ(μ j)

5. The series∑

j

Rμ(λ j)

R′λ(λ j)

and∑

j

Rλ(μ j)

R′μ(μ j)

diverge either to −∞ or +∞.6. The series

∑j

Rμ(λ j)(1 + λ2

j

)R′

λ(λ j)and

∑j

Rλ(μ j)(1 + μ2

j

)R′

μ(μ j)

are convergent.


Proof We begin by proving that if {λk}k and {μk}k are, respectively, the spectraof the self-adjoint extensions J(τ1) and J(τ2) of a Jacobi operator J, thenconditions 1–6 hold true.

Since {λk}k = σ(J(τ1)) and {μk}k = σ(J(τ2)), the functions Rτ1 and Rτ2 ,given by (2.10), have the sequences {λk}k and {μk}k, respectively, as their setsof zeros. These functions do not grow faster than an entire function of minimaltype of order one. By Proposition 3 [see (3.7)], the limits

limr→∞

∏|λk|�r

(1 − ζ

λk

), lim

r→∞∏

0<|μk|�r

(1 − ζ

μk

)

converge uniformly on compacts of C. This is condition 2. Moreover, by (3.6)and [25, Thm. 15 a Chap. 1], condition 1 holds.

The functions Rλ and Rμ, given by (4.10) and (4.11), coincide with Rτ ,given by (3.10), with τ = τ1 and τ = τ2, respectively. Thus (3.13) is rewrittenas follows

a(λ j) = MRμ(λ j)R′λ(λ j) , (4.12)

where M is given by (3.14). Analogously,

a(μ j) = −MRλ(μ j)R′μ(μ j) . (4.13)

On the basis of the positiveness of the normalizing constants, from (4.12)and (4.13), we obtain condition 3.

From what we discussed in Section 2 all the moments exist for the spectralfunctions of J(τ1) and J(τ2), which are, respectively,

∑λk�t

1

a(λk)and

∑μk�t

1

a(μk). (4.14)

Hence the series in both sides of condition 4 are convergent for m ∈ N ∪ {0}.Moreover, the spectral functions (4.14) are solutions of the same momentproblem associated with J [see the paragraph surrounding (2.11)], thereforethe equality of condition 4 holds.

Theorem 1 in the Addenda and Problems of [2, Chap. 4] tells us that∑

j

a(λ j)[R′

λ(λ j)]2 = +∞ (4.15)

is a necessary condition for the sequences {λ j} j and {a(λ j)} j to be the spectrumof J(τ1) and its corresponding normalizing constants. Thus, substituting (4.12)into (4.15), one establishes the divergence of the first series in condition 5.Similarly,

∑j

a(μ j)[R′

μ(μ j)]2 = +∞

must hold, which, by (4.13), implies the divergence of the second series incondition 5.


By the same theorem in [2] mentioned above, and taking into account (4.12)and (4.13), one obtains the convergence of the series in condition 6.

Let us now prove that the conditions 1–6 are sufficient. Using condition 2and the convergence of the series in the left hand side of condition 4 with m =0, we define the real constant

M :=∑

j

1


(4.16)

and the sequence of numbers

a j := MRμ(λ j)R′λ(λ j) (4.17)

By condition 3 and (4.16), it follows that a j > 0 for all j. Moreover, (4.16) and(4.17) imply that

∑j a−1

j = 1.With the aid of the sequences {λk} and {ak} define the function ρ : R → R+

as follows

ρ(t) :=∑λk�t

a−1k . (4.18)

Consider the self-adjoint operator of multiplication Aρ by the independentvariable in L2(R, dρ). We show below that this operator is the canonicalrepresentation (see Section 2) of a self-adjoint extension of a Jacobi matrix inthe limit circle case. The proof of this fact is similar to the proof of Theorem 2in Addenda and Problems of [2, Chap. 4]. Note, however, that our conditionsare slightly different.

Consider a function θk(t) ∈ L2(R, dρ) such that

θk(λ j) = √akδkj . (4.19)

Clearly, θk(t) is the normalized eigenvector of Aρ corresponding to λk. Letϕ(t) ∈ L2(R, dρ) be such that

〈ϕ, θ j〉L2(R,dρ) =√

a j

(λ j − i)R′λ(λ j)

. (4.20)

Taking into account (4.17), it is clear that the convergence of the first series incondition 6 ensures that ϕ(t) is indeed an element of L2(R, dρ). Define

D := {ξ ∈ L2(R, dρ) : ξ = (Aρ + iI )−1ψ, ψ ∈ L2(R, dρ), ψ ⊥ ϕ

}. (4.21)

Since D ⊂ dom(Aρ), we can consider the restriction of Aρ to the linear set D.Let us show that this restriction is a symmetric operator with deficiency indices(1, 1). First we verify that D is dense in L2(R, dρ). Suppose that a non-zeroη ∈ L2(R, dρ) is orthogonal to D. This would imply that there is a non-zeroconstant C ∈ C such that

η = C(Aρ − iI )ϕ .


Therefore,

〈η, θ j〉L2(R,dρ) = C√

a j

R′λ(λ j)

,

whence we easily conclude that η ∈ L2(R, dρ) would contradict condition 5.Consider now the restriction of Aρ to the set D, denoted henceforth by

Aρ �D, and let us find the dimension of ker((Aρ �D)∗ − iI ) which is character-ized as the set of all ω ∈ L2(R, dρ) for which the equation

⟨(Aρ + iI )ξ, ω

⟩L2(R,dρ)

= 0

is satisfied for any ξ ∈ D. It is not difficult to show that any such ω can bewritten as follows

ω = Cϕ , 0 �= C ∈ C.

Hence dim ker((Aρ �D)∗−iI )=1. Analogously, it can be shown that the dimen-sion of ker((Aρ �D)∗+iI ) also equals one. Indeed, if ω∈ker((Aρ �D)∗+iI ) then, up to a complex constant ω = (Aρ − iI )(Aρ + iI )−1ϕ .

Now we show that Aρ �D is the canonical representation of a Jacobioperator in the limit circle case and Aρ is the canonical representation of aself-adjoint extension of this Jacobi operator. We proceed stepwise.

1. We orthonormalize the sequence of functions {tn}∞n=0 with respect to theinner product of L2(R, dρ). Note that condition 4 guarantees that allelements of the sequence {tn}∞n=0 are in L2(R, dρ). We obtain thus a se-quence of polynomials {Pn−1(t)}∞n=1 which satisfy the three term recurrenceequation (2.15) and (2.16), where all the coefficients bk (k ∈ N) turn out tobe positive and qk (k ∈ N) are real numbers.

2. We verify that the polynomials are dense in L2(R, dρ), so the sequence wehave constructed is a basis in L2(R, dρ). Note first that the function Rλ(ζ )

is entire of at most minimal type of order one. Indeed, this follows fromProposition 4, in view of conditions 1 and 2. Now, for any element λk0 ofthe sequence {λk}k, we clearly have

|Rλ(it)| �∣∣∣∣∣1 + t2

λ2k0

∣∣∣∣∣ , t ∈ R .

This implies that Rλ satisfies (4.6). Hence the function Rλ and the se-quences {λ j} j, {a j} j satisfy the condition of Lemma 1. Thus, we have shownthe convergence of the series

∑j

λmj

R′λ(λ j)

(4.22)

for all m = 0, 1, 2, . . . , and that

1

Rλ(ζ )=∑

j

1

R′λ(λ j)(ζ − λ j)

. (4.23)


Taking into account Definitions 1 and 2 of the Addenda and Problemsof [2, Chap. 4], one obtains from Corollary 2 of [2, Addenda and ProblemsChap. 4], together with conditions 5 and 6, that {λk} is a canonical sequenceof nodes and {a−1

k } the corresponding sequence of masses for the momentproblem given by {sm}∞m=0 with

sm := 1

M∑

j

λmj


. (4.24)

Hence, for this moment problem, ρ is a canonical solution [2, Def. 3.4.1].By definition, a canonical solution is N-extremal and by [2, Thm. 2.3.3], thepolynomials are dense in L2(R, dρ).

3. We prove that the elements of the basis {Pn−1(t)}∞n=1 are in D. From (4.22)and (4.23), by Lemma 1 of the Addenda and Problems of [2, Chap. 4], onehas for m = 0, 1, 2, . . .

∑j

λmj

R′λ(λ j)

= 0 .

Then, if S(t) is a polynomial⟨(Aρ + iI )S, ϕ

⟩L2(R,dρ)

=∑

j

S(λ j)

R′λ(λ j)

= 0 .

Whence it follows that S ∈ D.

Now, by (2.15) and (2.16), it is straightforward to show that U−1 Aρ �D U (see(2.12)) is a Jacobi operator in the limit circle case.

Denote by J the Jacobi operator U−1 Aρ �D U . On the basis of what wasdiscussed in Section 2 one can find τ1 ∈ (R ∪ {∞}) \ {0} such that the self-adjoint operator of multiplication in L2(R, dρ) is the canonical representationof J(τ1). τ1 cannot be zero since then {λk} should contain the zero. If 0 �∈ {μk}k,we define

τ2 :=⎧⎨⎩M if τ1 = ∞∞ if τ1 = −MMτ1

τ1+M in all other cases,(4.25)

and if 0 ∈ {μk}k simply assign τ2 := 0.For the proof to be complete it remains to show that {μk}k are the eigenval-

ues of J(τ2). To this end we first show that {μk}k are the eigenvalues of someself-adjoint extension of J. Let M = −M and define

a j := MRλ(μ j)R′μ(μ j) .

From condition 4 with m = 0, it follows that a j > 0 for any j and∑

j a−1j = 1,

and that the function ρ(t) := ∑μk�t a−1

k is a solution of the moment problem{sk}∞k=0 with sk given by (4.24). Moreover, taking into account conditions 1and 6, one easily verifies as before that the sequences {μ j} j and {a j} j, and thefunction Rμ satisfy the conditions of Lemma 1. Therefore, by Definitions 1,


2 and Corollary 2 of the Addenda and Problems of [2, Chap. 4], as well asconditions 5 and 6, it turns out that the sequence {μ j} j is a canonical sequenceof nodes and {a j} j the corresponding sequence of masses for the momentproblem given by {sk}∞k=0 with sk satisfying (4.24). Hence ρ is a canonicalsolution of this moment problem. Denote by J(τ2) the self-adjoint extensionof J having ρ as its spectral function.

Let us consider now the functions Rτ2 and Rτ2 corresponding to J(τ2) andJ(τ2), respectively (see (3.10)). It is straightforward to verify that

Rτ2(λ j) = a j

MR′τ1(λ j)

= Rτ2(λ j) , (4.26)

where the first equality follows from (3.13), while the second follows from(3.11) and (4.17). By (3.14), (3.15), and (4.25), one easily concludes from (4.26)that τ2 = τ2. �

Remark 3 When 0 ∈ {μk}k, the signs of the real numbers Rμ(λ j)R′λ(λ j) and

Rλ(μ j)R′μ(μ j) are known. Thus, we can write condition 3 as follows

Rμ(λ j)R′λ(λ j) < 0 Rλ(μ j)R′

μ(μ j) > 0 for all j .

This is a consequence of (3.12) and (3.14) by which we know that in equation(4.12) M = −a(0) < 0.

Remark 4 Note that, by Proposition 2, conditions 1–6 imply the interlacing ofthe sequences {λk}k and {μk}k.

Acknowledgements We thank A. Osipov for drawing our attention to [14] and the anonymousreferees whose comments led to an improved presentation of our work.

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Anisotropic Lavine’s Formula and Symmetrised TimeDelay in Scattering Theory

Rafael Tiedra de Aldecoa

Received: 30 September 2007 / Accepted: 27 June 2008 /Published online: 6 August 2008© Springer Science + Business Media B.V. 2008

Abstract We consider, in quantum scattering theory, symmetrised time delaydefined in terms of sojourn times in arbitrary spatial regions symmetric withrespect to the origin. For potentials decaying more rapidly than |x|−4 at infinity,we show the existence of symmetrised time delay, and prove that it satisfiesan anisotropic version of Lavine’s formula. The importance of an anisotropicdilations-type operator is revealed in our study.

Keywords Time delay · Lavine’s formula · Scattering theory

Mathematics Subject Classifications (2000) 46N50 · 81Q10 · 35J10

1 Introduction and Main Results

It is known for quite some time that the definition of time delay (in termsof sojourn times) in scattering theory has to be symmetrised in the case ofmultichannel-type scattering processes (see e.g. [3, 4, 12, 13, 21, 22]). Morerecently [6] it has been shown that symmetrised time delay does exist, intwo-body scattering processes, for arbitrary dilated spatial regions symmetricwith respect to the origin (the usual time delay does exist only for sphericalspatial regions [20]). This leads to a generalised formula for time delay, whichreduces to the usual one in the case of spherical spatial regions. The aim ofthe present paper is to provide a reasonable interpretation of this formula

R. Tiedra de Aldecoa (B)CNRS (UMR 8088) and Department of Mathematics, University of Cergy-Pontoise,2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, Francee-mail: [email protected]

156 R. Tiedra de Aldecoa

for potential scattering by proving its identity with an anisotropic version ofLavine’s formula [11].

Let us recall the definition of symmetrised time delay for a two-body scatter-ing process in R

d, d ≥ 1. Consider a bounded open set � in Rd containing the

origin and the dilated spatial regions �r := {rx | x ∈ �}, r > 0. Let H0 := − 12�

be the kinetic energy operator in H := L2(Rd) (endowed with the norm ‖ · ‖and scalar product 〈·, ·〉). Let H be a selfadjoint perturbation of H0 such thatthe wave operators W± := s- limt→±∞ eitH e−itH0 exist and are complete (so thatthe scattering operator S := W∗+W− is unitary). Then one defines for somestates ϕ ∈ H and r > 0 two sojourn times, namely:

T0r (ϕ) :=

∫ ∞

−∞dt

∫x∈�r

ddx∣∣(e−itH0 ϕ

)(x)

∣∣2

and

Tr(ϕ) :=∫ ∞

−∞dt

∫x∈�r

ddx∣∣(e−itH W−ϕ

)(x)

∣∣2.

If the state ϕ is normalized to one the first number is interpreted as the timespent by the freely evolving state e−itH0 ϕ inside the set �r, whereas the secondone is interpreted as the time spent by the associated scattering state e−itH W−ϕ

within the same region. The usual time delay of the scattering process for �r

with incoming state ϕ is defined as

τ inr (ϕ) := Tr(ϕ) − T0

r (ϕ),

and the corresponding symmetrised time delay for �r is given by

τr(ϕ) := Tr(ϕ) − 1

2

[T0

r (ϕ) + T0r (Sϕ)

].

If � is spherical and some abstract assumptions are verified, the limits of τ inr (ϕ)

and τr(ϕ) as r → ∞ exist and satisfy [6, Sec. 4.3]

limr→∞ τr(ϕ) = lim

r→∞ τ inr (ϕ) = −1

2

⟨H−1/2

0 ϕ, S∗[D, S]H−1/20 ϕ

⟩, (1.1)

where D is the generator of dilations. If � is not spherical the limit of τ inr (ϕ) as

r → ∞ does not exist anymore [20], but the limit of τr(ϕ) as r → ∞ still exists,provided that � is symmetric with respect to the origin [6, Rem. 4.8].

In this paper we study τr(ϕ) in the setting of potential scattering. Forpotentials decaying more rapidly than |x|−4 at infinity, we prove the existenceof limr→∞ τr(ϕ) by using the results of [6]. In a first step we show that the limitsatisfies the equation

limr→∞ τr(ϕ) = −⟨

f (H0)−1/2ϕ, S∗[D�, S] f (H0)

−1/2ϕ⟩, (1.2)

where f is a real symbol of degree 1 and D� ≡ D�( f ) is an operator acting asan anisotropic generator of dilations. Then we prove that formula (1.2) can berewritten as an anisotropic Lavine’s formula. Namely, one has (see Theorem

Anisotropic Lavine’s formula and symmetrised time delay 157

4.5 for a precise statement)

limr→∞ τr(ϕ) =

∫ ∞

−∞ds

⟨e−isH W− f (H0)

−1/2ϕ,V�, f e−isH W− f (H0)−1/2ϕ

⟩, (1.3)

where the operator

V�, f = f (H) − f (H0) − i[V, D�]generalises the virial V := 2V − i[V, D]. Formula (1.3) provides an interestingrelation between the potential V and symmetrised time delay, which wediscuss.

Let us give a description of this paper. In Section 2 we introduce thecondition on the set � (see Assumption 2.1) under which our results areproved. We also define the anisotropic generator of dilations D� and establishsome of its properties. Section 3 is devoted to symmetrised time delay inpotential scattering; the existence of symmetrised time delay for potentialsdecaying more rapidly than |x|−4 at infinity is established in Theorem 3.5. InTheorem 4.5 of Section 4 we prove the anisotropic Lavine’s formula (1.3) forthe same class of potentials. Remarks and examples are to be found at the endof Section 4.

We emphasize that the extension of Lavine’s formula to non spherical sets� is not straightforward due, among other things, to the appearance of asingularity in the space of momenta not present in the isotropic case (seeEq. 2.7 and the paragraphs that follow). The adjunction of the symbol f inthe definition of the operator D� (see Definition 2.2) is made to circumventthe difficulty.

Finally we refer to [9] (see also [8, 11, 15–17]) for a related work on Lavine’sformula for time delay.

2 Anisotropic Dilations

In this section we define the operator D� and establish some of its propertiesin relation with the generator of dilations D and the shape of �. We start byrecalling some notations.

Given two Hilbert spaces H1 and H2, we write B(H1,H2) for the set ofbounded operators from H1 to H2 with norm ‖ · ‖H1→H2 , and put B(H1) :=B(H1,H1). We set Q := (Q1, Q2, . . . , Qd) and P := (P1, P2, . . . , Pd), whereQ j (resp. P j) stands for the j-th component of the position (resp. momentum)operator in H. N := {0, 1, 2, . . .} is the set of natural numbers. Hk, k ∈ N, arethe usual Sobolev spaces over R

d, and Hst , s, t ∈ R, are the weighted Sobolev

spaces over Rd [1, Sec. 4.1], with the convention that Hs := Hs

0 and Ht := H0t .

Given a set M ⊂ Rd we write 1lM for the characteristic function for M. We

always assume that � is a bounded open set in Rd containing 0, with boundary

∂� of class C4. Often we even suppose that � satisfies the following strongerassumption (see [6, Sec. 2]).


Assumption 2.1 � is a bounded open set in Rd containing 0, with boundary ∂�

of class C4. Furthermore � satisfies∫ ∞

0dμ

[1l�(μx) − 1l�(−μx)

] = 0, ∀x ∈ Rd.

If p ∈ Rd, then the number

∫ ∞0 dt 1l�(tp) is the sojourn time in � of a free

classical particle moving along the trajectory t → x(t) := tp, t ≥ 0. Obviously� satisfies Assumption 2.1 if � is symmetric with respect to 0 (i.e. � = −�).Moreover if � is star-shaped with respect to 0 and satisfies Assumption 2.1,then � = −�.

We recall from [6, Lemma 2.2] that the limit

R�(x) := limε↘0

(∫ +∞

ε

dμ

μ1l�(μx) + ln ε

)(2.4)

exists for each x ∈ Rd \ {0}, and we define the function G� : R

d \ {0} → R by

G�(x) := 1

2[R�(x) + R�(−x)] . (2.5)

The function G� : Rd \ {0} → R is of class C4 since ∂� is of class C4. Let x ∈

Rd \ {0} and t > 0, then formulas (2.4) and (2.5) imply that

G�(tx) = G�(x) − ln(t).

From this one easily gets the following identities for the derivatives of G� :

x · (∇G�)(x) = −1, (2.6)

t|α|(∂αG�

)(tx) = (

∂αG�

)(x), (2.7)

where α is a d-dimensional multi-index with |α| ≥ 1 and ∂α := ∂α11 · · · ∂αd

d . Thesecond identity suggests a way of regularizing the functions ∂ jG� which partlymotivates the following definition. We use the notation Sμ(R; R), μ ∈ R, forthe vector space of real symbols of degree μ on R (see [1, Sec. 1.1]).

Definition 2.2 Let f ∈ S1(R; R) be such that

(1) f (0) = 0 and f (u) > 0 for each u > 0,(2) for each j = 1, 2, . . . , d, the function x → (∂ jG�)(x) f (x2/2) (a priori only

defined for x ∈ Rd \ {0}) belongs to C3(Rd; R).

Then we define F� : Rd → R

d by F�(x) := −(∇G�)(x) f (x2/2).

Given a set � there are many appropriate choices for the function f . Forinstance if γ > 0 one can always take f (u) = 2(u2 + γ )−1u3, u ∈ R. But when� is equal to the open unit ball B := {x ∈ R

d | |x| < 1} one can obviously makea simpler choice. Indeed in this case one has [6, Rem. 2.3.(b)] (∂ jGB)(x) =−x jx−2, and the choice f (u) = 2u, u ∈ R, leads to the C∞-function F�(x) = x.


Remark 2.3 One can associate to each set � a unique set � symmetric and star-shaped with respect to 0 such that G� = G� [6, Rem. 2.3.(c)]. The boundary∂� of � satisfies

∂� = {eG�(x) x | x ∈ R

d \ {0}},and �r = {

rx | x ∈ �}, r > 0. Thus the vector field F� = F� is orthogonal to

the hypersurfaces ∂�r in the following sense: if v belongs to the tangent spaceof ∂�r at y ∈ ∂�r, then F�(y) is orthogonal to v. To see this let s → y(s) ≡r eG�(x(s)) x(s) be any differentiable curve on ∂�r. Then d

ds y(s) belongs to thetangent space of ∂�r at y(s), and a direct calculation using Eqs. 2.6–2.7 givesF�(y(s)) · d

ds y(s) = 0.

In the rest of the section we give a meaning to the expression

D� := 1

2[F�(P) · Q + Q · F�(P)],

and we establish some properties of D� in relation with the generator ofdilations

D := 1

2(P · Q + Q · P).

For the next lemma we emphasize that H2 is contained in the domainD

(f (H0)

)of f (H0). The notation 〈·〉 stands for

√1 + | · |2, and S is the

Schwartz space on Rd.

Lemma 2.4 Let � be a bounded open set in Rd containing 0, with boundary ∂�

of class C4. Then

(a) The operator D� is essentially selfadjoint on S . As a bounded operator,D� extends to an element of B

(Hs

t ,Hs−1t−1

)for each s ∈ R, t ∈ [−2, 0] ∪

[1, 3].(b) One has for each t ∈ R and ϕ ∈ D(D�) ∩ D

(f (H0)

)

e−itH0 D� eitH0 ϕ = [D� − t f (H0)]ϕ. (2.8)

In particular one has the equality

i[H0, D�] = f (H0) (2.9)

as sesquilinear forms on D(D�) ∩ H2.

The second claim of point (a) is sufficient for our purposes, even if it is onlya particular case of a more general result.

Proof

(a) The essential seladjointness of D� on S follows from the fact that F� isof class C3 (see e.g. [1, Prop. 7.6.3.(a)]).


Due to the hypotheses on F� one has for each ϕ ∈ S the bound∥∥(∂α F� j)(P)ϕ‖ � Const. ‖〈P〉ϕ‖ , (2.10)

where F� j is the j-th component of F� and α is a d-dimensional multi-index with |α| � 3. Furthermore

‖D�‖Hs3→Hs−1

2�

∑j�d

supϕ∈S ,‖ϕ‖Hs

3=1

∥∥〈P〉s−1〈Q〉2[F� j(P)Q j + i2 (∂ jF� j)(P)

]ϕ∥∥

for each s ∈ R. Since 〈Q〉2 acts as the operator 1 − � after a Fouriertransform, the inequalities above imply that D� extends to an element ofB(Hs

3,Hs−12 ). A similar argument shows that D� extends to an element

of B(Hs1,Hs−1) for each s ∈ R. The second part of the claim follows then

by using interpolation and duality.(b) Let ϕ ∈ e−itH0 S . Since e−itH0 Q j eitH0 ϕ = (Q j − tP j)ϕ, it follows by for-

mula (2.6) that

e−itH0 D� eitH0 ϕ = [D� + tP · (∇G�)(P) f (H0)]ϕ = [D� − t f (H0)]ϕ.

This together with the essential selfajointness of e−itH0 D� eitH0 one−itH0 S implies the first part of the claim. Relation (2.9) follows by takingthe derivative of (2.8) w.r.t. t in the form sense and then setting t = 0. ��

Remark 2.5 If � = B and f (u) = 2u, then F�(x) = x for each x ∈ Rd, and

the operators D� and D coincide. If � is not spherical it is still possible todetermine part of the behaviour of the group Wt := eitD� . Indeed let R × R

d �(t, x) → ξt(x) ∈ R

d be the flow associated to the vector field −F� , that is, thesolution of the differential equation

ddt

ξt(x) = (∇G�)(ξt(x)) f(ξt(x)2/2

), ξ0(x) = x. (2.11)

Then it is known (see e.g. the proof of [1, Prop. 7.6.3.(a)]) that the group Wt

acts in the Fourier space as(Wtϕ

)(x) := √

ηt(x)ϕ(ξt(x)), (2.12)

where ηt(x) ≡ det(∇ξt(x)) is the Jacobian at x of the mapping x → ξt(x). Takingthe scalar product of (2.11) with ξt(x) and then using formula (2.6) leads to theequation

ddt

ξt(x)2 = −2 f(ξt(x)2/2

), ξ0(x) = x.

If t < 0 and x �= 0, then ξt(x)2 � x2 > 0, and ξt(x)2 is given by the implicitformula

2t +∫ ξt(x)2

x2du f (u/2)−1 = 0.

This, together with the facts that x → f (x2/2) belongs to S2(R; R) and f (u) >

0 for u > 0, implies the estimate 〈ξt(x)〉 � e−ct 〈x〉 for some constant c > 0.


Since 〈ξt(x)〉 � 〈x〉 for each t � 0 it follows that

〈ξt(x)〉 � (1 + e−ct) 〈x〉 (2.13)

for all t ∈ R and x ∈ Rd (the case x = 0 is covered since ξt(0) = 0 for all t ∈ R).

Equation (2.13) implies that the domain H2 of H0 is left invariant by thegroup Wt.

The results of Remarks 2.3 and 2.5 suggest that Wt may be interpreted as ananisotropic version of the dilation group, which reduces to the usual dilationgroup in the case � = B and f (u) = 2u.

In the next lemma we show some properties of the mollified resolvent

Rλ := iλ(D� + iλ)−1, λ ∈ R \ {0}.We refer to [18, Lemma 6.2] for the same results for the usual generator of thedilation group D, that is, when � = B and f (u) = 2u. See also [5, Lemma 4.5]for a general result.


of class C4. Then

(a) One has for each t ∈ R and ϕ ∈ D(ξt(P)2

)

eitD� H0 e−itD� ϕ = 1

2ξt(P)2ϕ. (2.14)

(b) For each λ ∈ R with |λ| large enough, Rλ belongs to B(H2), and Rλ

extends to an element of B(H−2). Furthermore we have for each ϕ ∈ H2

and each ψ ∈ H−2

lim|λ|→∞ ‖(1 − Rλ)ϕ‖H2 = 0 and lim|λ|→∞ ‖(1 − Rλ)ψ‖H−2 = 0.

Proof

(a) Let ϕ ∈ eitD� S . A direct calculation using formula (2.12) gives

(F eitD� H0 e−itD� ϕ

)(k) = 1

2ξt(k)2(Fϕ)(k),

where F is the Fourier transformation. This together with the essentialselfajointness of eitD� H0 e−itD� on eitD� S implies the claim.

(b) Let ϕ ∈ H2 and take λ ∈ R with |λ| > c, where c is the constant in theinequality (2.13). Using the (strong) integral formula

(D� + iλ)−1 = i∫ ∓∞

0dt eλt e−itD� , sgn(λ) = ±1,


and Relation (2.14) we get the equalities

(D� + iλ)−1ϕ = (H0 + 1)−1(D� + iλ)−1(H0 + 1)ϕ +

+ i∫ ∓∞

0dt eλt [e−itD� , (H0 + 1)−1

](H0 + 1)ϕ

= (H0 + 1)−1(D� + iλ)−1(H0 + 1)ϕ −

− i∫ ∓∞

0dt eλt(H0 + 1)−1 e−itD�

[H0 − 1

2ξt(P)2

]ϕ

= (H0 + 1)−1(D� + iλ)−1ϕ +

+ i2(H0 + 1)−1

∫ ∓∞

0dt eλt e−itD� ξt(P)2ϕ.

It follows that

H0 Rλϕ = −λ

2

∫ ∓∞

0dt eλt e−itD� ξt(P)2ϕ, sgn(λ) = ±1.

Now |λ| > c, and∥∥ξt(P)2ϕ

∥∥ � (1 + e−ct)‖ϕ‖H2 due to the bound (2.13). Thus

‖H0 Rλϕ‖ � |λ|2

∫ ∞

0dt e−|λ|t ∥∥ξ− sgn(λ)t(P)2ϕ

∥∥

� |λ|2

∫ ∞

0dt

(e−|λ|t + e(sgn(λ)c−|λ|)t )‖ϕ‖H2

� Const.‖ϕ‖H2 . (2.15)

Using the estimate (2.15) and a duality argument one gets the bounds

‖Rλ‖H2→H2 � Const. and ‖Rλ‖H−2→H−2 � Const., (2.16)

which imply the first part of the claim. For the second part we remark that

1 − Rλ = (iλ)−1 D� Rλ

on H. Using this together with the bounds (2.16) one easily shows thatlim|λ|→∞ ‖(1 − Rλ)ϕ‖H2 = 0 for each ϕ ∈ H2 and that lim|λ|→∞ ‖(1 − Rλ)ψ

‖H−2 = 0 for each ψ ∈ H−2. ��

3 Symmetrised Time Delay

In this section we collect some facts on short-range scattering theory inconnection with the existence of symmetrised time delay. We always assumethat the potential V satisfies the usual Agmon-type condition:

Assumption 3.1 V is a multiplication operator by a real-valued function suchthat V defines a compact operator from H2 to Hκ for some κ > 1.


By using duality, interpolation and the fact that V commutes with theoperator 〈Q〉t, t ∈ R, one shows that V also defines a bounded operator fromH2s

t to H2(s−1)t+κ for any s ∈ [0, 1], t ∈ R. Furthermore the operator sum H :=

H0 + V is selfadjoint on D(H) = H2, the wave operators W± exist and arecomplete, and the projections 1l�r (Q) are locally H-smooth on (0, ∞) \ σpp(H)

(see e.g. [7, Sec. 3] and [19, Sec. XIII.8]).Since the first two lemmas are somehow standard, we give their proofs in

Appendix.

Lemma 3.2 Let V satisfy Assumption 3.1 with κ > 1, and take z ∈ C \ {σ(H0) ∪σ(H)}. Then the operator (H − z)−1 extends to an element of B

(H−2s

t ,H2(1−s)t

)for each s ∈ [0, 1], t ∈ R.

Alternate formulations of the next lemma can be found in [7, Lemma 4.6]and [22, Lemma 3.9]. For each s � 0 we define the dense set

Ds := {ϕ ∈ D(〈Q〉s) | η(H0)ϕ = ϕ for some η ∈ C∞

0 ((0, ∞) \ σpp(H))}.

Lemma 3.3 Let V satisfy Assumption 3.1 with κ > 2. Then one has for eachϕ ∈ Ds with s > 2

∥∥(W− − 1) e−itH0 ϕ∥∥ ∈ L1(R−, dt) (3.17)

and∥∥(W+ − 1) e−itH0 ϕ

∥∥ ∈ L1(R+, dt). (3.18)

Lemma 3.4 Let V satisfy Assumption 3.1 with κ > 4, and let ϕ ∈ Ds for somes > 2. Then there exists s′ > 2 such that Sϕ ∈ Ds′ , and the following conditionsare satisfied:∥∥(W− − 1) e−itH0 ϕ

∥∥ ∈ L1(R−, dt) and∥∥(W+ − 1) e−itH0 Sϕ

∥∥ ∈ L1(R+, dt).

Proof The first part of the claim follows by [10, Thm. 1.4.(ii)]. Since ϕ ∈ Ds andSϕ ∈ Ds′ with s, s′ > 2, the second part of the claim follows by Lemma 3.3. ��

Theorem 3.5 Let � satisfy Assumption 2.1. Suppose that V satisfies Assumption3.1 with κ > 4. Let ϕ ∈ Ds with s > 2. Then the limit of τr(ϕ) as r → ∞ exists,and one has

limr→∞ τr(ϕ) = −⟨

f (H0)−1/2ϕ, S∗[D�, S] f (H0)

−1/2ϕ⟩. (3.19)

Proof Due to Lemma 3.4 all the assumptions for the existence of limr→∞ τr(ϕ)

are verified (see [6, Sec. 4]), and we know by Theorem [6, Thm. 4.6] that

limr→∞ τr(ϕ) = −1

2

⟨ϕ, S∗ [

i[Q2, G�(P)], S]ϕ⟩.


It follows that

limr→∞ τr(ϕ) = 1

2

⟨ϕ, S∗[Q · (∇G�)(P) + (∇G�)(P) · Q, S]ϕ⟩

= 1

2

⟨f (H0)

−1/2ϕ, S∗[ f (H0)1/2

(Q · (∇G�)(P) +

+ (∇G�)(P) · Q)

f (H0)1/2, S

]f (H0)

−1/2ϕ⟩

= −⟨f (H0)

−1/2ϕ, S∗[D�, S] f (H0)−1/2ϕ

⟩.

��

Note that Theorem 3.5 can be proved with the function f (u) = 2u, even if� is not spherical. Indeed, in such a case, point (2) of Definition 2.2 is theonly assumption not satisfied by f , and a direct inspection shows that thisassumption does not play any role in the proof of Theorem 3.5.

Remark 3.6 Some results of the literature suggest that Theorem 3.5 may beproved under a less restrictive decay assumption on V if one modifies someof the previous definitions. Typically one proves the existence of (usual) timedelay for potentials decaying more rapidly than |x|−2 (or even |x|−1) at infinityby using a smooth cutoff in configuration space and by considering particularpotentials. The reader is referred to [2, 14, 15, 23, 24] for more information onthis issue.

4 Anisotropic Lavine’s Formula

In this section we prove the anisotropic Lavine’s formula (1.3). We first give aprecise meaning to some commutators.

Lemma 4.1 Let � be a bounded open set in Rd containing 0 with boundary ∂�

of class C4. Let V satisfy Assumption 3.1 with κ > 1. Then

(a) The commutator [V, D�], defined as a sesquilinear form on D(D�) ∩ H2,extends uniquely to an element of B(H2,H−2).

(b) For each t ∈ R the commutator [D�, e−itH], defined as a sesquilinearform on D(D�) ∩ H2, extends uniquely to an element [D�, e−itH]a ofB(H2,H−2) which satisfies

∥∥[D�, e−itH]a∥∥H2→H−2 � Const. |t|.

(c) For each η ∈ C∞0 (R) the commutator [D�, η(H)], defined as a sesquilinear

form on D(D�) ∩ H2, extends uniquely to an element of B(H). In partic-ular, the operator η(H) leaves D(D�) invariant.


Proof Point (a) follows easily from Lemma 2.4.(a) and the hypotheses on V.Given point (a) and Lemma 2.6.(b), one shows points (b) and (c) as in [18,Lemma 7.4]. ��

If V satisfies Assumption 3.1 with κ > 2, then the result of Lemma 4.1.(a)can be improved by using Lemma 2.4.(a). Namely, there exists δ > 1/2 suchthat the commutator [V, D�], defined as a sesquilinear form on D(D�) ∩ H2,extends uniquely to an element [V, D�]a of B

(H2−δ,H−2

δ

).

The next Lemma is a generalisation of [9, Lemmas 2.5 & 2.7]. It is provedunder the following assumption on the function f .

Assumption 4.2 For each t ∈ R there exists ρ > 1 such that the operator f (H) −f (H0), defined on H2, extends to an element of B

(H2

t ,Ht+ρ

).

We refer to Remark 4.4 for examples of admissible functions f . Here weonly note that the operator

V�, f := f (H) − i[H, D�]a = f (H) − f (H0) − i[V, D�]a.

belongs to B(H2−δ,H−2δ ) for some δ>1/2 as soon as f satisfies Assumption 4.2.


of class C4. Let V satisfy Assumption 3.1 with κ > 2. Suppose that Assumption4.2 is verified. Then

(a) One has for each η ∈ C∞0 ((0, ∞) \ σpp(H)) and each t ∈ R the inequality

∥∥(D� + i)−1 e−itH η(H)(D� + i)−1∥∥ � Const. 〈t〉−1 .

(b) For each η ∈ C∞0 ((0, ∞) \ σpp(H)) the operators [D�, W±η(H0)] and

[D�, W∗±η(H)], defined as sesquilinear forms on D(D�), extend uniquelyto elements of B(H). In particular, the operators W±η(H0) and W∗±η(H)

leave D(D�) invariant.

Proof

(a) Since the case t = 0 is trivial, we can suppose t �= 0. Let ϕ, ψ ∈ D(D�) ∩H2, then

⟨D�ϕ, e−itH ψ

⟩ − ⟨ϕ, e−itH D�ψ

⟩

= limλ→∞

∫ t

0ds

⟨ϕ, ei(s−t)H i[H, D� Rλ] e−isH ψ

⟩


due to Lemma 2.6.(b). By using Lemma 2.4.(b) and Lemma 4.1.(b) we getin B(H2,H−2) the equalities

[D�, e−itH]a = e−itH

∫ t

0ds eisH i[H, D�]a e−isH

= t e−itH f (H) − e−itH∫ t

0ds eisH V�, f e−isH . (4.20)

Take η, ϑ ∈ C∞0 ((0, ∞) \ σpp(H)) with ϑ identically one on the support

of η, and let ζ ∈ C∞0 ((0, ∞) \ σpp(H)) be defined by ζ(u) := f (u)−1ϑ(u).

Then η(H) = f (H)ζ(H)η(H) and

e−itH η(H) = 1

tζ(H)t e−itH f (H)η(H)

= 1

tζ(H) e−itH

∫ t

0ds eisH V�, f e−isH η(H) +

+ 1

tζ(H)

[D�, e−itH]a

η(H).

Since V�, f belongs to B(H2−δ,H−2δ ) for some δ > 1/2, a local H-

smoothness argument shows that the first term is bounded by Const.|t|−1

in H. Furthermore by using Lemma 4.1.(c) one shows that (D� +i)−1ζ(H)[D�, e−itH]aη(H)(D� + i)−1 is bounded in H by a constantindependent of t. Thus

∥∥(D� + i)−1 e−itH η(H)(D� + i)−1∥∥ � Const. |t|−1,

and the claim follows.(b) Consider first [D�, W+η(H0)]. Given η ∈ C∞

0 ((0, ∞) \ σpp(H)) let ζ ∈C∞

0 ((0, ∞) \ σpp(H)) be identically one on the support of η. Due toLemma 4.1.(c) one has on D(D�)

[D�, ζ(H) eitH η(H) e−itH0 ζ(H0)

]=ζ(H)

[D�, eitHη(H) e−itH0

]ζ(H0)+[D�, ζ(H)] eitHη(H) e−itH0 ζ(H0)+

+ ζ(H) eitH η(H) e−itH0[D�, ζ(H0)],


and the last two operators belong to B(H) with norm uniformly boundedin t. Let ϕ, ψ ∈ D(D�). Using Lemma 2.4.(b) and Lemma 2.6.(b) one getsfor the first operator the following equalities

⟨ϕ, ζ(H)

[D�, eitH η(H) e−itH0

]ζ(H0)ψ

⟩= ⟨

ϕ, ζ(H)[D�, eitH]

η(H) e−itH0 ζ(H0)ψ⟩ +

+ ⟨ϕ, ζ(H) eitH[D�, η(H)] e−itH0 ζ(H0)ψ

⟩ ++ ⟨

ϕ, ζ(H) eitH η(H)[D�, e−itH0

]ζ(H0)ψ

⟩

= −∫ t

0ds

⟨ϕ, ζ(H) ei(t−s)H i[H, D�]a eisH η(H) e−itH0 ζ(H0)

⟩++ ⟨

ϕ, ζ(H) eitH[D�, η(H)] e−itH0 ζ(H0)ψ⟩ +

+ t⟨ϕ, ζ(H) eitH η(H) e−itH0 f (H0)ζ(H0)ψ

⟩

=∫ t

0ds

⟨ϕ, ζ(H) ei(t−s)H V�, f eisH η(H) e−itH0 ζ(H0)

⟩++ ⟨

ϕ, ζ(H) eitH[D�, η(H)] e−itH0 ζ(H0)ψ⟩ −

− t⟨ϕ, η(H) eitH{ f (H) − f (H0)} e−itH0 ζ(H0)ψ

⟩.

The first two terms are bounded by c‖ϕ‖ · ‖ψ‖ with c > 0 independentof ϕ, ψ and t (use the local H-smoothness of V�, f for the first term).Furthermore, due to the local H- and H0-smoothness of f (H) − f (H0)

one can find a sequence tn → ∞ as n → ∞ such that

limn→∞ tn

⟨ϕ, η(H) eitn H{ f (H) − f (H0)} e−itn H0 ζ(H0)ψ

⟩ = 0.

This together with the previous remarks implies that

limn→∞

⟨ϕ, [D�, ζ(H) eitn H η(H) e−itn H0 ζ(H0)]ψ

⟩� c′‖ϕ‖ · ‖ψ‖,

with c′ > 0 independent of ϕ, ψ and t. Thus using the intertwiningrelation and the identity η(H0) = ζ(H0)η(H0)ζ(H0) one finds that∣∣ 〈D�ϕ, W+η(H0)ψ〉 − 〈ϕ, W+η(H0)ψ〉 ∣∣

= limn→∞

∣∣ ⟨ϕ, [D�, ζ(H) eitn H η(H) e−itn H0 ζ(H0)]ψ⟩ ∣∣

� c′‖ϕ‖ · ‖ψ‖.This proves the result for [D�, W+η(H0)]. A similar proof holdsfor [D�, W−η(H0)]. Since the wave operators are complete, one hasW∗±η(H) = s- limt→±∞ eitH0 e−itH η(H), and an analogous proof can begiven for the operators [D�, W∗±η(H)]. ��


Remark 4.4 In the case � = B the requirements of Definition 2.2 and Assump-tion 4.2 are satisfied by many functions f . A natural choice is f (u) = 2u, u ∈ R,since in such a case f (H) − f (H0) = 2V ∈ B

(H2

t ,Ht+κ

), t ∈ R, κ > 1. If � is

not spherical there are still many appropriate choices for f . For instance ifγ > 0, then the function f (u) = 2(u2 + γ )−1u3, u ∈ R, satisfies all the desiredrequirements. Indeed in such a case one has on H2 the following equalities

f (H) − f (H0)

= 2V − 2γ[(H2 + γ )−1 H − (

H20 + γ

)−1H0

]

= 2V − 2γ (H2 + γ )−1V + 2γ (H2 + γ )−1(H0V + V H0 + V2

) (H2

0 + γ)−1

H0,

and thus f (H) − f (H0) also extends to an element of B(H2

t ,Ht+κ

), t ∈ R, κ >1,

due to Lemma 3.2 and the assumptions on V.

The next Theorem provides a rigorous meaning to the anisotropic Lavine’sformula (1.3).

Theorem 4.5 Let � satisfy Assumption 2.1. Let V satisfy Assumption 3.1 withκ > 4. Suppose that Assumption 4.2 is verified. Then one has for each ϕ ∈ Ds

with s > 2

limr→∞ τr(ϕ) =

∫ ∞

−∞ds


−1/2ϕ,V�, f e−isH W− f (H0)−1/2ϕ

⟩2,−2,

(4.21)

where 〈 · , · 〉2,−2 : H2 × H−2 → C is the anti-duality map between H2 and H−2.

Proof

(1) Set W(t) := eitH e−itH0 , and let ψ := η(H)ψ , where η ∈ C∞0 ((0, ∞) \

σpp(H)) and ψ ∈ D(D�). We shall prove that ‖D�W(t)∗ψ‖ � c, with cindependent of t. Due to Lemma 2.4.(b) and Lemma 4.1.(c) one has

‖D�W(t)∗ψ‖ = ∥∥ e−itH0 D� eitH0 e−itH η(H)(D� + i)−1ψ1

∥∥� |t|∥∥{ f (H) − f (H0)} e−itH η(H)(D� + i)−1ψ1

∥∥++ ∥∥{D� − t f (H)} e−itH η(H)(D� + i)−1ψ1

∥∥, (4.22)

where ψ ≡ η(H)(D� + i)−1ψ1. Let z ∈ C \ {σ(H0) ∪ σ(H)} and setη(H) := (H − z)2η(H). Then Lemmas 2.4.(a), 3.2, and 4.3.(a) imply that

|t|∥∥{ f (H) − f (H0)} e−itH η(H)(D� + i)−1ψ1

∥∥� |t|∥∥{ f (H) − f (H0)}(H − z)−2(D� + i)

∥∥ ·· ∥∥(D� + i)−1 e−itH η(H)(D� + i)−1

∥∥� Const.


Calculations similar to those of Lemma 4.3.(a) show that the second termof (4.22) is also bounded uniformly in t.

(2) Let W(t) and ψ be as in point (1). Lemma 2.4.(b), Lemma 4.1.(c), andcommutator calculations as in (4.20) lead to⟨W(t)∗ψ, D�W(t)∗ψ

⟩ = ⟨ψ, eitH D� e−itH ψ

⟩ − t⟨ψ, eitH f (H0) e−itH ψ

⟩

= 〈ψ, D�ψ〉 −∫ t

0ds

⟨e−isH ψ,V�, f e−isH ψ

⟩2,−2 +

+ t⟨ψ, eitH{ f (H) − f (H0)} e−itH ψ

⟩.

The local H-smoothness of f (H) − f (H0) implies the existence of asequence tn → ∞ as n → ∞ such that

limn→∞ tn

⟨ψ, eitn H{ f (H) − f (H0)} e−itn H ψ

⟩ = 0.

This together with point (1) and the local H-smoothness of V�, f impliesthat

⟨W∗

+ψ, D�W∗+ψ

⟩ = 〈ψ, D�ψ〉 −∫ ∞

0ds


⟩2,−2.

Similarly, one finds

⟨W∗

−ψ, D�W∗−ψ

⟩ = 〈ψ, D�ψ〉 +∫ 0

−∞ds


⟩2,−2,

and thus

⟨W∗

+ψ, D�W∗+ψ

⟩−⟨W∗

−ψ, D�W∗−ψ

⟩=−∫ ∞

−∞ds


⟩2,−2.

(4.23)

Let ϕ ∈ Ds with s > 2. Due to Lemma 4.3.(b) the vector W− f (H0)−1/2ϕ is

of the form η(H)ψ , with η ∈ C∞0 ((0, ∞) \ σpp(H)) and ψ ∈ D(D�). Thus

one can set ψ = W− f (H0)−1/2ϕ in formula (4.23). This gives

⟨Sf (H0)

−1/2ϕ, D� Sf (H0)−1/2ϕ

⟩ − ⟨f (H0)

−1/2ϕ, D� f (H0)−1/2ϕ

⟩

= −∫ ∞

−∞ds


−1/2ϕ,V�, f e−isH W− f (H0)−1/2ϕ

⟩2,−2,

and the claim follows by Theorem 3.5. ��

Remark 4.6 Symmetrised time delay and usual time delay are equal when� is spherical (see formula (1.1)). Therefore in such a case formula (4.21)must reduce to the usual Lavine’s formula. This turns out to be true. Indeed


if � = B and f (u) = 2u, then f (H0) = 2H0, V�, f is equal to the virial V :=2V − i[V, D]a, and formula (4.21) takes the usual form


2

∫ ∞

−∞ds

⟨e−isH W− H−1/2

0 ϕ, V e−isH W− H−1/20 ϕ

⟩2,−2.

In the following remark we give some insight into the meaning of formula(4.21) when � is not spherical. Then we present two simple examples as anillustration.

Remark 4.7 Let V satisfy Assumption 3.1 with κ > 4, and choose a set � �= Bsatisfying Assumption 2.1. In such a case the function fγ (u) := 2(u2 + γ )−1u3,u ∈ R, fulfills the requirements of Definition 2.2 and Assumption 4.2 (seeRemark 4.4). Thus Theorem 4.5 applies, and one has for ϕ ∈ Ds with s > 2

limr→∞ τr(ϕ) = lim

γ↘0

∫ ∞

−∞ds

⟨e−isHW− fγ (H0)

−1/2ϕ,V�, fγ e−isHW− fγ (H0)−1/2ϕ

⟩2,−2.

Now fγ (H0)ϕ converges in norm to 2H0ϕ as γ ↘ 0, so formally one gets theidentity


2

∫ ∞

−∞ds

⟨e−isH W− H−1/2

0 ϕ,V� e−isH W− H−1/20 ϕ

⟩2,−2, (4.24)

where

V� := 2V − i[V, D�]a = 2V − i2

∑j�d

{[V, F� j(P)

] · Q j + Q j ·[V, F� j(P)

]},

and

F� j(P) = −(∂ jG�)(P)P2. (4.25)

The pseudodifferential operator V� generalises the virial V of the isotropiccase. It furnishes a measure of the variation of the potential V along the

Fig. 1 The vector field FEand the sets ∂Er


Fig. 2 The vector field FSand the sets ∂Sr

vector field −F� , which is orthogonal to the hypersurfaces ∂�r due toRemark 2.3. Therefore formula (4.24) establishes a relation between sym-metrised time delay and the variation of V along −F� . Moreover one canrewrite V� as

V� = V + i[V, D − D�]a

= V + i2

∑j�d

{[V,

(P j − F� j(P)

)] · Q j + Q j ·[V,

(P j − F� j(P)

)]},

where P − F�(P) is orthogonal to P due to formulas (4.25) and (2.6). Conse-quently there are two distinct contributions to symmetrised time delay. Thefirst one is standard; it is associated with the term V, and it is due to thevariation of the potential V along the radial coordinate (see [11, Sec. 6] fordetails). The second one is new; it is associated with the term i[V, D − D�]a

and it is due to the variation of V along the vector field x → x − F�(x).

Example 4.8 (Examples in R2) Set d = 2, suppose that V satisfies Assumption

3.1 with κ > 4, and let � be the superellipse E := {(x1, x2) ∈ R

2 | x41 + x4

2 < 1}.

Then one has GE(x) = − 14 ln

(x4

1 + x42

)and (∂ jGE)(x) = −x3

j

(x4

1 + x42

)−1. Thus,due to Remark 4.7 the symmetrised time delay associated with E is (formally)characterised by the pseudodifferential operator

VE = 2V − i2

∑j�d

{[V, FE j(P)

] · Q j + Q j ·[V, FE j(P)

]},

where FE j(P) = P3j P2

(P 4

1 + P 42

)−1 (see Fig. 1).When � is equal to the star-type set

S :={�(θ) eiθ ∈ R

2 | θ ∈ [0, 2π), �(θ) <[

cos(2θ)8 + sin(2θ)8]−1/2

},


one has GS(x) = 72 ln(x2

1 + x22) − 1

2 ln[(x2

1 − x22)

8 + 28(x1x2)8], and a direct cal-

culation using formula (4.25) gives the vector field FS . The result is plotted inFig. 2.

Acknowledgements The author thanks the Swiss National Science Foundation and the Depart-ment of Mathematics of the University of Cergy-Pontoise for financial support.

Appendix

Proof of Lemma 3.2 We first prove that (H − z)−1 extends to an element ofB

(H−2

t ,Ht)

for each t � 0. This clearly holds for t = 0. Since (H0 − z)−1 〈P〉2 =2 + (1 + 2z)(H0 − z)−1 one has by virtue of the second resolvent equation

〈Q〉t (H − z)−1 〈P〉2 〈Q〉−t

= 2 + (1 + 2z) 〈Q〉t (H0 − z)−1 〈Q〉−t −− 〈Q〉t (H0 − z)−1(〈Q〉 V) 〈Q〉−t · 〈Q〉t−1 (H − z)−1 〈P〉2 〈Q〉−t .

(4.26)

If we take t = 1 we find that each term on the r.h.s. of (4.26) is in B(H) due to[2, Lemmas 1 & 2]. Hence, by interpolation, 〈Q〉t (H − z)−1 〈P〉2 〈Q〉−t ∈ B(H)

for each t ∈ [0, 1]. Next we choose t ∈ (1, 2] and obtain, by using the precedingresult and (4.26), that 〈Q〉t (H − z)−1 〈P〉2 〈Q〉−t ∈ B(H) for these values of t.By iteration (take t ∈ (2, 3], then t ∈ (3, 4], etc.) one obtains that 〈Q〉t (H −z)−1 〈P〉2 〈Q〉−t ∈ B(H) for each t > 0. Thus (H − z)−1 extends to an elementof B

(H−2

t ,Ht)

for each t ≥ 0. A similar argument shows that (H − z)−1 alsoextends to an element of B

(H−2

t ,Ht)

for each t < 0. The claim follows then byusing duality and interpolation. ��

Proof of Lemma 3.3 For ϕ ∈ Ds and t ∈ R, we have (see the proof of [7,Lemma 4.6])

(W− − 1) e−itH0 ϕ = −i e−itH∫ t

−∞dτ eiτ H V e−iτ H0 ϕ,

where the integral is strongly convergent. Hence to prove (3.17) it is enough toshow that

∫ −δ

−∞dt

∫ t

−∞dτ

∥∥V e−iτ H0 ϕ∥∥ < ∞ (4.27)

for some δ > 0. If ζ := min{κ, s}, then∥∥ 〈Q〉ζ ϕ

∥∥ < ∞, and V 〈P〉−2 〈Q〉ζ be-longs to B(H) due to Assumption 3.1. Since η(H0)ϕ = ϕ for some η ∈C∞

0 ((0, ∞) \ σpp(H)), this implies that∥∥V e−iτ H0 ϕ

∥∥ � Const.∥∥ 〈Q〉−ζ 〈P〉2 η(H0) e−iτ H0 〈Q〉−ζ

∥∥.


For each ε > 0, it follows from [2, Lemma 9] that there exists a constant c > 0such that

∥∥V e−iτ H0 ϕ∥∥ � c (1 + |τ |)−ζ+ε. Since ζ > 2, this implies (3.17). The

proof of (3.18) is similar. ��

References

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2. Amrein, W.O., Cibils, M.B., Sinha, K.B.: Configuration space properties of the S-matrix andtime delay in potential scattering. Ann. Inst. Henri Poincaré 47, 367–382 (1987)

3. Amrein, W.O., Jacquet, Ph.: Time delay for one-dimensional quantum systems with steplikepotentials. Phys. Rev. A 022106 (2008)

4. Bollé, D., Osborn, T.A.: Time delay in N-body scattering. J. Math. Phys. 20, 1121–1134 (1979)5. Cycon, H.L., Froese, R.G., Kirsch, W., Simon, B.: Schrödinger Operators with Applications to

Quantum Mechanics and Global Geometry. Springer-Verlag, Berlin (1987)6. Gérard, C., Tiedra de Aldecoa, R.: Time-delay and Lavine’s formula. J. Math. Phys. 48, 122101

(2007)7. Jensen, A.: Time-delay in potential scattering theory. Comm. Math. Phys. 82, 435–456 (1981)8. Jensen, A.: A stationary proof of Lavine’s formula for time-delay. Lett. Math. Phys. 7(2),

137–143 (1983)9. Jensen, A.: On Lavine’s formula for time-delay. Math. Scand. 54(2), 253–261 (1984)

10. Jensen, A., Nakamura, S.: Mapping properties of wave and scattering operators for two-bodySchrödinger operators. Lett. Math. Phys. 24, 295–305 (1992)

11. Lavine, R.: Commutators and local decay. In: Lavita, J.A., Marchand, J.P. (eds.) ScatteringTheory in Mathematical Physics, pp. 141–156. D. Reidel, Dordrecht (1974)

12. Martin, P.A.: Scattering theory with dissipative interactions and time delay. Nuovo CimentoB 30, 217–238 (1975)

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14. Mohapatra, A., Sinha, K.B., Amrein, W.O.: Configuration space properties of the scatteringoperator and time delay for potentials decaying like |x|−α, α > 1. Ann. Inst. H. Poincaré Phys.Théor. 57(1), 89–113 (1992)

15. Nakamura, S.: Time-delay and Lavine’s formula. Comm. Math. Phys. 109(3), 397–415 (1987)16. Narnhofer, H.: Another definition for time delay. Phys. Rev. D 22(10), 2387–2390 (1980)17. Narnhofer, H.: Time delay and dilation properties in scattering theory. J. Math. Phys. 25(4),

987–991 (1984)18. Perry, P., Sigal, I.M., Simon, B.: Spectral analysis of N-body Schrödinger operators. Ann. of

Math. (2) 114(3), 519–567 (1981)19. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators.

Academic Press, New York (1978)20. Sassoli de Bianchi, M., Martin, P.A.: On the definition of time delay in scattering theory. Helv.

Phys. Acta 65(8), 1119–1126 (1992)21. Smith, F.T.: Lifetime matrix in collision theory. Phys. Rev. 118, 349–356 (1960)22. Tiedra de Aldecoa, R.: Time delay and short-range scattering in quantum waveguides. Ann.

Henri Poincaré 7(1), 105–124 (2006)23. Wang, X.P.: Time-delay operator for a class of singular potentials. Helv. Phys. Acta 60(4),

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Differential Equations 13(2), 223–259 (1988)


Estimates for Entries of Matrix Valued Functionsof Infinite Matrices

M. I. Gil’

Received: 5 March 2008 / Accepted: 15 July 2008 /Published online: 14 August 2008© Springer Science + Business Media B.V. 2008

Abstract Sharp upper estimates for the absolute values of entries of matrixvalued functions of infinite matrices, as well as two sided estimates for theentries of matrix valued functions of infinite M-matrices (monotone matrices)are derived. They give us bounds for the lattice norms of matrix valuedfunctions and positivity conditions for functions of M-matrices. In addition,some results on perturbations and comparison of matrix functions are proved.Applications of the obtained estimates to the Hille-Tamarkin matrices anddifferential equations are also discussed.

Keywords Infinite matrices · Matrix valued functions · Monotone matrices ·Positivity · Hille-Tamarkin matrices · Norm estimates · Perturbations ·Comparison

Mathematics Subject Classifications (2000) 47A56 · 47A60

1 Introduction and Statement of the Main Result

In the book [8], I.M. Gel’fand and G.E. Shilov have established an estimatefor the norm of a regular matrix valued function in connection with theirinvestigations of partial differential equations. However that estimate is notsharp, it is not attained for any matrix. The problem of obtaining a preciseestimate for the norm of a matrix function has been repeatedly discussed in

This research was supported by the Kamea fund of the Israel

M. I. Gil’ (B)Department of Mathematics, Ben Gurion University of the Negev,P.O. Box 653, Beer-Sheva 84105, Israele-mail: [email protected]

176 M. I. Gil’

the literature, cf. [5]. In the paper [10] (see also [13]) the author has deriveda precise estimate for the Euclidean norm which is attained in the case ofnormal matrices. But that estimate requires bounds for the eigenvalues. Inthis paper we derive sharp estimates for the absolute values of entries ofmatrix valued functions of infinite matrices. In addition, two sided estimatesfor the entries of functions of infinite monotone matrices are derived. Theseestimates give us bounds for the lattice norms of matrix valued functions, andenable us to investigate perturbations of matrix valued functions and comparethem. Besides, bounds for the eigenvalues are not required. Applications ofthe obtained estimates to differential equations are also discussed.

Our results supplement the very interesting recent investigations of matrixvalued functions [7, 17, 24], M-matrices [1, 2, 23], infinite matrices and theirapplications [3, 11, 16, 19, 20, 25]. Of course we cannot survey the whole subjecthere and refer the reader to the above pointed papers and references giventherein.

Everywhere below, X is a complex Banach space of scalar sequences h ={hk} with a norm ‖.‖. In particular, X = l p (p � 1) with the norm

‖h‖l p =[ ∞∑

k=1

|hk|p

]1/p

(1 � p < ∞) and ‖h‖l∞ = supk

|hk|.

The unit operator in X is denoted by I. Let σ(A) be the spectrum of a linearoperator A and Rz(A) = (A − zI)−1 (z �∈ σ(A)) be the resolvent of A; rs(A)

denotes the spectral radius of A.Everywhere below, A = (a jk)

∞j,k=1 is a matrix representing a bounded

linear operator in the standard basis of X whose diagonal part is D =diag [a11, a22, ... ] and off diagonal is V := A − D. That is, the entries v jk ofV are v jk = a jk ( j �= k) and v jj = 0 ( j, k = 1, 2, ...). Clearly,

rs(D) = supj=1,2,....

|a jj|.

Denote by co(D) the closed convex hull of the diagonal entries a11, a22, ... andput

�(A) := {z ∈ C : |z| � rs(D) + ‖V‖}.Clearly co(D) ⊂ �(A). In the sequel f (λ) is a scalar function holomorphic ona neighborhood of �(A). The matrix valued function f (A) is defined by

f (A) = − 1

2π i

∫�

f (λ)Rλ(A)dλ, (1.1)

where � ⊂ �(A) is a closed contour surrounding σ(A). We put |A| =(|a jk|)∞j,l=1, i.e. |A| is the matrix whose entries are absolute values of A inthe standard basis. We also write C � 0 if all the entries of a matrix C arenonnegative. If C and B are two matrices, then we write C � B if C − B � 0.The same sense have the symbols |h|, h � 0 and h � g for sequences h, g ∈ X.

Estimates for entries of matrix valued functions of infinite matrices 177

Moreover, it is assumed that any considered number series converges andany operator series strongly converges. Now we are in a position to formulateour main result.

Theorem 1.1 With the notation

γk(A) := supz∈co (D)

| f (k)(z)|k! (k = 0, 1, 2, ...),

the inequality

| f (A)| �∞∑

k=0

γk|V|k (1.2)

holds.

This theorem is proved in the next section. It generalizes the main resultfrom [14].

In the sequel the norm in X is a lattice norm. That is, ‖ f‖ � ‖h‖ whenever| f | � |h| for f, h ∈ X, cf. [18, p. 6]. By ‖A‖ the operator norm of A is denoted:‖A‖ := supx∈X ‖Ax‖/‖x‖. So ‖h‖ = ‖|h|‖ and ‖A‖ � ‖|A|‖. Theorem 1.1implies

Corollary 1.2 The inequality

‖ f (A)‖ �∞∑

k=0

γk‖|V|k‖ (1.3)

is valid.

Theorem 1.1 and Corollary 1.2 are sharp: inequalities (1.2) and (1.3) becomeequalities, provided A is diagonal: V = 0 and the set of the diagonal entries{a11, a22, ...} is convex.

For instance, let X = l p for some finite p > 1, and V be a Hille-Tamarkinmatrix, cf. [22]. Namely,

Np(V) :=⎛⎝ ∞∑

j=1

⎡⎣ ∞∑

k=1, k�= j

|a jk|q⎤⎦

p/q⎞⎠

1/p

< ∞ (1.4)

with 1/p + 1/q = 1. So under (1.4), A = (a jk) represents a linear operator inl p which is bounded, provided D is bounded. As it is well-known and ‖|V|‖ �Np(V), cf. [22]. Now (1.3) yields

Corollary 1.3 Let X = l p and condition (1.4) hold. Then

‖ f (A)‖l p �∞∑

k=0

γk Nkp(V).

178 M. I. Gil’

Note that in the paper [12], estimates for the norm of the powers of Hille-Tamarkin quasinilpotent matrices have been established.

2 Proof of Theorem 1.1

First assume that A is a finite matrix. By the equality A = D + V we get

Rλ(A) ≡ (A − Iλ)−1 = (D + V − λI)−1 = (I + Rλ(D)V)−1 Rλ(D),

provided the norm of Rλ(D)V is less than one. But

‖Rλ(D)V‖ � ‖V‖infk |λ − akk| � ‖V‖

|λ| − rs(D)< 1

for any λ with |λ| > rA := ‖V‖ + rs(D). Thus,

f (A) = − 1

2π i

∫|λ|=r

f (λ)Rλ(A)dλ =∞∑

k=0

Ck (r = rA + ε, ε > 0), (2.1)

where

Ck = (−1)k+1 1

2π i

∫|λ|=r

f (λ)(Rλ(D)V)k Rλ(D)dλ.

Since D is a diagonal matrix with respect to basis {ek}, we can write out

Rλ(D) =n∑

j=1

Q j

λ j − λ(λ j = a jj),

where Qk = (., ek)ek. We thus have

Ck =n∑

j1=1

Q j1 Vn∑

j2=1

Q j2 V . . . Vn∑

jk=1

Q jk+1 I j1 j2... jk+1 . (2.2)

Here

I j1... jk+1 = (−1)k+1

2π i

∫|λ|=r

f (λ)dλ

(λ j1 − λ) . . . (λ jk+1 − λ).

Lemma 1.5.1 from [13] gives us the inequalities

|I j1... jk+1 | � γk ( j1, j2, ..., jk+1 = 1, ..., n).

Hence, by (2.2)

|Ck| � γk

n∑j1=1

Q j1 |V|n∑

j2=1

Q j2 |V| . . . |V|n∑

jk=1

Q jk+1 .

Butn∑

j1=1

Q j1 |V|n∑

j2=1

Q j2 |V| . . . |V|n∑

jk=1

Q jk+1 = |V|k.


Thus

Ck � γk|V|k.Now (2.1) implies

| f (A)| �∞∑

k=0

|Ck| �∞∑

k=0

γk|V|k.

So in the finite dimensional case the theorem is proved.Now let A be infinite dimensional and Pn the projection onto subspace

generated by the first n elements of the standard basis. Then the finitedimensional matrices An = Pn APn strongly converge to A. Let Dn = Pn Dand Vn = PnV Pn be the diagonal and off-diagonal parts of An, respectively.Then as it is above proved, the required relations hold with A = An. Butf (An) → f (A) in the strong topology [6]. So each entry of f (An) convergesto the corresponding entry of f (A). This proves the result.

3 Functions of M-Matrices

A real matrix A in X is said to be an M-matrix (a monotone matrix) if its offdiagonal part V is nonnegative, cf. [4]. Put

a = infj=1,2,...

a jj, b = supj=1,2,...

a jj.

In this section A is an M-matrix, f (λ) is holomorphic on a neighborhood of�(A), as above, and, in addition, it is real on [a, b ].

Theorem 3.1 With the notations

αk := infa�x�b

f (k)(x)

k! , βk = supa�x�b

f (k)(x)

k! (k = 0, 1, 2, ...),

the inequalities

f (A) �∞∑

k=0

αkVk, (3.1)

and

f (A) �∞∑

k=0

βkVk (3.2)

are valid. In particular, if αk � 0 (k = 0, 1, 2, ...), then f (A) � 0.

Proof First let A be n-dimensional. Again use relations (2.1), (2.2). Lemma1.5.2 from [13] gives us the equality

I j1... jk+1 = f (k)(θ)

k! (a � θ � b). (3.3)

180 M. I. Gil’

So

αk � I j1... jk+1 � βk.

Since V � 0, Q jV Qk = a jk(., ek)e j, andn∑

j1=1

Q j1 Vn∑

j2=1

Q j2 V . . . Vn∑

jk=1

Q jk = Vk,

we have

Ck � αk

n∑j1=1

Q j1 Vn∑

j2=1

Q j2 V . . . Vn∑

jk=1

Q jk = αkVk.

Similarly, Ck � βkVk. Hence (2.1) proves the theorem in the finite dimensionalcase.

Now let A be infinite dimensional and Pn the projection onto subspacegenerated by the first n elements of the standard basis. Then the finitedimensional matrices An = Pn APn strongly converge to A. Let Dn = Pn Dand Vn = PnV Pn be the diagonal and off-diagonal parts of An, respectively.Now taking into account the above proved result and that f (An) → f (A) inthe strong topology [6] we can assert that each entry of f (An) converges to thecorresponding entry of f (A). This proves the result.

Recall that the norm in X is assumed to be lattice. Theorem 3.1 yields thefollowing result.

Corollary 3.2 The inequalities

| f (A)| �∞∑

k=0

νkVk (νk := max{|αk|, |βk|}) (3.4)

and

‖ f (A)‖ �∞∑

k=0

νk‖Vk‖

are true.

Denote by R+(l1) the cone of vectors from l1, whose coordinates in thestandard basis are nonnegative. Thanks to (3.1) we get

Corollary 3.3 Let X = l1 and αk � 0, k = 0, 1, 2, . . . . Then for any h ∈ R+(l1)

we have the inequality

‖ f (A)h‖l1 �∞∑

k=0

αk‖Vkh‖l1 .

Furthermore, inequality (3.4) yields


Corollary 3.4 Let X = l p (1 < p < ∞) and V satisfy condition (1.4). Then

‖ f (A)‖l p �∞∑

k=0

νk Nkp(V).

4 Perturbations of Entire Functions of Hille-Tamarkin Matrices

We need the following result.

Lemma 4.1 Let Z and Y be complex normed spaces with norms ‖.‖Z and‖.‖Y, respectively, and h a Y-valued function defined on Z . Assume thath(C + λB) (λ ∈ C) is an entire function for all C, B ∈ Z . That is, for any φ fromthe space adjoint to Y, the functional < φ, h(C + λB) > is an entire function. Inaddition, let there be a monotone non-decreasing function G : [0, ∞) → [0, ∞),such that ‖h(C)‖Y � G(‖C‖Z ) (C ∈ Z ). Then

‖h(C)−h(B)‖Y �‖C−B‖Z G(1+1/2‖C+B‖Z +1/2‖C − B‖Z ) (C, B ∈ Z ).

For the proof see [15].Let A = (a jk) and A = (a jk) be matrices representing bounded operators in

a Banach space X with a lattice norm ‖.‖. Thanks to Corollary 1.2, for anyentire f ,

‖ f (A)‖ �∞∑

k=0

γk(A)‖|V|‖k �∞∑

k=0

γk(A)‖|A|‖k. (4.1)

Now let X = l p for some finite p > 1, and A be a Hille-Tamarkin matrix:

Np(A) =⎛⎝ ∞∑

j=1

[ ∞∑k=1

|a jk|q]p/q

⎞⎠

1/p

< ∞.

Then ‖|A|‖l p � Np(A) and

γk(A) � sup|z|�Np(A)

| f (k)(z)|k! (k = 0, 1, 2, ...).

The previous lemma and (4.1) imply

Theorem 4.2 Let A and A be Hille-Tamarkin matrices in l p, 1 < p < ∞. As-sume that f (λ) (λ ∈ C) is an entire function. Then

‖ f (A)− f (A)‖l p � Np(A− A)

∞∑k=0

ηk(A, A)(1+Np(A+ A)/2+Np(A− A)/2)k

182 M. I. Gil’

where

ηk(A, A) := 1

k! sup|z|�1+Np(A+A)/2+Np(A−A)/2

| f (k)(z)|.

5 Comparison of Functions of M-Matrices

In this section A = (a jk) and A = (a jk) are M-matrices. So A = D + V, whereD and V are the diagonal and off-diagonal parts of A, respectively. Recall thatD and V are the diagonal and off-diagonal parts of A, respectively.

Lemma 5.1 Let

a � akk = akk � b but a jk � a jk ( j �= k, j, k = 1, 2, ...). (5.1)

In addition, let f be holomorphic on a neighborhood of �(A) and positive on[a, b ]: αk � 0, k = 1, 2, . . . . Then f (A) � f (A) � 0. Moreover,

∞∑k=0

αk(Vk − Vk) � f (A) − f (A) �∞∑

k=0

βk(Vk − Vk).

Proof First, let A, A be n-dimensional (n < ∞). We have by (2.1),

f (A) − f (A) =∞∑

k=0

Ck,

where

Ck =n∑

j1, j2,... jk+1=1

(Q j1 V Q j2 V . . . V Q jk+1 − Q j1 V Q j2 V . . . V Q jk+1)I j1 j2... jk+1

Under assumption (5.1), according to (3.3) we have I j1 j2... jk+1 � 0. Thus Ck � 0.Moreover, since V � V,

Ck �βk

n∑j1, j2,... jk+1=1

(Q j1 VQ j2 V . . . VQ jk+1 −Q j1 VQ j2 V . . . VQ jk+1)=βk(Vk−Vk),

and Ck � αk(Vk − Vk). So in the finite dimensional case, the lemma is proved.Taking into account that any bounded operator is a strong limit of finitedimensional operators, we arrive at the required result.

Furthermore, for real constants a and b, let

b � akk � akk � a but a jk = a jk ( j �= k, j, k = 1, 2, ...). (5.2)

Put r := max{rs(D), rs(D)} + ‖V‖ and

dk := max1� j�k

a jj − a jj.


Lemma 5.2 Under conditions (5.2), let f be holomorphic on a neighborhood of{z ∈ C : |z| � r} and positive on [a, b ]:

αk := infa�x�b

f (k+1)(x)

k! � 0 (k = 0, 1, 2, ...).

Then f (A) � f (A) � 0 and

∞∑k=0

αkdkVk � f (A) − f (A) �∞∑

k=0

dkβkVk

where

βk := supa�x�b

f (k+1)(x)

k! (k = 0, 1, 2, ...).

Proof First, let A, A be n-dimensional. We have by (2.1),

f (A) − f (A) =∞∑

k=0

Tk,

where

Tk :=n∑

j1, j2,... jk+1=1

Q j1 V Q j2 V . . . V Q jk+1(I j1 j2... jk+1 − I j1 j2... jk+1).

Here

I j1, j2,..., jk+1 = (−1)k

2π i

∫|λ|=r

f (λ)dλ

(a j1 j1 − λ) . . . (a jk+1 jk+1 − λ)(r > r).

As it is well-known [9, Section I.4.3], I j1, j2,..., jk+1 is the k-order divided differ-ence of f in the points a j1 j1 , . . . , a jk+1 jk+1 . By the Herimit integral representation[21, p. 4], we have

I j1, j2,..., jk+1 =∫ 1

0

∫ t1

0...

∫ tk

0f (k)[(1 − t1)a j1 j1 + (t1 − t2)a j2 j2 + ...

+ a jk+1 jk+1 tk+1]dtk+1 ... dt1. (5.3)

The same representation has I j1, j2,..., jk+1 with akk instead akk. By (5.3) we obtain

I j1, j2,..., jk+1 � I j1, j2,..., jk+1 ,

provided f n+1(x) � 0, x ∈ [a, b ]. So Tk � 0 and thus f (A) � f (A).Furthermore, for real points x1, ..., xk, y1, ..., yk with x j � y j, j � k, we can

write out

f (k)[(1 − t1)x1 + (t1 − t2)x2 + ... + xktk] − f (k)[(1 − t1)y1 + (t1 − t2)y2 + ...

... + yktk] = f (k+1)(θ)� (minj�k

y j � θ � maxj�k

x j),

184 M. I. Gil’

where

� = (1 − t1)(x1 − y1) + (t1 − t2)(x2 − y2) + ... + (xk − yk)tk � dk (t j � 1).

Hence, taking into account that∫ 1

0

∫ t1

0...

∫ tk−1

0dtk ... dt1 = 1

k!we get

αkdk � I j1, j2,..., jk+1 − I j1, j2,..., jk+1 � dkβk.

This implies the required result in the finite dimensional case. Reducing theobtained result to the infinite dimensional case as in the previous lemma, weget the required result.

6 Examples

Example 6.1 Let f (A) = eAt (t � 0). With the notations of Section 1 we canwrite out

f (k)(λ) = tkeλt; γk = tk

k!eα(D)t

where α(D) = maxk Re akk. So by Theorem 1.1,

|eAt| � eα(D)t∞∑

k=0

tk

k! |V|k = e(α(D)I+|V|)t (t � 0).

Now let A be an M-matrix. Then with the notations of Section 3,

αk = tk

k!eat, βk = tk

k!ebt, k = 0, 1, 2, ....

So by Theorem 3.1,

∞∑k=0

tk

k! Vk � eAt � ebt∞∑

k=0

tk

k! Vk (t � 0).

Or

e(aI+V)t � eAt � e(bI+V)t (t � 0).

Example 6.2 Let f (A) = sin (At) (t � 0). Then

f (2k)(λ) = t2k(−1)ksin (λt); f (2k+1)(λ) = t2k+1(−1)kcos (λt) (k = 0, 1, 2, ...).


Let the diagonal matrix D be real. Then γk � 1k! t

k. So by Theorem 1.1,

|sin (At)| �∞∑

k=0

1

k! tk|V|k = et|V|. (6.1)

Consider the second order nonlinear differential equation

d2xdt2

+ A2x(t) = F(x(t)) (t > 0) (6.2)

dx(0)

dt= x(0) = 0, (6.3)

where A is a real matrix, F : Rn → R

n is a continuous function, satisfying

‖F(h)‖ � v + q‖h‖ (v, q = const; h ∈ X)

with some lattice norm. Problem (6.2), (6.3) is equivalent to the followingequation:

x(t) =∫ t

0sin A(t − s)F(x(s))ds.

So

‖x(t)‖ �∫ t

0‖sin A(t − s)‖(q‖x(s)‖ + v)ds

with an arbitrary ideal norm. Put y(t) = ‖x(t)‖. Then by (6.1),

y(t) � v f (t) + q∫ t

0e‖|V|‖(t−s)y(s)ds

where

f (t) =∫ t

0e‖|V|‖sds.

Hence taking into account that f monotonically increases, we get by theGronwall inequality,

y(t) � v f (t)exp [qf (t)].Such estimates are important, in particular, in the theory of oscillations, cf. [5].

References

1. Ameur, Y., Kaijser, S., Silvestrov, S.: Interpolation classes and matrix monotone functions. J.Operator Theory 57(2), 409–427 (2007)

2. Bapat, R.B., Catral, M., Neumann, M.: On functions that preserve M-matrices and inverseM-matrices. Linear and Multilinear Algebra 53(3), 193–201 (2005)

3. Candan, M., Solak, I.: On some difference sequence spaces generated by infinite matrices. Int.J. Pure Appl. Math. 25(1), 79–85 (2005)

4. Collatz, L.: Functional Analysis and Numerical Mathematics. Academic, New York (1966)

186 M. I. Gil’

5. Daleckii, Y.L., Krein, M.G.: Stability of solutions of differential equations in Banach space.American Mathematical Society, Providence (1974)

6. Dunford, N., Schwartz, J.T.: Linear Operators, part I. Interscience, New York (1966)7. Fritzsche, B., Kirstein, B., Lasarow, A.: Orthogonal rational matrix-valued functions on the

unit circle: recurrence relations and a Favard-type theorem. Math. Nachr. 279(5–6), 513–542(2006)

8. Gel’fand, I.M., Shilov, G.E.: Some Questions of Theory of Differential Equations. Nauka,Moscow (in Russian) (1958)

9. Gel’fond, A.O.: Calculations of Finite Differences. Nauka, Moscow (in Russian) (1967)10. Gil’, M.I.: Estimates for norm of matrix-valued functions. Linear and Multilinear Algebra 35,

65–73 (1993)11. Gil’, M.I.: Spectrum localization of infinite matrices. Math. Phys. Anal. Geom. 4(4), 379–394

(2001)12. Gil’, M.I.: Invertibility and spectrum of Hille-Tamarkin matrices. Math. Nachr. 244, 1–11

(2002)13. Gil’, M.I.: Operator functions and localization of spectra. In: Lectures Notes In Mathematics

vol. 1830. Springer, Berlin (2003)14. Gil’, M.I.: Estimates for absolute values of matrix functions. Electron. J. Linear Algebra 16,

444–450 (2007)15. Gil’, M.I.: Inequalities of the Carleman type for Schatten-von Neumann operators. Asian-

European J. Math. 1(2), 1–11 (2008)16. Golinskii, L.: On the spectra of infinite Hessenberg and Jacobi matrices. Mat. Fiz. Anal. Geom.

7(3), 284–298 (2000)17. Lasarow, A.: Dual Szego pairs of sequences of rational matrix-valued functions. Int. J. Math.

Math. Sci. 2006(5), 37 (2006)18. Meyer-Nieberg, P.: Banach Lattices. Springer, Berlin (1991)19. Mittal, M.L., Rhoades, B.E., Mishra, V.N., Singh, U.: Using infinite matrices to approximate

functions of class Lip(α, p) using trigonometric polynomials. J. Math. Anal. Appl. 326(1), 667–676 (2007)

20. Nagar, D.K., Tamayo-Acevedo, A.C.: Integrals involving functions of Hermitian matrices. FarEast J. Appl. Math. 27(3), 461–471 (2007)

21. Ostrowski, A.M.: Solution of Equations in Euclidean and Banach Spaces. Academic, NewYork (1973)

22. Pietsch, A.: Eigenvalues and s-Numbers. Cambridge University Press, Cambridge (1987)23. Singh, M., Vasudeva, H.L.: Monotone matrix functions of two variables. Linear Algebra Appl.

328(1–3), 131–152 (2001)24. Werpachowski, R.: On the approximation of real powers of sparse, infinite, bounded and

Hermitian matrices. Linear Algebra Appl. 428(1), 316–323 (2008)25. Zhao, X., Wang, T.: The algebraic properties of a type of infinite lower triangular matrices

related to derivatives. J. Math. Res. Exposition 22(4), 549–554 (2002)

Math Phys Anal Geom (2008) 11:187–364DOI 10.1007/s11040-008-9042-y

Rational Functions with a General Distributionof Poles on the Real Line Orthogonal with Respectto Varying Exponential Weights: I

K. T.-R. McLaughlin · A. H. Vartanian · X. Zhou

Received: 19 January 2008 / Accepted: 10 April 2008 / Published online: 18 October 2008© Springer Science + Business Media B.V. 2008

Abstract Orthogonal rational functions are characterized in terms of a familyof matrix Riemann–Hilbert problems on R, and a related family of energyminimisation problems is presented. Existence, uniqueness, and regularityproperties of the equilibrium measures which solve the energy minimisationproblems are established. These measures are used to derive a family of‘model’ matrix Riemann–Hilbert problems which are amenable to asymptoticanalysis via the Deift–Zhou non-linear steepest-descent method.

Keywords Asymptotics · Equilibrium measures · Orthogonal rationalfunctions · Riemann–Hilbert problems · Variational problems

Mathematics Subject Classifications (2000) Primary 42C05;Secondary 30E20 · 30E25 · 30C15

K. T.-R. McLaughlinDepartment of Mathematics, The University of Arizona,617 N. Santa Rita Ave., P. O. Box 210089,Tucson, AZ 85721-0089, USAe-mail: [email protected]

A. H. Vartanian (B)Department of Mathematics, College of Charleston,66 George Street, Charleston, SC 29424, USAe-mail: [email protected]

X. ZhouDepartment of Mathematics, Duke University,Box 90320, Durham, NC 27708-0320, USAe-mail: [email protected]

188 K. T.-R. McLaughlin et al.

1 Introduction, Background, and Summary of Results

1.1 Introduction

This manuscript considers a number of questions related to general orthogonalrational functions (ORFs). These may be thought of as a natural generalisationof orthogonal polynomials. Just as orthogonal polynomials are obtained via theGram-Schmidt orthogonalisation procedure (with respect to a given measure)applied to the sequence {1, z, z2, . . . }, ORFs may be obtained via the Gram-Schmidt procedure, but applied to a more general, pre-determined sequenceof simple rational functions.

A basic example follows. Using the measure defined on R,

dψ(z)=e−z2e−z−2

e−(z−1)−2e−(z−2)−2

dz,

one may apply the Gram-Schmidt procedure to a sequence of rational func-tions with poles at 0, 1, and 2. Clearly, the order of this initial sequence ofrational functions matters. This infinite sequence of rational functions will bedescribed by first specifying a finite pole sequence, {0, 0, 1, 0, 1, 2}, which yieldsthe following seven rational functions:

{1, z−1, z−2, (z−1)−1, z−3, (z−1)−2, (z−2)−1}.The infinite sequence of rational functions is specified by repeating this polesequence, and augmenting the order of the pole at 0, 1, and 2 every timethat pole is encountered. Thus, after the first seven rational functions specifiedabove, the next six members of the infinite sequence are

{z−4, z−5, (z−1)−3, z−6, (z−1)−4, (z−2)−2},and this continues, with the rational functions appearing in six-tuples. Theresult of applying the Gram-Schmidt procedure to this sequence of functions,using the aforementioned measure, is a sequence of ORFs. More precisely,the nth term in the new sequence is in the linear span of the first n membersof the original sequence of simple rational functions, and is orthogonal to all‘previous’ members of the simple sequence.

The general definition of a sequence of ORFs is quite cumbersome. Thereason is that one must specify the pole sequence, and if a pole should happento repeat, then the associated ‘simple rational function’ must be linearlyindependent from all previous members of the sequence; so typically, the orderof the pole increases with each occurrence of that pole. Although one mightmerely describe the sequence of rational functions abstractly, it is useful, andnecessary for a Riemann–Hilbert characterization, to explicitly annotate thislevel of detail. This manuscript only deals with the case of a finite sequence ofpoles, which must necessarily repeat: the more general situation in which thepole sequence is infinite, is not considered. In Subsection 1.2, then, the readerwill find a description of the pole sequence that requires effort to absorb.It is important to note that the Riemann–Hilbert problems described here,

Rational Functions with a General Distribution of Poles on the Real Line Orthogonal... 189

which characterize the ORFs, depend on this level of detail regarding thepole sequence.

The uninitiated reader might wonder why one would consider such ageneralisation. Although such questions may be delegated to a matter of taste,we describe here one application [1], and refer the reader to the monograph[2] for a complete list of applications.

Suppose one would like to find a multi-point rational approximant to afunction which is given as the Stieltjes transform of the measure defined above:

Fψ(z)=∫

R

dψ(ξ)z−ξ .

Now, in a vicinity of each of the points z=0, z=1, and z=2 the functionFψ(z) possesses a complete asymptotic description. The (n−1,n) 3-pointPadé approximation to Fψ(z) is a rational function of the form Un(z)/Vn(z),with deg(Un)�n−1 and deg(Vn)=n, which matches each of the asymptoticexpansions to specified degrees at each of the points 0, 1, and 2. A challengingquestion is to describe the asymptotic behaviour of the sequence of Padéapproximants as n→∞.

The amazing connection is this: the approximants themselves, as well as theerror in approximation, are described explicitly in terms of the ORFs! For aprecise description of these (and more) connections, we refer the reader to[1]. The goal of describing the asymptotic behaviour of the sequence of Padéapproximants may be achieved if one has a complete asymptotic description(as n→∞) of the ORFs.

Toward the ultimate goal of a complete asymptotic description of ORFswith respect to varying exponential weights, the purpose of this manuscriptis to lay the foundation: we describe families of Riemann–Hilbert problemswhich characterize the ORFs. In addition, we derive variational problemswhich are central to the subsequent asymptotic analysis of the Riemann–Hilbert problems. Since the variational problems contain external fields withsingular points, we also establish existence, uniqueness, and regularity of theassociated equilibrium measures.

Historically, the analysis of ORFs played a major rôle in a variety of so-called moment problems, in which one is given a list of p pole locations, and p

infinite sequences of real numbers, and one asks for the existence of a measurewhose moments relative to each pole are prescribed by one of the infinitesequences of real numbers. Two examples are as follows:

(i) the extended Hamburger moment problem (EHMP) [3, 4]: given pdistinct (fixed) real numbers a1, a2, . . . , ap and p sequences of finite real

numbers{

EHc (i)k

}k∈N

, i=1, 2, . . . , p, find necessary and sufficient condi-

tions for the existence of a distribution function1 μEHMP on (−∞,+∞)

1A real-valued, bounded, non-decreasing function with infinitely many points of increase on itsdomain of definition.


such that∫ +∞−∞ dμEH

MP(τ )=1 andEHc (i)k =∫ +∞

−∞ (τ−ai)−k dμEH

MP(τ ), k∈N, i=1, 2, . . . , p;

(ii) the extended Stieltjes moment problem (ESMP) [5]: given q distinct(fixed) real numbers a1, a2, . . . , aq ordered by size (e.g., a1< a2 <

· · ·< aq), agree to call the real interval [c,d] a Stieltjes interval for thefinite point set {a1, a2, . . . , aq} if (c,d) ∩ {a1, a2, . . . , aq}=∅. Given q

sequences of finite real numbers{

ESc (r)j

}j∈N

, r=1, 2, . . . ,q, and a finite real

numberESc0, find necessary and sufficient conditions for the existence of a

distribution functionμESMP, with all its points of increase on a given Stieltjes

interval [c,d], such thatESc0=

∫ dc dμES

MP(τ ) andESc (r)j =∫ d

c (τ−ar)− j dμES

MP(τ ),j∈N, r=1, 2, . . . ,q.

Two other, related problems of analysis are the Pick–Nevanlinna problem andthe frequency analysis problem, which are now described:

(iii) consider a sequence of complex points {zn}n∈N such that zn coalesce intoa finite number, p, say, of distinct real points a1, a2, . . . , ap according tothe prescription zpq+1= a1, zpq+2= a2, . . . , zpq+p= ap, q∈Z

+0 :=N ∪ {0}.

The corresponding Pick–Nevanlinna problem [6] can be formulated

thus: given the p sequences of numbers{γ(m)l

}l∈Z

+0

, m=1, 2, . . . , p,

find a Nevanlinna function2 X(z) which has the asymptotic expansionsX(z)=Rm,δ�z→am

∑l∈Z

+0γ(m)l (z−am)

l, m=1, 2, . . . , p, where Rm,δ :={z∈C;δ<Arg(z−am)<π−δ}, δ>0. (As shown in [6], this Pick–Nevanlinnaproblem is related to the EHMP and certain (weak) multi-point Padéapproximation problems.);

(iv) the so-called frequency analysis problem (see, e.g., the review article[7]) consists of determining the unknown frequencies ω j, j=1, 2, . . . , J,

J∈N, from a discrete time signal{

xN(m)=∑Jj=−J α jeimω j

}N−1

m=0of ob-

served values, where (N�) N is the ‘sample size’, α0 �0, 0 �=α− j=α j∈C,0=:ω0<ω1<ω2< · · ·<ωJ<π , and ω− j=−ω j∈R, j=1, 2, . . . , J. Thefrequency analysis problem has been dealt with by exploiting the factthat, under various conditions, certain roots of the Szegö polynomials[8] on the unit circle (T :={z∈C; |z|=1}) converge, as N→∞, to the‘frequency points’ eiω j and 1, j=±1,±2, . . . ,±J, and the ‘remaining—uninteresting—roots’ are bounded away from T as N→∞. Recently,however, the question of the generalisation, or extension, of this theory,where, in lieu of Szegö polynomials, certain rational functions replacepolynomials, has been raised [9–11].

2A function X(z) which is analytic for z∈C+ :={z∈C; Im(z)>0} with Im(X(z))�0 is called aNevanlinna function.


As shown in [1, 3–6, 9–11], the key ingredient subsumed in the analysis of themulti-point rational approximation problem and the above-mentioned prob-lems (i)–(iv) (by no means an exhaustive list!) is to consider, in lieu of (Szegö)orthogonal polynomials, suitably orthogonalised rational functions with nopoles in the extended complex plane (C :=C ∪ {∞}) outside of a prescribed(fixed) pole set, called, generically, ORFs. Historically, and to the best of theauthors’ knowledge as at the time of the presents, M. M. Djrbashian instigatedthe study of ORFs; in particular, the investigation of ORFs on T (see, e.g.,[12–16]). Since then, a monumental, systematic and comprehensive study ofORFs has been undertaken by A. Bultheel, P. González-Vera, E. Hendriksen,and O. Njåstad (see the recent monograph [2] and the plethora of referencestherein; see, also, [17–23]). There has also been concomitant recent progressin the matrix generalisation of the scalar ORF theory on T (see, e.g., [24, 25]).ORFs have applications, and potential applications, to generalised problems inmoment theory (see, e.g., [26, 27]), interpolation theory and numerical analysis(see, e.g., [28–35]), control theory [36], multi-point Padé-type approximants[37–39], the theory of Cauchy and Cauchy-type integrals and their derivativeswith meromorphic density (see, e.g., [40–47]), an extended Toda lattice [48],uniform approximation of sgn(x) (see, e.g., [49]), and spectral theory (see,e.g., [50]).

Thus far, the bulk of the analyses, asymptotic or otherwise, of ORFs on T

[2, 51, 52] assume that the poles of the ORFs lie in the interior of the openunit disc (O :={z∈C; |z|<1}) (see, however, Chapter 11 of [2], and [17, 19]).For the scarce number of analyses, asymptotic or otherwise, of ORFs on theextended real line (R :=R ∪ {±∞}), the poles of the ORFs are assumed tobe real and disjoint from the support of the orthogonality measure [18, 53],to lie in C \ R, or, due to so-called ‘special technical considerations’, some‘forbidden value’ of a pole must be excluded from the analyses. Since thegeneralisation of orthogonal polynomials on T and on R requires the polesto be in the exterior of the closed unit disc (C \ (O ∪ T)) or in R, respectively,the lacunae described above must be addressed in order to have an exhaustiveunderstanding of the general ORF theory. For the case of ORFs on T withpoles in C \ (O ∪ T), and for the case of ORFs on R with poles in the openlower half-plane (C− :={z∈C; Im(z)<0}), Velázquez [50] has recently pre-sented an operator-theoretic approach to address non-asymptotic aspects ofthese issues.

Most poignantly, for the case of ORFs on R, there is a dearth of analysis.In addition, a technically challenging component of the analysis, asymptoticor otherwise, is where the poles of the ORFs lie on the support of theorthogonality measure. In fact, the present paper, which is Part I (the ‘finite-pole case’, FPC) of a two-part study of ORFs on R, considers the case wherethe finite-in-number and not necessarily distinct real poles of the ORFs arebounded and lie in R. Part II (the ‘mixed-pole case’, MPC) considers the moregeneral case where at least one, but not all, of the finite-in-number and notnecessarily distinct poles of the ORFs is the point at infinity.

Hereafter, only the FPC ORFs will be considered.


1.2 Background and Notation

There is a plethora of definitions which must be presented in this Subsection 1.2in order to completely describe the FPC ORFs. Since these definitions mayappear quite daunting, we have chosen to intersperse, at various points, a basicexample of an FPC ORF real pole sequence. This is intended to parse thedefinitions in a digestible manner, and to elucidate the ideas presented.

One starts with the real pole sequence denoted by α1, α2, . . . , αK. Thesequence is assumed to be bounded; but, the individual pole locations are notnecessarily distinct. The example which we will return to frequently in thisSubsection 1.2 is the following real pole sequence of ‘length’ K=6:

{α1, α2, α3, α4, α5, α6}={0, 0, 1, 0, 1, 2}.

Notational Remark 1.2.1 The reader may mistakenly interpret the notation{α1, α2, . . . , αK} to be standard notation from set theory. In this work, however,such notation shall be thought of as a lexicographic listing of elements whichmay, therefore, repeat.

The next ingredient is the measure of orthogonality, which will be taken tobe a probability measure μ of the form

dμ(z)= w(z)dz, (1)

with varying exponential weight function

w(z)=exp(−N V(z)), N∈N, (2)

where the external field V : R \ {α1, α2, . . . , αK}→R satisfies the followingconditions:

V(z) is real analytic on R \ {α1, α2, . . . , αK}; (3)

lim|x|→+∞

(V(x)

ln(x2+1)

)=+∞; (4)

limx→αk

(V(x)

ln((x−αk)−2+1)

)=+∞, k=1, 2, . . . ,K. (5)

Next, the definition of a collection of spaces of rational functions R

n,k

is needed. For n∈N and k=1, 2, . . . ,K, R

n,k is defined to be the set of allrational functions with poles restricted to the real pole sequence α1, α2, . . . , αK;more precisely,

R

n,k :=⎧⎨⎩ f : C \ {α1, α2, . . . , αK}→C; f (z)=d�0 +

n−1∑i=1

K∑j=1

d�i, j(z−α j)

κij

+k∑

r=1

d n,r(z−αr)κnr

, d�0, d�i, j, d n,r ∈R

},


with the convention∑0

m=1 ∗∗∗:=0, where

κnk : N × {1, 2, . . . ,K}→N, (n,k) →κnk :=(n−1)γk+ρk

denotes the multiplicity of the pole αk, k=1, 2, . . . ,K, in the repeated realpole sequence

1{α1, α2, . . . , αK} ∪ · · · ∪ n−1{α1, α2, . . . , αK} ∪n{α1, α2, . . . , αk},

with γk the ‘repeating number’ of pole αk in the set {α1, α2, . . . , αK}, and ρk

the ‘repeating index’ of pole αk in the set {α1, α2, . . . , αK} up to, and includ-ing, ‘position k’. As an illustration, for the pole set {α1, α2, α3, α4, α5, α6}={0, 0, 1, 0, 1, 2},

γ1=γ2=γ4=3, γ3=γ5=2, γ6=1

and

ρ1=1, ρ2=2, ρ3=1, ρ4=3, ρ5=2, ρ6=1,

and, for n∈N,

κn1 = (n−1)γ1+ρ1=3(n−1)+1=3n−2,

κn2 = (n−1)γ2+ρ2=3(n−1)+2=3n−1,

κn3 = (n−1)γ3+ρ3=2(n−1)+1=2n−1,

κn4 = (n−1)γ4+ρ4=3(n−1)+3=3n,

κn5 = (n−1)γ5+ρ5=2(n−1)+2=2n,

κn6 = (n−1)γ6+ρ6=(n−1)+1=n.

Denote the linear space over R spanned by a constant and the rationalfunctions {(z−αk)

−κnk} n∈Nk=1,2,...,K

by R; more precisely,

R :=⋃n∈N

K⋃k=1

R

n,k.

A function element 0 �= f ∈R is called a rational function corresponding to thereal pole set {α1, α2, . . . , αK}. The ordered base of rational functions for R is

B ∼

⎧⎪⎨⎪⎩const.,

n=1︷︸︸︷(z−α1)

−κ11, (z−α2)−κ12, . . . , (z−αK)

−κ1K︸︷︷︸k=1,2,...,K

,

n=2︷︸︸︷(z−α1)

−κ21, (z−α2)−κ22, . . . , (z−αK)

−κ2K︸︷︷︸k=1,2,...,K

, . . . . . .

. . . . . . ,

n=m︷︸︸︷(z−α1)

−κm1, (z−α2)−κm2, . . . , (z−αK)

−κmK︸︷︷︸k=1,2,...,K

, . . . . . .

⎫⎪⎬⎪⎭ ,


corresponding, respectively, to the cyclically repeated real pole sequence

P ∼⎧⎨⎩no pole,

n=1︷︸︸︷α1, α2, . . . , αK︸︷︷︸

k=1,2,...,K

,

n=2︷︸︸︷α1, α2, . . . , αK︸︷︷︸

k=1,2,...,K

, . . . . . . ,

n=m︷︸︸︷α1, α2, . . . , αK︸︷︷︸

k=1,2,...,K

, . . . . . .

⎫⎬⎭.

For 0 �= f ∈R, there exists a unique pair (n,k)∈N × {1, 2, . . . ,K} suchthat f ∈R

n,k. For (n,k)∈N × {1, 2, . . . ,K} and 0 �= f ∈R

n,k, define the leadingcoefficient of f , symbolically LC( f ), as

LC( f ) :=d n,k.

For (n,k)∈N × {1, 2, . . . ,K}, 0 �= f ∈R

n,k is called monic if LC( f )=1.A linear functional L will now be defined; however, in order for the integrals

appearing in its definition to be well defined, it will be assumed that theprobability measure μ∈M1(R), where

M1(R) :={μ;∫

R

dμ(ξ)=1,∫

R

ξm dμ(ξ)<∞,∫

R

(ξ−αk)−m dμ(ξ)<∞,

m∈N,k=1, 2, . . . ,K}

denotes the set of all non-negative unit Borel measures on R for which allmoments at αk, k=1, 2, . . . ,K, and at the point at infinity exist. Define, for(n,k)∈N × {1, 2, . . . ,K}, the linear functional L by its action on the (rational)basis elements of R as follows:

L : R→R, f = d�0+n−1∑i=1

K∑j=1

d�i, j(z−α j)

κij+

k∑r=1

d n,r(z−αr)κnr

→ L( f ) := d�0+n−1∑i=1

K∑j=1

d�i, jci, j+k∑

r=1

d n,rcn,r,

where

ci, j=L((z−α j)−κij) :=

∫R

(ξ−α j)−κij dμ(ξ), (i, j)∈N × {1, 2, . . . ,K}.

(Of course, since M1(R)�μ, c0,0 :=L(1)=∫R

dμ(ξ)=1.)Define the real bilinear form 〈···, ···〉L as follows:

〈···, ···〉L : R ×R→R, ( f, g) →〈 f, g〉L :=L( f (z)g(z))=∫

R

f (ξ)g(ξ)dμ(ξ).

It is a fact that the bilinear form 〈···, ···〉L thus defined is an inner product (seeSection 2, the proof of Lemma 2.1); and this fact is used, with little or no furtherreference, throughout this work.

If f ∈R, then || f (···)||L :=(〈 f, f 〉)1/2 is called the norm of f with respectto L (note that || f (···)||L �0 for all f ∈R, and || f (···)||L>0 if 0 �= f ∈R). A


sequence of rational functions will now be defined. {φnk} n∈N

k=1,2,...,Kis called a—

real—orthonormal rational function sequence with respect to L if, for n∈N

and k=1, 2, . . . ,K:

(i) φnk ∈R

n,k;(ii) 〈φn

k , φn′k′ 〉L=δnn′δkk′ , where δij is the Kronecker delta;

(iii) 〈φnk , φ

nk〉L=:||φn

k(···)||2L=1.

(For consistency of notation, set φ00(z)≡1.)

In order to elucidate the precise structure of the orthogonality conditionsfor the FPC ORFs, and to state the results of this work (see Subsection 1.3),the following notational prelude is requisite.

What follows next is an ordered partitioning of {1, 2, . . . ,K} and the realpole set {α1, α2, . . . , αK}. In order to proceed, though, the appearance ofa parameter, designated s, must first be explained. For the real pole set{α1, α2, . . . , αK} (described above), let s denote the number of distinct poles;e.g., for {0, 0, 1, 0, 1, 2}, s=3.

For k=1, 2, . . . ,K, a decomposition of the index set corresponding to polesdistinct from αk will be needed, that is:

{k′ ∈{1, 2, . . . ,K}; αk′ �=αk}.In order to decompose this set, one needs to consider the (possibly smaller) col-lection of distinct poles, ordered consistently with the original pole sequence,and with the pole αk excised: the size of this set is s−1. For the jth member ofthe reduced collection of poles (also referred to as the ‘residual’ pole set), thenumber of times that that pole appears in the original pole sequence shall bedenoted k j, j=1, 2, . . . , s−1. (Note that this integer should be thought of as afunction of k; but, for simplicity of notation, this dependence is suppressed.)

One then decomposes the set of integers written above into a disjointunion so that the first subset is the collection of all integers (from 1 to K)corresponding to the first pole in this reduced collection of poles, the secondsubset is the collection of all integers corresponding to the second pole in thisreduced collection of poles, etc. The precise definition of this decomposition isas follows. Write the ordered disjoint integer partition

{k′ ∈{1, 2, . . . ,K}; αk′ �=αk} :={

i(1)1, i(1)2, . . . , i(1)k1︸︷︷︸k1

}

∪ { i(2)1, i(2)2, . . . , i(2)k2︸︷︷︸k2

} ∪ · · ·

· · · ∪ { i(s−1)1, i(s−1)2, . . . , i(s−1)ks−1︸︷︷︸ks−1

}

:=s−1⋃q=1

{i(q)1, i(q)2, . . . , i(q)kq︸︷︷︸

kq

},


where, for q∈{1, 2, . . . , s−1}, 1� i(q)1< i(q)2< · · ·< i(q)kq � K, and {i( j )1,i( j)2, . . . , i( j)k j} ∩ {i(l)1, i(l)2, . . . , i(l)kl }=∅ ∀ l �= j∈{1, 2, . . . , s−1}, which in-duces, on the real pole set {α1, α2, . . . , αK}, the following disjoint ordering,

{αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk} :={αi(1)1 , αi(1)2 , . . . , αi(1)k1︸︷︷︸

k1

}

∪ {αi(2)1 , αi(2)2 , . . . , αi(2)k2︸︷︷︸k2

} ∪ · · ·

· · · ∪ {αi(s−1)1 , αi(s−1)2 , . . . , αi(s−1)ks−1︸︷︷︸ks−1

}

:=s−1⋃q=1

{αi(q)1 , αi(q)2 , . . . , αi(q)kq︸︷︷︸

kq

},

where, for q∈{1, 2, . . . , s−1}, αi(q)1 ≺αi(q)2 ≺· · ·≺αi(q)kq, where the notation

‘a≺b ’ means “a precedes b” or “a is to the left of b”, {αi( j)1 , αi( j)2 , . . . , αi( j)k j} ∩

{αi(l)1 , αi(l)2 , . . . , αi(l)kl}=∅ ∀ l �= j∈{1, 2, . . . , s−1}, such that

αi(q)1 =αi(q)2 =· · ·=αi(q)kq, q=1, 2, . . . , s−1,

#{αi(q)1 , αi(q)2 , . . . , αi(q)kq

}=kq, q=1, 2, . . . , s−1.

(Note, also, the interesting relation #{k′ ∈{1, 2, . . . ,K}; αk′ �=αk}=∑s−1q=1 kq=

K−γk, k=1, 2, . . . ,K.)3

In order to illustrate the above notation, consider the real pole sequence (of‘length’ K=6) {α1, α2, α3, α4, α5, α6}={0, 0, 1, 0, 1, 2}, for which s=3:

(i) k=1

{k′ ∈{1, 2, . . . , 6}; αk′ �=α1=0} = {3, 5, 6}={3, 5} ∪ {6}:={i(1)1, i(1)k1} ∪ {i(2)k2} ⇒

k1=2, i(1)1=3, i(1)k1=2 = 5, k2=1, i(2)k2=1=6,

3If all the real poles are distinct, that is, αi �=α j ∀ i �= j∈{1, 2, . . . ,K}, then, for k=1, 2, . . . ,K,{k′ ∈{1, 2, . . . ,K};αk′ �=αk} is the ordered disjoint union of singeltons, that is, ∪K−1

q=1 {i(q)kq },with kq=1, q=1, 2, . . . ,K−1, 1�i(1)1< i(2)1< · · ·< i(K−1)1 �K, and {i(q)kq } ∩ {i(r)kr }=∅ ∀ q �=r∈{1, 2, . . . ,K−1}, which induces, on the real pole set {α1, α2, . . . , αK}, the follow-ing disjoint ordering, {αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk} :=∪K−1

q=1 {αi(q)kq}, with αi(1)1 ≺αi(2)1 ≺· · ·≺

αi(K−1)1 , #{αi(q)kq}=kq=1, q=1, 2, . . . ,K−1, {αi(q)kq

} ∩ {αi(r)kr}=∅ ∀ q �=r∈{1, 2, . . . ,K−1}, and

#{αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk}=∑K−1q=1 kq=K−1.


which induces the ordering (on the ‘residual’ pole set){αk′ ; k′ ∈ {1, 2, . . . , 6}, αk′ �=α1=0

} := {αi(1)1 , αi(1)k1

} ∪ {αi(2)k2

}= {α3, α5} ∪ {α6} = {1, 1} ∪ {2}.

(ii) k=2

{k′ ∈{1, 2, . . . , 6}; αk′ �=α2=0} = {3, 5, 6}={3, 5} ∪ {6}:= {i(1)1, i(1)k1} ∪ {i(2)k2} ⇒

k1=2, i(1)1=3, i(1)k1=2 = 5, k2=1, i(2)k2=1=6,

which induces the ordering (on the ‘residual’ pole set){αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α2=0

} := {αi(1)1 , αi(1)k1

} ∪ {αi(2)k2

}= {α3, α5} ∪ {α6}={1, 1} ∪ {2}.

(iii) k=3

{k′ ∈{1, 2, . . . , 6}; αk′ �=α3=1} = {1, 2, 4, 6}={1, 2, 4} ∪ {6}:= {i(1)1, i(1)2, i(1)k1} ∪ {i(2)k2} ⇒

k1=3, i(1)1=1, i(1)2=2, i(1)k1=3=4, k2 = 1, i(2)k2=1=6,


} := {αi(1)1 , αi(1)2 , αi(1)k1

} ∪ {αi(2)k2

}= {α1, α2, α4} ∪ {α6} = {0, 0, 0} ∪ {2}.

(iv) k=4

{k′ ∈{1, 2, . . . , 6}; αk′ �=α4=0} = {3, 5, 6} ={3, 5} ∪ {6}:= {i(1)1, i(1)k1} ∪ {i(2)k2} ⇒

k1=2, i(1)1=3, i(1)k1=2 = 5,k2 = 1, i(2)k2=1=6,


} :={αi(1)1 , αi(1)k1

} ∪ {αi(2)k2

}= {α3, α5} ∪ {α6}={1, 1} ∪ {2}.

(v) k=5

{k′ ∈{1, 2, . . . , 6}; αk′ �=α5=1} = {1, 2, 4, 6}={1, 2, 4} ∪ {6}:={i(1)1, i(1)2, i(1)k1} ∪ {i(2)k2

} ⇒

k1=3, i(1)1=1, i(1)2=2, i(1)k1=3=4,k2 = 1, i(2)k2=1=6,



} := {αi(1)1 , αi(1)2 , αi(1)k1

} ∪ {αi(2)k2

}= {α1, α2, α4} ∪ {α6} = {0, 0, 0} ∪ {2}.

(vi) k=6{k′ ∈{1, 2, . . . , 6}; αk′ �=α6=2

} = {1, 2, 3, 4, 5}={1, 2, 4} ∪ {3, 5}:= {

i(1)1, i(1)2, i(1)k1

} ∪ {i(2)1, i(2)k2

} ⇒

k1=3, i(1)1=1, i(1)2=2, i(1)k1=3=4,k2=2, i(2)1=3, i(2)k2=2=5,

which induces the ordering (on the ‘residual’ pole set)

{αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α6=2} := {αi(1)1 , αi(1)2 , αi(1)k1

} ∪ {αi(2)1 , αi(2)k2

}= {α1, α2, α4} ∪ {α3, α5}={0, 0, 0} ∪ {1, 1}.

This concludes the example.In order to introduce the notion of the residual multiplicity (see below), a

notational remark is requisite. For k=1, 2, . . . ,K and a given set of positiveintegers i1, i2, . . . , iM, let

j= ind{i1, i2, . . . , iM|k}denote the largest positive integer j, if it exists, from the collection i1, i2, . . . , iM

that is less than k; e.g.,

j= ind{1, 3, 7, 11| 8} ⇒ j=7 and j= ind{5, 6, 9| 3} ⇒ no such j exists.

Recall that we have defined, for each member of the reduced collection ofreal poles, the index set {i(q)1, i(q)2, . . . , i(q)kq}, q=1, 2, . . . , s−1. It will beimportant to know if, within each of these index sets, there is a positive integerless than k. Towards this end, define, for each choice of k=1, 2, . . . ,K, thefollowing set:

Jq(k) :={

j= ind{i(q)1, i(q)2, . . . , i(q)kq |k

}}, q=1, 2, . . . , s−1,

and denote by

mq(k), q=1, 2, . . . , s−1,

the unique element of the set Jq(k), if it is not empty.For n∈N and k=1, 2, . . . ,K, define, with the above orderings and

definitions,

κnkkq:={(n−1)γi(q)kq

, Jq(k)=∅, q∈{1, 2, . . . , s−1},(n−1)γmq(k)+ρmq(k), Jq(k) �=∅, q∈{1, 2, . . . , s−1},


where κnkkq: N × {1, 2, . . . ,K}→Z

+0 , q=1, 2, . . . , s−1, is the residual multi-

plicity of pole αi(q)kqin the repeated real pole sequence

Pn,k :={ 1α1, α2, . . . , αK︸︷︷︸

K

} ∪ · · · ∪ { n−1α1, α2, . . . , αK︸︷︷︸

K

} ∪ { nα1, α2, . . . , αk︸︷︷︸

k

}.

More precisely, for n∈N and k=1, 2, . . . ,K, as all occurrences of the realpole αk, k=1, 2, . . . ,K, are excised from the repeated real pole sequencePn,k, where the multiplicity, or number of occurrences, of the real pole αk

is κnk=(n−1)γk+ρk, one is left with the ‘residual’ real pole set (via the aboveinduced ordering on the real poles)

Pn,k \{αk, αk, . . . , αk︸︷︷︸

κnk

} :=s−1⋃q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κnkkq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κnkk1

}

∪ {αi(2)k2, αi(2)k2

, . . . , αi(2)k2︸︷︷︸κnkk2

} ∪ · · ·

· · · ∪ {αi(s−1)ks−1, αi(s−1)ks−1

, . . . , αi(s−1)ks−1︸︷︷︸κnkks−1

},

where the number of times the real pole αi(q)kq(�=αk) occurs is (its multiplicity)

κnkkq, q=1, 2, . . . , s−1. It can occur that, for n=1, κ1kkq

=0, for some valuesof q∈{1, 2, . . . , s−1}: in such cases, one defines, {αi(q)kq

, αi(q)kq, . . . , αi(q)kq

} :=∅,q∈{1, 2, . . . , s−1}; however, for (N�) n�2 and k=1, 2, . . . ,K, κnkkq

�1, q∈{1, 2, . . . , s−1}.

For n∈N and k=1, 2, . . . ,K, a counting-of-residual-multiplicities argumentgives rise to the following ordered sum formula:

s−1∑q=1

κnkkq:=κnkk1

+κnkk2+· · ·+κnkks−1

=(n−1)K+k−κnk.

In order to illustrate the latter notation, consider, again, the real polesequence (of ‘length’ K=6) {α1, α2, α3, α4, α5, α6}={0, 0, 1, 0, 1, 2}, for whichs=3 (recall, also, the above example):

(i) k=1

J1(1) :={

j= ind{i(1)1, i(1)k1 | 1

}}={ j= ind{3, 5| 1}}=∅,

J2(1) :={

j= ind{i(2)k2 | 1

}}={ j= ind{6| 1}}=∅,


hence

κn1k1= (n−1)γi(1)k1

=(n−1)γ5=2(n−1),

κn1k2= (n−1)γi(2)k2

=(n−1)γ6=n−1,

that is, as one moves from left to right across the repeated real polesequence

Pn,1 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1︸︷︷︸1

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0︸︷︷︸1

}

and removes all occurrences of the real pole α1=0, which occurs κn1=(n−1)γ1+ρ1=3(n−1)+1=3n−2 times, one is left with the residual realpole set (via the above induced ordering)

Pn,1\{α1, α1, . . . , α1︸︷︷︸κn1

}=Pn,1\{0, 0, . . . , 0︸︷︷︸3n−2

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn1kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn1k1

}

∪ {αi(2)k2, αi(2)k2

, . . . , αi(2)k2︸︷︷︸κn1k2

}

= {1, 1, . . . , 1︸︷︷︸2(n−1)

}∪{2, 2, . . . , 2︸︷︷︸n−1

},

where the number of times the real pole αi(1)k1=1 (�=0=α1) occurs is

κn1k1=2(n−1), and the number of times the real pole αi(2)k2

=2 (�=0=α1)

occurs is κn1k2=n−1. For n=1, since κ11k1

=κ11k2=0, one sets, as per

the convention above, {αi(q)kq, αi(q)kq

, . . . , αi(q)kq} :=∅, q=1, 2, in which

case, as κ11=ρ1=1, P1,1 \ {α1}=P1,1 \ {0}=∅ ∪ ∅=∅. In this case, theordered sum formula reads

2∑q=1

κn1kq:=κn1k1

+κn1k2=2(n−1)+(n−1)=3(n−1).

(ii) k=2

J1(2) :={

j= ind{i(1)1, i(1)k1 | 2

}}={ j= ind{3, 5| 2}}=∅,

J2(2) :={

j= ind{i(2)k2 | 2

}}={ j= ind{6| 2}}=∅,


hence

κn2k1= (n−1)γi(1)k1

=(n−1)γ5=2(n−1),

κn2k2= (n−1)γi(2)k2

=(n−1)γ6=n−1,

that is, as one moves from left to right across the repeated realpole sequence

Pn,2 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2︸︷︷︸

2

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0︸︷︷︸

2

}


Pn,2\{α2, α2, . . . , α2︸︷︷︸κn2

} = Pn,2\{0, 0, . . . , 0︸︷︷︸3n−1

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn2kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn2k1

}

∪ {αi(2)k2, αi(2)k2

, . . . , αi(2)k2︸︷︷︸κn2k2

}

= {1, 1, . . . , 1︸︷︷︸2(n−1)

}∪{2, 2, . . . , 2︸︷︷︸n−1

},


κn2k1=2(n−1), and the number of times the real pole αi(2)k2

=2 (�=0=α2)


=κ12k2=0, one sets, as per the

convention above, {αi(q)kq, αi(q)kq

, . . . , αi(q)kq} :=∅, q=1, 2, in which case,

as κ12=ρ2=2, P1,2 \ {α2, α2}=P1,2 \ {0, 0}=∅ ∪ ∅=∅. In this case, theordered sum formula reads

2∑q=1

κn2kq:=κn2k1

+κn2k2=2(n−1)+(n−1)=3(n−1).

(iii) k=3

J1(3) := { j= ind{i(1)1, i(1)2, i(1)k1 | 3}} ={ j= ind{1, 2, 4| 3}}={2} ⇒ m1(3)=2,

J2(3) := { j= ind{i(2)k2 | 3}}={ j= ind{6| 3}}=∅,


hence

κn3k1= (n−1)γm1(3)+ρm1(3)=(n−1)γ2+ρ2=3(n−1)+2=3n−1,

κn3k2= (n−1)γi(2)k2

=(n−1)γ6=n−1,


Pn,3 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3︸︷︷︸

3

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1︸︷︷︸

3

}


Pn,3\{α3, α3, . . . , α3︸︷︷︸κn3

}=Pn,3\{1, 1, . . . , 1︸︷︷︸2n−1

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn3kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn3k1

}

∪ {αi(2)k2

, αi(2)k2, . . . , αi(2)k2︸︷︷︸

κn3k2

}

= {0, 0, . . . , 0︸︷︷︸3n−1

} ∪ {2, 2, . . . , 2︸︷︷︸n−1

},


κn3k1=3n−1, and the number of times the real pole αi(2)k2

=2 (�=1=α3) occurs is κn3k2

=n−1. For n=1, since κ13k2=0, one sets, as per

the convention above, {αi(2)k2, αi(2)k2

, . . . , αi(2)k2} :=∅, in which case, as

κ13=ρ3=1, P1,3 \ {α3}=P1,3 \ {1}={0, 0} ∪ ∅={0, 0}. In this case, theordered sum formula reads

2∑q=1

κn3kq:=κn3k1

+κn3k2=(3n−1)+(n−1)=4n−2.

(iv) k=4

J1(4) :={

j= ind{i(1)1, i(1)k1 | 4

}}={ j= ind{3, 5| 4}}={3} ⇒ m1(4)=3,

J2(4) :={

j= ind{i(2)k2 | 4

}}={ j= ind{6| 4}}=∅,


hence

κn4k1= (n−1)γm1(4)+ρm1(4)=(n−1)γ3+ρ3=2(n−1)+1=2n−1,

κn4k2= (n−1)γi(2)k2

=(n−1)γ6=n−1,


Pn,4 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4︸︷︷︸

4

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0︸︷︷︸

4

}

and removes all occurrences of the real pole α4=0, which occurs κn4=(n−1)γ4+ρ4=3(n−1)+3=3n times, one is left with the residual realpole set (via the above induced ordering)

Pn,4\{α4, α4, . . . , α4︸︷︷︸κn4

} = Pn,4\{0, 0, . . . , 0︸︷︷︸3n

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn4kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn4k1

}

∪ {αi(2)k2

, αi(2)k2, . . . , αi(2)k2︸︷︷︸

κn4k2

}

= {1, 1, . . . , 1︸︷︷︸2n−1

}∪{2, 2, . . . , 2︸︷︷︸n−1

},


κn4k1=2n−1, and the number of times the real pole αi(2)k2

=2 (�=0=α4)


=0, one sets, as per theconvention above, {αi(2)k2

, αi(2)k2, . . . , αi(2)k2

} :=∅, in which case, as κ14=ρ4=3, P1,4 \ {α4, α4, α4}=P1,4 \ {0, 0, 0}={1} ∪ ∅={1}. In this case, theordered sum formula reads

2∑q=1

κn4kq:=κn4k1

+κn4k2=(2n−1)+(n−1)=3n−2.

(v) k=5

J1(5) :={

j= ind{i(1)1, i(1)2, i(1)k1 | 5

}} ={ j= ind{1, 2, 4| 5}}={4} ⇒ m1(5)=4,

J2(5) :={

j= ind{i(2)k2 | 5

}}={ j= ind{6| 5}}=∅,


hence

κn5k1= (n−1)γm1(5)+ρm1(5)=(n−1)γ4+ρ4=3(n−1)+3=3n,

κn5k2= (n−1)γi(2)k2

=(n−1)γ6=n−1,


Pn,5 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4, α5︸︷︷︸

5

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0, 1︸︷︷︸

5

}

and removes all occurrences of the real pole α5=1, which occurs κn5=(n−1)γ5+ρ5=2(n−1)+2=2n times, one is left with the residual realpole set (via the above induced ordering)

Pn,5\{α5, α5, . . . , α5︸︷︷︸κn5

}=Pn,5\{1, 1, . . . , 1︸︷︷︸2n

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn5kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn5k1

}

∪ {αi(2)k2

, αi(2)k2, . . . , αi(2)k2︸︷︷︸

κn5k2

}

= {0, 0, . . . , 0︸︷︷︸3n

}∪{2, 2, . . . , 2︸︷︷︸n−1

},


κn5k1=3n, and the number of times the real pole αi(2)k2

=2 (�=1=α5)


=0, one sets, as per theconvention above, {αi(2)k2

, αi(2)k2, . . . , αi(2)k2

} :=∅, in which case, as κ15=ρ5=2, P1,5 \ {α5, α5}=P1,5 \ {1, 1}={0, 0, 0} ∪ ∅={0, 0, 0}. In this case,the ordered sum formula reads

2∑q=1

κn5kq:=κn5k1

+κn5k2=3n+(n−1)=4n−1.

(vi) k=6

J1(6) :={

j= ind{i(1)1, i(1)2, i(1)k1 | 6

}} = { j= ind{1, 2, 4| 6}}={4} ⇒ m1(6)=4,

J2(6) :={

j= ind{i(2)1, i(2)k2 | 6

}}={ j= ind{3, 5| 6}}={5} ⇒ m2(6)=5,


hence

κn6k1= (n−1)γm1(6)+ρm1(6)=(n−1)γ4+ρ4=3(n−1)+3=3n,

κn6k2= (n−1)γm2(6)+ρm2(6)=(n−1)γ5+ρ5=2(n−1)+2=2n,

that is, as one moves from left to right across the cyclically repeated realpole sequence

Pn,6 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4, α5, α6︸︷︷︸

6

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0, 1, 2︸︷︷︸

6

}

and removes all occurrences of the real pole α6=2, which occurs κn6=(n−1)γ6+ρ6=(n−1)+1=n times, one is left with the residual real poleset (via the above induced ordering)

Pn,6\{α6, α6, . . . , α6︸︷︷︸κn6

}=Pn,6\{2, 2, . . . , 2︸︷︷︸n

} :=2⋃

q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κn6kq

}

= {αi(1)k1

, αi(1)k1, . . . , αi(1)k1︸︷︷︸

κn6k1

}

∪ {αi(2)k2

, αi(2)k2, . . . , αi(2)k2︸︷︷︸

κn6k2

}

= {0, 0, . . . , 0︸︷︷︸3n

}∪{1, 1, . . . , 1︸︷︷︸2n

},


κn6k1=3n, and the number of times the real pole αi(2)k2

=1 (�=2=α6)

occurs is κn6k2=2n. For n=1, since κ16=ρ6=1, it follows that

P1,6 \ {α6}=P1,6 \ {2}={0, 0, 0} ∪ {1, 1}. In this case, the ordered sumformula reads

2∑q=1

κn6kq:=κn6k1

+κn6k2=3n+2n=5n.

This concludes the example.For simplicity of notation, set, hereafter,

αi(q)kq:=αpq , q=1, 2, . . . , s−1.

Recall that, for each choice of k (from 1 to K), we have previously decom-posed the index set corresponding to poles distinct from αk into an ordereddisjoint union of subsets so that the jth subset (for j running from 1 to s−1) isthe collection of all integers (from 1 to K) corresponding to the j th pole in thereduced collection of poles. Now, we will define the next subset in this ordering


containing all integers (now from 1 to k) corresponding to the pole αk. Thus,for each choice of k from 1 to K, write the ordered integer partition

{{1, 2, . . . ,K}�k′�k; αk′ =αk} :={

i(s)1, i(s)2, . . . , i(s)ks︸︷︷︸ks

},

with i(s)ks:=k, 1� i(s)1< i(s)2< · · ·< i(s)ks

� K, #{i(s)1, i(s)2, . . . , i(s)ks} =

ks(=κ1i(s)ks=κ1k) =ρi(s)ks

=ρk, and {i( j )1, i( j )2, . . . , i( j )k j} ∩ {i(s)1, i(s)2, . . . ,i(s)ks

}=∅, j=1, 2, . . . , s−1, which induces, by the definition above, the fol-lowing real pole ordering,

{αk′ ; k′ ∈{1, 2, . . . ,K}, k′�k, αk′ =αk} :={αi(s)1 , αi(s)2 , . . . , αi(s)ks

},

with αi(s)ks:=αk, αi(s)1 ≺αi(s)2 ≺· · ·≺αi(s)ks

, {αi( j )1, αi( j )2, . . . , αi( j )k j} ∩ {αi(s)1 ,

αi(s)2 , . . . , αi(s)ks}=∅, j=1, 2, . . . , s−1, such that

αi(s)1 =αi(s)2 =· · ·=αi(s)ks:= αps

:=αk,

#{αi(s)1 , αi(s)2 , . . . , αi(s)ks

} = ks=κ1i(s)ks=κ1k=ρk.

For (N�) n�2 and k=1, 2, . . . ,K,

#{αi(s)ks

, αi(s)ks, . . . , αi(s)ks

}=κni(s)ks=κnk=(n−1)γk+ρk.

In order to illustrate this latter notation, consider, again, the real polesequence (of ‘length’ K=6) {α1, α2, α3, α4, α5, α6}={0, 0, 1, 0, 1, 2}, forwhich s=3:

(i) k=1

{{1, 2, . . . , 6} � k′�1; αk′ =α1=0}={1} :={i(3)k3

}⇒k3 = 1, i(3)k3=1=1,

which induces the real pole ordering

{αk′ ; k′ ∈{1, 2, . . . , 6}, k′�1, αk′ =α1=0} :={αi(3)k3}={α1}={0},

hence

κni(s)ks=κni(3)1 =κn1=(n−1)γ1+ρ1=3(n−1)+1=3n−2,


Pn,1 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1︸︷︷︸1

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0︸︷︷︸1

}

and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α1=0}, one is left with the set (via the above-induced


ordering) that consists of all occurrences of the real pole α1=0, whichoccurs κn1=3n−2 times,

Pn,1 \2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸κn1kq

} :={αi(s)ks, αi(s)ks

, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn1

}={0, 0, . . . , 0︸︷︷︸3n−2

}.

(ii) k=2

{{1, 2, . . . , 6} � k′�2; αk′ =α2=0}={1, 2} :={i(3)1, i(3)k3} ⇒k3 = 2, i(3)1=1, i(3)k3=2=2,


{αk′ ;k′∈{1, 2, . . . , 6}, k′�2, αk′ =α2=0} :={αi(3)1 , αi(3)k3

}= {α1, α2}={0, 0},

hence



Pn,2 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2︸︷︷︸

2

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0︸︷︷︸

2

}

and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α2=0}, one is left with the set (via the above-inducedordering) that consists of all occurrences of the real pole α2=0, whichoccurs κn2=3n−1 times,

Pn,2 \2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κn2kq


, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn2

}={0, 0, . . . , 0︸︷︷︸3n−1

}.

(iii) k=3

{{1, 2, . . . , 6}�k′�3; αk′ =α3=1}={3} :={i(3)k3} ⇒k3=1, i(3)k3=1=3,


{αk′ ; k′ ∈{1, 2, . . . , 6}, k′�3, αk′ =α3=1} :={αi(3)k3}={α3}={1},

hence




Pn,3 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3︸︷︷︸

3

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1︸︷︷︸

3

}

and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α3=1}, one is left with the set (via the above-inducedordering) that consists of all occurrences of the real pole α3=1, whichoccurs κn3=2n−1 times,

Pn,3\2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κn3kq


, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn3

}={1, 1, . . . , 1︸︷︷︸2n−1

}.

(iv) k=4

{{1, 2, . . . , 6} � k′�4; αk′ =α4=0}={1, 2, 4} :={i(3)1, i(3)2, i(3)k3} ⇒k3 = 3, i(3)1=1, i(3)2=2, i(3)k3=3=4,


{αk′ ; k′ ∈{1, 2, . . . , 6}, k′�4, αk′ =α4=0} :={αi(3)1 , αi(3)2 , αi(3)k3

}= {α1, α2, α4}={0, 0, 0},

hence

κni(s)ks=κni(3)3 =κn4=(n−1)γ4+ρ4=3(n−1)+3=3n,


Pn,4 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4︸︷︷︸

4

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0︸︷︷︸

4

}

and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α4=0}, one is left with the set (via the above-inducedordering) that consists of all occurrences of the real pole α4=0, whichoccurs κn4=3n times,

Pn,4\2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κn4kq


, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn4

}={0, 0, . . . , 0︸︷︷︸3n

}.


(v) k=5 {{1, 2, . . . , 6} � k′�5; αk′ =α5=1}={3, 5} :={i(3)1, i(3)k3

}⇒k3 = 2, i(3)1=3, i(3)k3=2=5,

which induces the real pole ordering{αk′ ;k′∈{1, 2, . . . , 6}, k′�5, αk′ =α5=1

} :={αi(3)1, αi(3)k3

}= {α3, α5}={1, 1},

hence

κni(s)ks=κni(3)2 =κn5=(n−1)γ5+ρ5=2(n−1)+2=2n,


Pn,5 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4, α5︸︷︷︸

5

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0, 1︸︷︷︸

5

}

and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α5=1}, one is left with the set (via the above-inducedordering) that consists of all occurrences of the real pole α5=1, whichoccurs κn5=2n times,

Pn,5\2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κn5kq


, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn5

}={ 1, 1, . . . , 1︸︷︷︸2n

}.

(vi) k=6

{{1, 2, . . . , 6} � k′�6; αk′ =α6=2}={6} :={i(3)k3} ⇒k3 = 1, i(3)k3=1=6,


{αk′ ; k′ ∈{1, 2, . . . , 6}, k′�6, αk′ =α6=2} :={αi(3)k3}={α6}={2},

hence

κni(s)ks=κni(3)1 =κn6=(n−1)γ6+ρ6=(n−1)+1=n,


Pn,6 = { 1α1, α2, . . . , α6︸︷︷︸

6

} ∪ · · · ∪ { n−1α1, α2, . . . , α6︸︷︷︸

6

} ∪ { nα1, α2, α3, α4, α5, α6︸︷︷︸

6

}

= { 10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ · · · ∪ { n−10, 0, 1, 0, 1, 2︸︷︷︸

6

} ∪ { n0, 0, 1, 0, 1, 2︸︷︷︸

6

}


and removes the residual pole set (recall the examples above) {αk′ ; k′ ∈{1, 2, . . . , 6}, αk′ �=α6=2}, one is left with the set (via the above-inducedordering) that consists of all occurrences of the real pole α6=2, whichoccurs κn6=n times,

Pn,6\2⋃

q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κn6kq


, . . . , αi(s)ks︸︷︷︸κni(s)ks

=κn6

}={2, 2, . . . , 2︸︷︷︸n

}.

This concludes the example.With the above conventions and ordered disjoint partitions, one writes, for

n∈N and k=1, 2, . . . ,K, the repeated real pole sequence Pn,k as the followingordered disjoint partition:

s−1⋃q=1

{αi(q)kq

, αi(q)kq, . . . , αi(q)kq︸︷︷︸

κnkkq

} ∪ {αi(s)ks, αi(s)ks

, . . . , αi(s)ks︸︷︷︸κni(s)ks

}

:=s−1⋃q=1

{αpq , αpq , . . . , αpq︸︷︷︸

κnkkq

} ∪ {αk, αk, . . . , αk︸︷︷︸κnk

},

where, by convention, the set {αk, αk, . . . , αk} is written as the right-most set.With the above notational preamble concluded, one now returns to the

precise formulation of the orthogonality conditions for the FPC ORFs.We have a nested sequence of (rational) base sets. Fixing n∈N and

k=1, 2, . . . ,K, we determine one member of this nested sequence of (rational)base sets:

⎧⎨⎩const.,

1

(z−α1)−κ11, (z−α2)

−κ12, . . . , (z−αK)−κ1K︸︷︷︸

K

,

2

(z−α1)−κ21, (z−α2)

−κ22, . . . , (z−αK)−κ2K︸︷︷︸

K

, . . . . . .

. . . . . . ,n

(z−α1)−κn1, (z−α2)

−κn2, . . . , (z−αk)−κnk︸︷︷︸

k

⎫⎬⎭

:= {const.}⋃

∪s−1q=1 ∪

κnkkq

r=1

{(z−αpq)

−r}⋃∪κnkm=1

{(z−αk)

−m}

:= {const.}⋃

∪s−1q=1

{(z−αpq)

−1, (z−αpq)−2, . . . , (z−αpq)

−κnkkq}

⋃{(z−αk)

−1, (z−αk)−2, . . . , (z−αk)

−κnk},


corresponding, respectively, to the ordered repeated real pole sequence⎧⎨⎩no pole,

1α1, α2, . . . , αK︸︷︷︸

K

,2

α1, α2, . . . , αK︸︷︷︸K

, . . . ,n

α1, α2, . . . , αk︸︷︷︸k

⎫⎬⎭

:= {no pole}⋃

∪s−1q=1{αpq , αpq , . . . , αpq︸︷︷︸

κnkkq

}⋃

{αk, αk, . . . , αk︸︷︷︸κnk

}.

Orthonormalisation with respect to 〈···, ···〉L, via the Gram-Schmidt orthogonali-sation method, leads to the FPC orthonormal rational functions, {φn

k(z)} n∈Nk=1,2,...,K

(for consistency of notation, set φ00(z)≡1), which can be written as

φnk(z)=φ�0(n,k)+

n−1∑m=1

K∑j=1

ν�

m, j(n,k)

(z−α j)κmj

+k∑

r=1

μ�n,r(n,k)(z−αr)κnr

.

Using the above orderings, it is convenient to express the FPC orthonormalrational functions as follows:

φnk(z) :=φ0(n,k)+

s−1∑q=1

κnkkq∑r=1

νr,q(n,k)(z−αpq)

r+

κnk∑m=1

μn,m(n,k)(z−αk)m

.

The φnk ’s are normalised so that they all have real coefficients; in particular, for

n∈N and k=1, 2, . . . ,K,

LC(φnk)=μ�n,k(n,k)=μn,κnk(n,k)>0.

(For consistency of notation, set φ�0(0, 0)=φ0(0, 0)≡1.) Furthermore, notethat, for n∈N and k=1, 2, . . . ,K, by construction:

(1)⟨φn

k , (z−αk)− j⟩

L=∫

R

φnk(ξ)(ξ−αk)

− j dμ(ξ)=0, j=0, 1, . . . ,κnk−1;

(2)⟨φn

k , μn,κnk(n,k)(z−αk)−κnk

⟩L=μn,κnk(n,k)

∫R

φnk(ξ)(ξ−αk)

−κnk dμ(ξ)=1;

(3)⟨φn

k , (z−αpq)−r⟩

L=∫

R

φnk(ξ)(ξ−αpq)

−r dμ(ξ)=0,

q=1, 2, . . . , s−1, r=1, 2, . . . ,κnkkq.

(Note: if, for k=1, 2, . . . ,K, the residual real pole set {αk′ ; k′ ∈{1, 2, . . . ,K},αk′ �=αk}=∅, then the corresponding orthogonality conditions (3) above arevacuous; actually, this can only occur for n=1.)


For n∈N and k=1, 2, . . . ,K, the orthogonality conditions (1)–(3) abovegive rise to a total of

κnk+1+s−1∑q=1

κnkkq

︸︷︷︸= (n−1)K+k−κnk

=(n−1)K+k+1

(linear) equations determining the (n−1)K+k+1 real (n- and k-dependent)coefficients.

It is convenient to introduce, at this stage, the main object of study ofthis work, namely, the monic FPC ORFs, {πππn

k(z)} n∈Nk=1,2,...,K

(for consistency of

notation, set πππ00(z)≡1). For n∈N and k=1, 2, . . . ,K:

πππnk(z) :=

φnk(z)

LC(φnk)

= φ�

0(n,k)

μ�

n,k(n,k)+ 1

μ�

n,k(n,k)

n−1∑m=1

K∑j=1

ν�

m, j(n,k)

(z−α j)κmj

+ 1

μ�

n,k(n,k)

k−1∑r=1

μ�n,r(n,k)(z−αr)κnr

+ 1

(z−αk)κnk

:= φ0(n,k)μn,κnk(n,k)

+ 1

μn,κnk(n,k)

s−1∑q=1

κnkkq∑r=1


r

+ 1

μn,κnk(n,k)

κnk−1∑m=1


+ 1

(z−αk)κnk.

(Recall that LC(φnk) denotes the leading coefficient of the FPC orthonormal

rational functions.) The monic FPC ORFs, {πππnk(z)} n∈N

k=1,2,...,K, possess the following

orthogonality properties:

(1′)⟨πππn

k, (z−αk)− j⟩

L=∫

R

πππnk(ξ)(ξ−αk)

− j dμ(ξ)=0, j=0, 1, . . . ,κnk−1;

(2′)⟨πππn

k, (z−αk)−κnk

⟩L=∫

R


−κnk dμ(ξ)=(LC(φnk))

−2

=(μn,κnk(n,k))−2;

(3′)⟨πππn

k, (z−αpq)−r⟩

L=∫

R

πππnk(ξ)(ξ−αpq)

−r dμ(ξ)=0,

q=1, 2, . . . , s−1, r=1, 2, . . . ,κnkkq.


(Note: if, for k=1, 2, . . . ,K, the residual real pole set {αk′ ; k′ ∈{1, 2, . . . ,K},αk′ �=αk}=∅, then the corresponding orthogonality conditions (3′) above arevacuous; actually, this can occur only for n=1.)

For n∈N and k=1, 2, . . . ,K, it follows from the monic FPC ORF orthogo-nality conditions (1′)–(3′) above that

⟨πππn

k,πππnk

⟩L=:||πππn

k(···)||2L=(LC(φnk))

−2=(μn,κnk(n,k))−2,

whence ||πππnk(···)||L=(μn,κnk(n,k))

−1>0.

Remark 1.2.1 There is a circuitous connection between FPC ORFs and themore standard orthogonal polynomials sequence(s). For n∈N and k=1, 2,. . . ,K, an FPC ORF has numerator which is a polynomial of degree(n−1)K+k that is orthogonal to lower-degree polynomials (of degrees0, 1, . . . , (n−1)K+k−1), but with respect to the non-standard (exponentiallyvarying) measure of orthogonality dμ(z)=∏s−1

q=1(z−αpq)−2κnkkq (z−αk)

−2κnk

·(z−αk) exp(−N V(z))dz, N∈N, which changes signs (due to the factor z−αk).(Note: the latter measure represents, in fact, a doubly-indexed family ofmeasures.) Intuition from weighted approximation and this doubly-indexedfamily of measures provides an alternative understanding for the importanceof the associated family of variational (minimisation) problems, which aresummarised in Subsection 1.3, (9) (see, also, Section 3, Lemma 3.8). The above-mentioned connection (to orthogonal polynomials) also has the potential toyield an alternative approach to the asymptotic analysis (in the double-scalinglimit N,n→∞ such that N/n=1+o(1)) of the FPC ORFs; however, in theopinion of the author’s, it is more convenient to present RHPs that are directlyassociated to the FPC ORFs. �

1.3 Summary of Results

Having defined, heretofore, and in considerable detail, the principal objectsof this study, that is, the monic FPC ORFs, {πππn

k(z)} n∈Nk=1,2,...,K

, the ‘norming

constant’, μn,κnk(n,k), (n,k)∈N × {1, 2, . . . ,K}, and the FPC orthonormalrational functions, φn

k(z) :=μn,κnk(n,k)πππnk(z), (n,k)∈N × {1, 2, . . . ,K}, it must

be mentioned that the ultimate goal of this multi-fold study of ORFs (FPCand MPC) is to obtain precise, and uniform, asymptotics, in the double-scaling limit N,n→∞ such that N/n=1+o(1), of πππn

k(z), z∈C, μn,κnk(n,k),and, subsequently, φn

k(z), z∈C.In order to follow through with the above-mentioned asymptotic pro-

gramme, however, a correct formulation of the ORF problem, for an a prioriprescribed, not necessarily distinct, real pole set {α1, α2, . . . , αK} lying on thesupport of the orthogonality measure, is a seminal necessity. In fact, thepresent work, which is the first installment of a multi-fold series of works ded-icated to a detailed study of the above-described ORFs, serves a dual purpose,namely: (i) to address the above-mentioned ‘formulation problem’ for the FPC


ORFs; and (ii) to prepare the groundwork for subsequent asymptotic analyses,in the double-scaling limit N,n→∞ such that N/n=1+o(1).

The genesis of our ORF studies (FPC and MPC) consists in reformulating, inthe spirit of Fokas et al. [54, 55] (see, also, [56]), the ORF problem as K familiesof matrix Riemann–Hilbert problems (RHPs) on R, and then to study thelarge-n (as N,n→∞ such that N/n=1+o(1)) behaviour of the correspondingK solution families, wherein the latter family of K asymptotic analyses consistsof a union of the Deift–Zhou (DZ) non-linear steepest-descent method forundulatory—matrix—RHPs [57, 58] and the extension of Deift et al. [59].

Given the (N�) K arbitrary, bounded and not necessarily distinct realpoles α1, α2, . . . , αK lying on the support of the orthogonality measure (withvarying exponential weight) dμ(z)=exp(−N V(z))dz, N∈N, where V: R \{α1, α2, . . . , αK}→R is characterised by conditions (3)–(5), the first set of re-sults of this work (Part I) can be summarised thus; for n∈N and k=1, 2, . . . ,K:

• an equivalent reformulation of the monic FPC ORF problem as a familyof K matrix RHPs on R (see Section 2, Lemma RHPFPC);

• explicit solution formulae for each of these K matrix RHPs, constructedexplicitly from the monic FPC ORFs and their Cauchy transforms (seeSection 2, Lemma 2.1);

• the subsequent establishment of the existence and the uniqueness of themonic FPC ORFs via a detailed analysis of a novel family of K generalisedHankel determinants associated with rational moments of the orthogonal-ity measure with respect to the given real pole set {α1, α2, . . . , αK} (seeSection 2, Lemma 2.1);

• explicit multi-integral representation for the ‘norming constant’,μn,κnk(n,k) (see Section 2, Corollary 2.1);

• explicit multi-integral representation for the monic FPC ORFs, πππnk(z),

z∈C, and, via the relation φnk(z) :=μn,κnk(n,k)πππ

nk(z), explicit multi-integral

representation for the FPC orthonormal rational functions, φnk(z), z∈C

(see Section 2, Lemma 2.2, and Remark 2.3, respectively).

The ultimate goal of this multi-fold study of ORFs is to prepare thefoundation for the asymptotic analysis (in the double-scaling limit N,n→∞such that N/n=1+o(1)) of πππn

k(z), z∈C, and μn,κnk(n,k) (subsequently,φn

k(z) :=μn,κnk(n,k)πππnk(z), z∈C). The present work (Part I) deals exclusively

with the FPC ORFs (see the follow-up work, Part II, for the MPC ORFs).The proceeding discussion, which summarises the remaining results of thiswork, while valid in its own right for finite n (∈N), is germane, principally, totransforming the family of K matrix RHPs on R into a family of K equivalent(‘model’) matrix RHPs on R suitable for asymptotic analysis.

It is a well-established mathematical fact that variational conditions forminimisation problems in logarithmic potential theory, via the equilibriummeasure (see, e.g., [56, 60–63]), play an absolutely crucial rôle in asymp-totic analyses of (matrix) RHPs associated with (continuous and discrete)


orthogonal polynomials, their roots, and corresponding recurrence relationcoefficients (see, e.g., [64–67]). The situation with respect to the large-nasymptotic analysis for the monic FPC ORFs is analogous; however, unlikeasymptotic analyses for the orthogonal polynomials case, the asymptoticanalysis for the monic FPC ORFs requires the consideration of K differentfamilies of matrix RHPs on R, one for each k=1, 2, . . . ,K (see Section 2,Lemmata RHPFPC and 2.1). Thus, one must consider, for n∈N, K sets ofvariational conditions for K suitably posed minimisation problems.

Remark 1.3.1 Before proceeding, a minor notational preamble is requisite.Write (cf. (1) and (2)) dμ(z)=exp(−N V(z))dz=exp(−nV(z))dz=:dμ(z),n∈N, where

V(z)=zoV(z),

with

zo : N × N→R+, (N,n) →zo :=N/n,

where R+ :={x∈R; x>0}, and where the ‘scaled’ external field V: R \{α1, α2, . . . , αK}→R satisfies the following conditions:

V(z) is real analytic on R \ {α1, α2, . . . , αK}; (6)

lim|x|→+∞

(V(x)

ln(x2+1)

)=+∞; (7)

limx→αk

(V(x)

ln((x−αk)−2+1)

)=+∞, k=1, 2, . . . ,K. (8)

(E.g., a rational function of the form V: R \ {α1, α2, . . . , αK}�z →∑Kk=1∑−1

q=−2mkςq,k(z−αk)

q+∑2m∞q=0 ςq,∞zq, where, for k=1, 2, . . . ,K, mk∈N, m∞∈N,

ς−2mk,k>0, and ς2m∞,∞>0, would satisfy conditions (6)–(8).)4 �

The following discussion summarises, succinctly, the remaining, principalresults of this work, all of which are seminal ingredients for the subsequentasymptotic analysis (in the double-scaling limit N,n→∞ such that zo=1+

4Since the double-scaling limit of interest is N,n→∞ such that zo=1+o(1), the monic FPC ORFsare now orthogonal with respect to the varying exponential measure dμ(z)=exp(−nV(z)) dz,n∈N, with V(z) (=zoV(z)) satisfying conditions (6)–(8), where the large parameter, n, enterssimultaneously into the order (=(n−1)K+k) of the monic FPC ORFs and the (varying expo-nential) weight; thus, asymptotics of the monic FPC ORFs are studied along a ‘diagonal strip’ of adoubly-indexed sequence.


o(1)) of the monic FPC ORFs,πππnk(z), z∈C, the ‘norming constant’,μn,κnk(n,k),

and the FPC orthonormal rational functions, φnk(z) :=μn,κnk(n,k)πππ

nk(z), z∈C.

• Let V: R \ {α1, α2, . . . , αK}→R satisfy conditions (6)–(8). Let IV : N ×{1, 2, . . . ,K} × M1(R)→R denote the energy functional

IV[n,k;μEQ] := IV[μEQ]=∫∫

R2ln

⎛⎝|ξ−τ | 1

n

( |ξ−τ ||ξ−αk||τ−αk|

)κnk−1n

×s−1∏q=1

( |ξ−τ ||ξ−αpq ||τ−αpq |

)κnkkqn

⎞⎠

−1

dμEQ(ξ)dμEQ(τ )

+2∫

R

V(ξ)dμEQ(ξ),

and consider, for n∈N and k=1, 2, . . . ,K, the associated minimisationproblem

EV(n,k) :=EV = inf{IV[μEQ]; μEQ∈M1(R)

}.

For n∈N and k=1, 2, . . . ,K, the infimum is finite, and there exists aunique measure μV(n,k) :=μV , called the equilibrium measure, achiev-ing this minimum, that is, M1(R)�μV = inf{IV[μEQ]; μEQ ∈M1(R)} (seeSection 3, Lemmata 3.1, 3.2, and 3.3).

For n∈N and k=1, 2, . . . ,K, the equilibrium measure,μV , has the following‘regularity properties’:

• the equilibrium measure has support which consists of the disjoint union ofa finite number, N+1 (∈N), of bounded real (compact) intervals. In fact,as shown in Section 3, Lemma 3.7, item (1)(1)(1), supp(μV)=: J=∪N+1

j=1 [bj−1, a j](⊂ R \ {α1, α2, . . . , αK}). The end-points of the support of μV , that is,{bj−1, a j}N+1

j=1 , as well as the non-negative integer N, depend on n and k;e.g., bj−1=bj−1(n,k) and a j=a j(n,k), j=1, 2, . . . , N+1. It is instructive tonote that the real poles do not lie within the support of the equilibriummeasure: [bj−1, a j] ∩ {α1, α2, . . . , αK}=∅, j=1, 2, . . . , N+1. The compactreal intervals have been enumerated so that −∞<b0<a1<b1<a2< · · ·<bN<aN+1<+∞. Note: all of the quantities above also depend on zo :=N/n; but, for notational simplicity, this dependence is suppressed;


• the end-points, {bj−1, a j}N+1j=1 , satisfy the locally solvable system of 2(N+1)

real moment equations (transcendental equations)

∫J

ξ j

(R(ξ))1/2+

⎛⎝2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ=0,

j=0,1, . . . ,N,

∫J

ξN+1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ

=−2

((n−1)K+k

n

),

∫ bj

a j

⎛⎝(R(ς))1/2

∫J(R(ξ))−1/2

+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ V ′(ξ)iπ

)dξξ−ς

)dς = 2

(κnk−1

n

)ln

∣∣∣∣bj−αk

a j−αk

∣∣∣∣+2s−1∑q=1

κnkkq

nln

∣∣∣∣bj−αpq

a j−αpq

∣∣∣∣

+ (V(bj)−V(a j)), j=1, 2, . . . , N,

where (R(z))1/2 :=(∏N+1j=1 (z−bj−1)(z−a j)

)1/2, with (R(z))1/2± := limε↓0(R(z±iε))1/2, and the branch of the square root is chosen so thatz−(N+1)(R(z))1/2∼C±�z→∞±1 (see Section 3, Lemma 3.7, item (1)(1)(1)). In fact,in the double-scaling limit N,n→∞ such that zo=1+o(1), the end-points,{bj−1, a j}N+1

j=1 , are real-analytic functions of zo;• the density of the equilibrium measure is given by

dμV(x) :=ψV(x)dx= 1

2π i(R(x))1/2+ hV(x)111J(x)dx,

where

hV(z) =1

2

((n−1)K+k

n

)−1 ∮CV

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ iV ′(ξ)π

⎞⎠ (R(ξ))−1/2

ξ−zdξ

(real analytic for z∈R \ {α1, α2, . . . , αK}), with CV (⊂ C \ {α1, α2, . . . , αK})the disjoint union of s+1 circular contours, one outer one of large radius


R� traversed clockwise, and s inner ones, one about each real pole αpq ,q=1, 2, . . . , s, of small radii rq, q=1, 2, . . . , s, traversed counter-clockwise,with the numbers 0<rq<R�<+∞, q=1, 2, . . . , s, chosen so that, for (any)non-real z in the domain of analyticity of V, int(CV) ⊃ J ∪ {z}, and 111J(x)is the characteristic function of the compact set J. Note that ψV(x)�0(resp., ψV(x)>0) for x∈ J (resp., x∈ int(J)); in fact, ψV(x) behaves likea square root at the end-points of the support of the equilibrium mea-sure, that is, ψV(x)=x↓bj−1 O((x−bj−1)

1/2) and ψV(x)=x↑a j O((a j−x)1/2), j=1, 2, . . . , N+1 (see Section 3, Lemma 3.7, item (2)(2)(2));

• the equilibrium measure and its—compact—support are uniquely charac-terised by the following Euler–Lagrange variational equations: there exists�(n,k) :=�∈R, the Lagrange multiplier, such that

2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)

dμV(ξ)−2

(κnk−1

n

)ln|z−αk|

−2s−1∑q=1

κnkkq

nln|z−αpq | + 2

s−1∑q=1

κnkkq

nln|αpq−αk| −V(z)−�=0, z∈ J,

(9)

2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)

dμV(ξ)−2

(κnk−1

n

)ln|z−αk|

−2s−1∑q=1

κnkkq

nln|z−αpq |+2

s−1∑q=1

κnkkq

nln|αpq−αk|−V(z)−��0, z∈R\ J

(see Section 3, Lemma 3.8);• the Euler–Lagrange variational conditions can be conveniently recast in

terms of the complex potential (the ‘g-function’) g: N × {1, 2, . . . ,K} × C \(−∞,max{max j=1,2,...,K{α j},max{supp(μV)}})→C (of μV):

g(n,k; z) :=g(z)=∫

Jln

⎛⎝(z−ξ) 1

n

((z−ξ)

(z−αk)(ξ−αk)

)κnk−1n

×s−1∏q=1

((z−ξ)

(z−αpq)(ξ−αpq)

)κnkkqn

⎞⎠dμV(ξ)

(see Section 3, Lemma 3.4). For n∈N and k=1, 2, . . . ,K, g(z) satisfies:

(G1) g(z) is analytic for z∈C\(−∞,max{maxj=1,2,...,K{αj},max{supp(μV)}})(see Section 3, Lemma 3.4);


(G2) g(z)=− (κnk−1n

)ln(z−αk)+P±

0 +O(z−αk) as z→αk, with z∈C±,where

P±0 :=

∫J

ln

⎛⎜⎝|ξ−αk| 1

n

s−1∏q=1

(|ξ−αk|

|ξ−αpq ||αpq −αk|

)κnkkqn

⎞⎟⎠dμV(ξ)

− iπ(

κnk−1

n

)∫J∩{x∈R; x<αk}

dμV(ξ)− iπs−1∑q=1

κnkkq

n

×∫

J∩{x∈R; x<αpq }dμV(ξ)±iπ

((n−1)K+k

n

)∫J∩{x∈R; x>αk}

dμV(ξ)

∓ iπ∑

q∈{ j∈{1,2,...,s−1};αp j>αk}

κnkkq

n

(see Section 3, Lemma 3.4);(G3) g+(z)+g−(z)−P+

0 −P−0 −V(z)−�=0, z∈ J, where g±(z) := limε↓0

g(z±iε) (see Section 3, Lemma 3.8);(G4) g+(z)+ g−(z)− P+

0 − P−0 − V(z)− � � 0, z∈R \ J, where equal-

ity holds for at most a finite number of points (see Section 3,Lemma 3.8);

(G5) g+(z)− g−(z)+ P−0 − P+

0 = i f R

g (z), z ∈ R, where f R

g (z) is a piece-wise-continuous, real-valued bounded function (see Section 3,Lemma 3.8);

(G6) i(g+(z) – g−(z) + P−0 – P+

0 + 2π i∑

q∈{ j∈{1,2,...,s−1};αp j>z}κnkkq

n +

2π i(κnk−1

n

)111{x∈R; x<αk}(z))′ = �(n,k; z), with

�(n,k; z)={

2π((n−1)K+k

n

)ψV(z)�0, z∈ J,

0, z ∈ R \ J,

where equality holds for at most a finite number of points (seeSection 3, Lemma 3.8);

• for k=1, 2, . . . ,K, an equivalent family of K asymptotic, ‘model’ matrixRHPs on R is derived (see Section 4, Lemma 4.3). From these K modelmatrix RHPs, asymptotics (in the double-scaling limit N,n→∞ such thatzo=1+o(1)) of πππn

k(z), z∈C, μn,κnk(n,k), and φnk(z) :=μn,κnk(n,k)πππ

nk(z),

z∈C, will be derived, in a future publication. Note: the equilibrium mea-sure and the corresponding variational problems emerge naturally in theasymptotic analysis of this family of K equivalent ‘model’ matrix RHPson R.

In this work on the characterisation and asymptotics (in the double-scalinglimit N,n→∞ such that zo=1+o(1)) of the FPC ORFs and related quantities,


the so-called ‘regular case’ is studied. For n∈N and k=1, 2, . . . ,K, one saysthat dμV , or V: R \ {α1, α2, . . . , αK}→R, is regular if:

(i) hV(x) �=0, x∈ J;

(ii) 2((n−1)K+k

n

)∫J ln(∣∣∣ z−ξξ−αk

∣∣∣)

dμV(ξ)−2(

κnk−1n

)ln|z−αk|−2

∑s−1q=1

κnkkq

n ln|z−αpq|+2

∑s−1q=1

κnkkq

n ln|αpq−αk|−V(z)−�<0, z∈R \ J;(iii) inequalities (G4) and (G6) are strict, that is, � (in (G4)) and � (in (G6))

are replaced by < and >, respectively.5

Remark 1.3.2 The following correspondence should also be noted: for n∈N

and k=1, 2, . . . ,K, g(z) solves the phase conditions (G1)–(G6)⇔ M1(R)�μVsolves the variational conditions (9). �

Remark 1.3.3 In the next section, the RHP formulation for the monic FPCORFs is presented. The reader may wonder whence came the RHP. Forvarious values of n (∈N) and different choices of the pole set, FPC ORFs wereconstructed; and, based on these calculations, and in the spirit of the RHPformulation for the orthogonal polynomials problem, an integral representa-tion for the ‘solution matrix’ was conjectured, but with the Cauchy kernelsnormalised at the poles. Then, the following question was posed: “what kind ofRHP do the FPC ORFs solve?” The ‘full’ RHP formulation for the FPC ORFs(more precisely, for the monic FPC ORFs) then follows from a careful analysisof the asymptotic behaviour of the ‘solution matrix’ in open neighbourhoodsof the poles, supplemented with a computation (which uses the Sokhotski–Plemelj formula) of the corresponding jump, or discontinuity, matrix. �

Remark 1.3.4 A heuristic explanation for the origin of the (doubly-indexedfamily of) energy—minimisation—problems appearing in (9) is as follows.Starting with the RHP for the monic FPC ORFs (see Section 2, LemmaRHPFPC), one makes a transformation involving a so-called g-function (orcomplex logarithmic potential), whose properties are determined at a laterstage: the result of this transformation is a ‘new’ RHP (see Section 3,Lemma 3.4). Then, one asks if there exists a choice of this g-function so that

5For n∈N and k=1, 2, . . . ,K, there are three distinct situations in which these conditions (i)–

(iii) may fail: (1) for at least one z∈R \ J, 2( (n−1)K+kn )

∫J ln(| z−ξ

ξ−αk|) dμV(ξ)−2

(κnk−1

n

)ln|z−αk|−

2∑s−1

q=1

κnkkqn ln|z−αpq |+2

∑s−1q=1

κnkkqn ln|αpq −αk|−V(z)−�=0, that is, equality is attained for at

least one point z in the complement of the support of the equilibrium measure, which correspondsto the situation in which a ‘band’ has just closed, or is about to open, about z; (2) for at least onez, hV(z)=0, that is, the function hV(z) vanishes for at least one point z within the support of theequilibrium measure, which corresponds to the situation in which a ‘gap’ is about to open, or close,about z; and (3) there exists at least one j∈{1, 2, . . . , N+1}, denoted j∗, such that hV(bj∗−1)=0 orhV(a j∗ )=0. Each of these three cases can occur only a finite number of times due to the fact thatV: R \ {α1, α2, . . . , αK}→R satisfies conditions (6)–(8) [63, 66].


the ‘new’, or transformed, RHP is in a form suitable for subsequent asymp-totic analysis (in the double-scaling limit N,n→∞ such that N/n=1+o(1)).Based on experience from other (asymptotic) RHP analyses, one arrives at acollection of equations and inequalities, which, if satisfied, achieve the desiredresult of obtaining an RHP in so-called ‘standard form’. These equations andinequalities, in turn, are shown to be equivalent to Euler–Lagrange variationalconditions associated with the energy—minimisation—problems. �

Remark 1.3.5 A general energy minimisation problem for all n∈N and k=1, 2, . . . ,K has been presented. As has been mentioned heretofore, the prin-cipal interest of this multi-fold study of ORFs is asymptotics as n→∞. Inthis limit, the family of energy minimisation problems stabilizes in the sensethat one may write the energy functional as a small perturbation (o(1)) of thefollowing ‘core’ energy functional:

∞IV[μEQ] :=

∫∫R2

ln

(|ξ−τ |K∏s

q=1(|ξ−αpq ||τ−αpq |)γi(q)kq

)−1

dμEQ(ξ)dμEQ(τ )

+ 2∫

R

V(ξ)dμEQ(ξ),

where (cf. Subsection 1.2) αpq :=αi(q)kq, q=1, 2, . . . , s, and the factor

∏s

q=1(|ξ−αpq ||τ−αpq |)γi(q)kq is independent of, or invariant with respect to, k (=1, 2,. . . ,K); e.g., for the real pole sequence (of ‘length’ K=6) {α1, α2, α3, α4,

α5, α6}={0, 0, 1, 0, 1, 2}, for which s=3, the above formula reads

∞IV[μEQ] =

∫∫R2

ln

( |ξ−τ |6(|ξ ||τ |)3(|ξ−1||τ−1|)2(|ξ−2||τ−2|)1

)−1

dμEQ(ξ)dμEQ(τ )

+ 2∫

R

V(ξ)dμEQ(ξ).

Note: if all the poles in the sequence {α1, α2, . . . , αK} are distinct, that is, αi �=α j

∀ i �= j∈{1, 2, . . . ,K}, then the ‘core’ energy functional is given by

∞IV[μEQ] =

∫∫R2

ln

(|ξ−τ |K∏K

j=1|ξ−α j||τ−α j|

)−1

dμEQ(ξ)dμEQ(τ )

+ 2∫

R

V(ξ)dμEQ(ξ).

�

Remark 1.3.6 The energy minimisation problem described herein requiresa new existence and regularity theory because of the presence of the realpoles α1, α2, . . . , αK, as well as the singular behaviour of V at each αk, k=1, 2, . . . ,K. �


2 The Monic FPC ORF Family of Riemann–Hilbert Problems:Existence and Uniqueness

In this section, the family of K matrix Riemann–Hilbert problems (RHPs)on R characterising the monic FPC ORFs, {πππn

k(z)} n∈Nk=1,2,...,K

, z∈C, is stated,

whence the existence and the uniqueness of the monic FPC ORFs and theassociated norming constants, μn,κnk(n,k), is established via a generalisedHankel determinant analysis (see Lemmata RHPFPC and 2.1), and explicitmulti-integral representations for the norming constant (see Corollary 2.1) andthe monic FPC ORFs (see Lemma 2.2) are obtained.

Before launching into the theory proper, however, and in order to prunethe usual foliage attendant upon this topic, it is convenient to summarise thenotation used throughout this work.

Notational Conventions

(1) I=( 1 00 1

)is the 2 × 2 identity matrix, σ1=

(0 11 0

), σ2=

(0 −ii 0

)and σ3=

(1 00 −1

)are the Pauli matrices, σ+=

(0 10 0

)and σ−=

(0 01 0

)are, respectively, the

raising and lowering matrices, R± :={x∈R;±x>0}, R≷x0 :={x∈R; x≷x0},

C± :={z∈C; ± Im(z)>0}, C∗ :=C \ {0}, C :=C ∪ {∞}, and sgn(x) :=0 if

x=0 and x|x|−1 if x �=0;(2) for a scalar ω and a 2×2 matrix ϒ , ωad(σ3)ϒ :=ωσ3ϒω−σ3 ;(3) a contour D which is the finite union of piecewise-smooth, simple curves

(as closed sets) is said to be orientable if its complement C \ D canalways be divided into two, possibly disconnected, disjoint open sets�+ and �

−, either of which has finitely many components, such thatD admits an orientation so that it can either be viewed as a positivelyoriented boundary D+ for �

+ or as a negatively oriented boundary D−for �

− [68], that is, the (possibly disconnected) components of C \ Dcan be coloured by + or by − in such a way that the + regions do notshare boundary with the − regions, except, possibly, at finitely manypoints [69];

(4) for each segment of an oriented contour D , according to the givenorientation, the “+” side is to the left and the “–” side is to the rightas one traverses the contour in the direction of orientation, that is,for a matrix Aij(···), i, j=1, 2, (Aij(···))± denote the non-tangential limits(Aij(z))± := lim z′ → z

z′ ∈± side of D

Aij(z′);(5) for 1� p<∞ and D some point set,

Lp

M2(C)(D) :=

{f :D→M2(C); || f (···)||L p

M2(C)(D) :=

(∫D| f (z)|p |dz|

) 1p

<∞},


where, for A (···)∈M2(C), |A (···)| :=(∑2

i, j=1 Aij(···)Aij(···))1/2 (the Hilbert-

Schmidt norm), with ∗∗∗ denoting complex conjugation of ∗∗∗, for p=∞,

L ∞M2(C)(D) :=

{g : D→M2(C); ||g(···)||L ∞

M2(C)(D) := max

i, j=1,2supz∈D

|gij(z)|<∞},

and, for f ∈I+L 2M2(C)(D) :={I+h; h∈L 2

M2(C)(D)},

|| f (···)||I+L 2M2(C)

(D) :=(|| f (∞)||2L ∞

M2(C)(D)+|| f (···)− f (∞)||2

L 2M2(C)

(D)

) 12 ;

(6) for a matrix Aij(···), i, j=1, 2, to have boundary values in theL 2

M2(C)(D) sense on an oriented contour D , it is meant that

lim z′ → zz′ ∈± side of D

∫D |A (z′)−(A (z))±|2 |dz|=0; e.g., if D=R is oriented

from +∞ to −∞, then A (···) has L 2M2(C)(D) boundary values on D

means that limε↓0∫

R|A (x∓iε)−(A (x))±|2 dx=0;

(7) ||F(···)||∩p∈JLp

M2(C)(∗) :=

∑p∈J ||F(···)||L p

M2(C)(∗), with card(J)<∞;

(8) M1(R) denotes the set of all non-negative unit Borel measures on R

for which all moments at αk, k=1, 2, . . . ,K, and at the point at infinityexist, that is,

M1(R) :={μ;∫

R

dμ(ξ)=1,∫

R

ξm dμ(ξ)<∞,∫

R

(ξ−αk)−m dμ(ξ)<∞,

m∈N, k=1, 2, . . . ,K};

(9) for (ν1, ν2)∈R × R, the function (•−ν1)iν2 : C \ (−∞, ν1)→C,• →exp(iν2 ln(• − ν1)), with the branch cut taken along (−∞, ν1),and with the principal branch of the logarithm chosen;

(10) for a 2 × 2 matrix-valued function T(z), the notation T(z)=z→z0 O(∗)(resp., o(∗)) means Tij(z)=z→z0 O(∗ij) (resp., o(∗ij)), i, j=1, 2.

Remark 2.1 The superscripts ±, and sometimes the subscripts ±, in this sectionshould not be confused with the subscripts ± appearing in the various RHPs(this is a general comment which applies throughout the entire text, unlessstated otherwise). Although C :=C ∪ {∞} (resp., R :=R ∪ {−∞} ∪ {+∞}) isthe standard notation for the (closed) Riemann sphere (resp., closed real line),the simplified, and somewhat abusive, notation C (resp., R) is used to denoteboth the (closed) Riemann sphere, C (resp., closed real line, R), and the (open)complex field, C (resp., open real line, R), and the context(s) should make clearwhich object(s) the notation C (resp., R) represents. �

Remark 2.2 Throughout the remainder of this work, the notations ofSubsection 1.2 will be used extensively, with little, or no, explanation. �


The family of K matrix RHPs characterising the monic FPC ORFs is nowstated.

RHPFPC. Let V: R \ {α1, α2, . . . , αK}→R satisfy conditions (3)–(5). For n∈N

and k=1, 2, . . . ,K, find Y : N × {1, 2, . . . ,K} × C \ R→SL2(C) solving: (i)Y (n,k; z) :=Y (z) is holomorphic for z∈C \ R; (ii) the boundary valuesY±(z) := limε↓0 Y (z±iε) satisfy the jump condition

Y+(z)=Y−(z)υ(z), z∈R,

where

υ: R→GL2(R), z →υ(z) := I+exp(−N V(z)) σ+, N∈N;(iii)

Y (z)(z−αk)(κnk−1)σ3 =

C\R� z→αk

I+O(z−αk);

(iv)

Y (z)z−σ3 =C\R� z→∞

O(1);

(v) for q∈{1, 2, . . . , s−1},Y (z)(z−αpq)

κnkkqσ3 =C\R� z→αpq

O(1).

Lemma 2.1 Let Y : N×{1, 2, . . . ,K}×C\R→SL2(C) solve RHPFPC. For n∈N

and k=1, 2, . . . ,K, RHPFPC possesses a unique solution given by

Y (z)=

⎛⎜⎜⎝(z−αk)πππ

nk(z) (z−αk)

∫R

((ξ−αk)πππnk(ξ)) exp(−N V(ξ))

(ξ − αk)(ξ − z)dξ2π i

Y21(z) (z−αk)∫

R

Y21(ξ) exp(−N V(ξ))(ξ − αk)(ξ − z)

dξ2π i

⎞⎟⎟⎠, z∈C\R,

where Y21 : N × {1, 2, . . . ,K} × C \ {α1, α2, . . . , αK}→C denotes the (2 1)-element of Y , and πππn

k : N × {1, 2, . . . ,K} × C \ {α1, α2, . . . , αK}→C is themonic FPC ORF defined in Subsection 1.2.

Proof Set w(z) :=exp(−N V(z)), N∈N, where V: R \ {α1, α2, . . . , αK}→R

satisfies conditions (3)–(5). If, for n∈N and k=1, 2, . . . ,K, (recall thatY (n,k; z) :=Y (z)) Y : C \ R→SL2(C) solves RHPFPC, then, from the jumpcondition (ii) of RHPFPC, it follows that, for the elements of the first columnof Y (z),

(Y j1(z))+=(Y j1(z))− :=Y j1(z), j=1, 2,


and, for the elements of the second column of Y (z),

(Y j 2(z))+−(Y j 2(z))−=Y j1(z)w(z), j=1, 2. (10)

Via the normalisation condition (iii) of RHPFPC, and the boundedness condi-tions (iv) and (v) of RHPFPC, it follows that, for n∈N and k=1, 2, . . . ,K,

Y11(z)(z−αk)κnk−1 =

C\R� z→αk

1 + O(z−αk),

Y12(z)(z−αk)−(κnk−1) =

C\R� z→αk

O(z−αk),

(11)Y21(z)(z−αk)

κnk−1 =C\R� z→αk

O(z−αk),

Y22(z)(z−αk)−(κnk−1) =

C\R� z→αk

1 + O(z−αk),

Y11(z)z−1 =C\R� z→∞

O(1), Y12(z)z =C\R� z→∞

O(1),

Y21(z)z−1 =C\R� z→∞

O(1), Y22(z)z =C\R� z→∞

O(1), (12)

and, for q∈{1, 2, . . . , s−1},Y11(z)(z−αpq)

κnkkq =C\R� z→αpq

O(1), Y12(z)(z−αpq)−κnkkq =

C\R� z→αpq

O(1),

Y21(z)(z−αpq)κnkkq =

C\R� z→αpq

O(1), Y22(z)(z−αpq)−κnkkq =

C\R� z→αpq

O(1),

(13)

whence, temporarily re-inserting explicit n- and k-dependencies (Yij(n,k; z) :=Yij(z), i, j=1, 2), one notes that, for m=1, 2, Ym1 : N × {1, 2, . . . ,K} × C \{α1, α2, . . . , αK}→C and Ym2 : N × {1, 2, . . . ,K} × C \ R→C; in particular,for n∈N and k=1, 2, . . . ,K, Y11(z) and Y21(z) have no jumps throughoutthe z-plane, Y11(z) is a monic (that is, coeff((z−αk)

−κnk)=1) meromorphicfunction with poles at α1, α2, . . . , αK, and Y21(z) is a meromorphic functionwith poles at α1, α2, . . . , αK. Application of the Sokhotski–Plemelj formulato the jump condition (10), with the Cauchy kernel normalised at αk, k=1, 2, . . . ,K, gives rise to the following Cauchy-type integral representation:

Y j 2(z)=∫

R

(z−αk)Y j1(ξ)w(ξ)

(ξ−αk)(ξ−z)dξ2π i, z∈C \ R, j=1, 2; (14)

hence, for n∈N and k=1, 2, . . . ,K, Y : C \ R→SL2(C) has the integralrepresentation

Y (z)=

⎛⎜⎜⎝

Y11(z)∫

R

(z − αk)Y11(ξ)w(ξ)


Y21(z)∫

R

(z − αk)Y21(ξ)w(ξ)


⎞⎟⎟⎠ , z∈C \ R.


A detailed analysis for the elements of the first row of Y (z), that is, Y1m(z),m=1, 2, is presented first; then, the corresponding analysis for the elements ofthe second row of Y (z), that is, Y2m(z), m=1, 2, is presented. Recalling that(cf. Subsection 1.2)

∫Rξmw(ξ)dξ <∞ and

∫R(ξ−α j)

−(m+1)w(ξ)dξ <∞, m∈Z+0 ,

j=1, 2, . . . ,K, it follows via the expansion 1z1−z2

=∑li=0

zi2

zi+11+ zl+1

2

zl+11 (z1−z2)

, l∈Z+0 ,

the integral representation (cf. (14)) Y12(z)=∫

R

(z−αk)Y11(ξ)w(ξ)

(ξ−αk)(ξ−z)dξ2π i , z∈C \ R,

and the first of each of the asymptotic conditions (11), (12), and (13) that, forn∈N and k=1, 2, . . . ,K,∫

R

(Y11(ξ)

ξ−αk

)w(ξ)

(ξ−αk)pdξ=0, p=0, 1, . . . ,κnk−1, (15)

∫R

(Y11(ξ)

ξ−αk

)w(ξ)

(ξ−αk)κnkdξ �=0, (16)

∫R

(Y11(ξ)

ξ−αk

)w(ξ)

(ξ−αpq)r

dξ=0, q=1, 2, . . . , s−1, r=1, 2, . . . ,κnkkq. (17)

(Note: if, for n∈N and k=1, 2, . . . ,K, the set {αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk}=∅, then the corresponding (17) are vacuous; actually, this can onlyoccur if n=1.) Recalling from the analysis preceding the integral represen-tation (14) that, for n∈N and k=1, 2, . . . ,K, Y11 : N × {1, 2, . . . ,K} × C \{α1, α2, . . . , αK}→C is a monic (coeff((z−αk)

−κnk)=1) meromorphic functionwith pole set {α1, α2, . . . , αK} and with no jumps throughout the z-plane, andthat, for n∈N and k=1, 2, . . . ,K, the monic FPC ORFs, πππn

k(z), satisfy theorthogonality conditions stated in Subsection 1.2, it follows from these lattertwo observations and (15), (16), and (17) that, for n∈N and k=1, 2, . . . ,K,

πππnk(z)=

Y11(z)z−αk

.

Via (15) and (17), and this latter formula, one writes, for n∈N and k=1,2, . . . ,K, (16) in a more transparent form:

∫R

(Y11(ξ)

ξ−αk

)w(ξ)

(ξ−αk)κnk

dξ

=∫

R

(Y11(ξ)

ξ−αk

)μn,κnk (n,k)(ξ−αk)

κnk

w(ξ)

μn,κnk (n,k)dξ =

∫R

(Y11(ξ)

ξ−αk

)

×⎛⎝φ�0(n,k)+

n−1∑m=1

K∑j=1

ν�m, j(n,k)

(ξ−α j)κmj

+k−1∑r=1

μ�n,r(n,k)(ξ−αr)κnr

+μn,κnk (n,k)(ξ−αk)

κnk

⎞⎠

︸︷︷︸=φn

k (ξ)

× w(ξ)

μn,κnk (n,k)dξ =

∫R

πππnk(ξ)︸︷︷︸

= (μn,κnk (n,k))−1φn

k (ξ)

φnk(ξ)w(ξ)

μn,κnk (n,k)dξ

= (μn,κnk (n,k))−2∫

R

(φnk(ξ))

2w(ξ) dξ︸︷︷︸

= 1

=(μn,κnk (n,k))−2;


hence, via this latter relation, (15)–(17) are written in the following, moreconvenient, form: for n∈N and k=1, 2, . . . ,K,

∫R


−p w(ξ)dξ=0, p=0, 1, . . . ,κnk−1, (18)

∫R


−κnk w(ξ)dξ=(μn,κnk(n,k))−2, (19)

∫R

πππnk(ξ)(ξ−αpq)

−r w(ξ)dξ=0, q=1, 2, . . . , s−1, r=1, 2, . . . ,κnkkq. (20)

(Note: if, for n∈N and k=1, 2, . . . ,K, the set {αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk}=∅, then the corresponding (20) are vacuous; in fact, this can only occurif n=1.) For n∈N and k=1, 2, . . . ,K, (18) gives rise to κnk conditions, (19)gives rise to 1 condition, and (20) give rise to

∑s−1q=1 κnkkq

=(n−1)K+k−κnk

conditions, for a total of (n−1)K+k+1 conditions, which is precisely thenecessary count in order to determine, uniquely (see below), the (n- and k-dependent) ‘norming constant’, μn,κnk(n,k), and the (n- and k-dependent)coefficients of the partial fraction expansion of πππn

k(z).One now examines, for n∈N and k=1, 2, . . . ,K, (18)–(20) in detail. Pro-

ceeding as per the detailed discussion of Subsection 1.2, write, for n∈N andk=1, 2, . . . ,K, the ordered disjoint partition for the repeated real pole se-quence (with the convention ∪0

m=1{∗∗∗,∗∗∗, . . . ,∗∗∗} :=∅):

1{α1, α2, . . . , αK︸︷︷︸K

} ∪ · · · ∪ n−1{α1, α2, . . . , αK︸︷︷︸K

} ∪ n{α1, α2, . . . , αk︸︷︷︸k

}

:=s−1⋃q=1

{αi(q)kq, αi(q)kq

, . . . , αi(q)kq︸︷︷︸lq=κnkkq


, . . . , αi(s)ks︸︷︷︸ls=κni(s)ks

}

:=s−1⋃q=1

{αpq , αpq , . . . , αpq︸︷︷︸lq=κnkkq

} ∪ {αk, αk, . . . , αk︸︷︷︸ls=κnk

},

where∑s

q=1 lq=∑s−1q=1 lq+ls=∑s−1

q=1 κnkkq+κnk=(n−1)K+k. Hence, via this

notational preamble, and the analysis leading up to the orthogonality


conditions (18)–(20), one writes, in the indicated order, for n ∈ N andk = 1, 2, . . . ,K, (cf. Subsection 1.2)

πππnk(z) =

φ0(n,k)μn,κnk(n,k)

+ 1

μn,κnk(n,k)

s−1∑q=1

κnkkq∑r=1

νr,q(n,k)(z−αi(q)kq

)r

+ 1

μn,κnk(n,k)

κni(s)ks−1∑

m=1

μn,m(n,k)(z−αi(s)ks

)m+ 1

(z−αk)κnk

= φ0(n,k)μn,κnk(n,k)

+ 1

μn,κnk(n,k)

s−1∑q=1

κnkkq∑r=1


r

+ 1

μn,κnk(n,k)

κnk−1∑m=1


+ 1

(z−αk)κnk

= φ0(n,k)μn,κnk(n,k)

+ 1

μn,κnk(n,k)

κnkk1∑r=1

νr,1(n,k)(z−αp1)

r+· · · + 1

μn,κnk(n,k)

×κnkks−1∑

r=1

νr,s−1(n,k)(z−αps−1)

r+ 1

μn,κnk(n,k)

κnk−1∑m=1


+ 1

(z−αk)κnk

:= φ0(n,k)+s−1∑m=1

lm=κnkkm∑q=1

νm,q(n,k)(z−αpm)

q+

ls=κnk∑r=1

νs,r(n,k)(z−αk)r

, νs,ls(n,k)≡1.

Substituting the latter partial fraction expansion of πππnk(z) into the orthogo-

nality conditions (18)–(20), one arrives at, for n∈N and k=1, 2, . . . ,K, theorthogonality conditions (recall that νs,ls(n,k)≡1)

∫R

⎛⎝φ0(n,k)+

s−1∑m=1

lm=κnkkm∑q=1

νm,q(n,k)(ξ−αpm)

q+

ls=κnk∑j=1

νs, j(n,k)(ξ−αk) j

⎞⎠ w(ξ)

(ξ−αk)rdξ=0,

r=0, 1, . . . ,κnk−1,

∫R

⎛⎝φ0(n,k)+

s−1∑m=1

lm=κnkkm∑q=1


q+

ls=κnk∑j=1


⎞⎠ w(ξ)

(ξ−αk)κnkdξ

=(μn,κnk(n,k))−2,

∫R

⎛⎝φ0(n,k)+

s−1∑m=1

lm=κnkkm∑q=1


q+

ls=κnk∑j=1


⎞⎠ w(ξ)

(ξ−αpq)r

dξ=0,

q=1, 2, . . . , s−1, r=1, 2, . . . , li.


For n∈N and k=1, 2, . . . ,K, the above orthogonality conditions give riseto a total of (n−1)K+k+1 linear inhomogeneous algebraic equationsfor the (n−1)K+k+1 real unknowns φ0(n,k), ν1,1(n,k), . . . , ν1,l1(n,k), . . . ,νs−1,1(n,k), . . . , νs−1,ls−1(n,k), νs,1(n,k), . . . , νs,ls−1(n,k), (μn,κnk(n,k))

−2, thatis, with dμ(z) := w(z)dz:

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∫R

dμ(ξ1)(ξ1−αp1 )

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )l1

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )···

∫R

dμ(ξ2)

(ξ2−αp1 )2

∫R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αp1 )··· ∫

R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αp1 )l1

∫R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αp2 )···

......

......

......

∫R

dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

···∫

R

dμ(ξl1+1)

(ξl1+1−αp2 )

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )l1

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )···

......

......

......

∫R

dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

l1

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

···...

......

......

...∫R

dμ(ξ0)

(ξ0−αk)0∫

R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp1 )

··· ∫R

(ξ0−αk)0 dμ(ξ0)

(ξ0−αp1 )l1

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp2 )

···...

......

......

...∫R

dμ(ξm1 )

(ξm1 −αk)κnk−1

∫R

(ξm1 −αk)−κnk+1 dμ(ξm1 )

(ξm1 −αp1 )··· ∫

R


(ξm1 −αp1 )l1

∫R


(ξm1 −αp2 )···

∫R

dμ(ξm2 )

(ξm2 −αk)κnk

∫R

(ξm2 −αk)−κnk dμ(ξm2 )

(ξm2 −αp1 )··· ∫

R


(ξm2 −αp1 )l1

∫R


(ξm2 −αp2 )···

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )l2

··· ··· ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk) ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)κnk−1 0

∫R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αp2 )l2

··· ··· ··· ∫R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αk) ··· ∫R

(ξ2−αp1 )−2 dμ(ξ2)


............

......

......∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

l2··· ··· ··· ∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk) ··· ∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk)κnk−1 0

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )l2

··· ··· ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk) ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk)κnk−1 0

............

......

......∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2··· ··· ··· ∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk) ··· ∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk)κnk−1 0

............

......

......∫

R


(ξ0−αp2 )l2

··· ··· ··· ∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αk) ··· ∫

R



............

......

......∫

R


(ξm1 −αp2 )l2

··· ··· ··· ∫R


(ξm1 −αk)··· ∫

R


(ξm1 −αk)κnk−1 0

∫R


(ξm2 −αp2 )l2

··· ··· ··· ∫R


(ξm2 −αk)··· ∫

R


(ξm2 −αk)κnk−1 −1

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠


×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

φ0(n,k)ν1,1(n,k)...

ν1,l1(n,k)ν2,1(n,k)...

ν2,l2(n,k)...

νs,1(n,k)...

νs,ls−1(n,k)(μn,κnk(n,k))

−2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

− ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)κnk

− ∫R

(ξ2−αp1 )−2 dμ(ξ2)

(ξ2−αk)κnk

...

− ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk)κnk

− ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk)κnk

...

− ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk)κnk

...− ∫

R

(ξ0−αk)0 dμ(ξ0)(ξ0−αk)κnk

...

− ∫R


(ξm1−αk)κnk

− ∫R


(ξm2−αk)κnk

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (21)

where

m1 :=(n−1)K+k−1 and m2 :=(n−1)K+k.

For n∈N and k=1, 2, . . . ,K, the linear system (21) of (n−1)K+k+1inhomogeneous algebraic equations for the (n−1)K+k+1 real unknownsφ0(n,k), ν1,1(n,k), . . . , ν1,l1(n,k), . . . , νs−1,1(n,k), . . . , νs−1,ls−1 (n,k), νs,1 (n,k), . . . , νs,ls−1(n,k), (μn,κnk(n,k))

−2 admits a unique solution if, and only if,the determinant of the coefficient matrix is non-zero: this fact will now beestablished; and, en route, an explicit multi-integral representation for the(n- and k-dependent) ‘norming constant’, (μn,κnk(n,k))

−2, will be derived. Forn∈N and k=1, 2, . . . ,K, one uses Cramer’s rule to show that

(μn,κnk(n,k))−2= cN

cD,

where, by the multi-linearity property of the determinant,

cN =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0)dμ(ξ1) · · · dμ(ξl1)dμ(ξl1+1) · · · dμ(ξl1+l2) · · ·

× ··· dμ(ξm2−κnk+1) ··· dμ(ξm2 )

(ξ0−αk)0(ξ1−αp1 )1···(ξl1−αp1 )

l1 (ξl1+1−αp2 )1···(ξl1+l2−αp2 )

l2 ···(ξm2−κnk+1−αk)1···(ξm2−αk)κnk


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk

1 1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk

......

......

......

......

......

...1 1

(ξl1−αp1 )

··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1

(ξl1−αk)κnk

1 1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1

(ξl1+1−αk)κnk

......

......

......

......

......

...1 1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk) ··· 1

(ξl1+l2−αk)κnk

......

......

......

......

......

...1 1(ξm1 −αp1 )

··· 1

(ξm1 −αp1 )l1

1(ξm1 −αp2 )

··· 1

(ξm1 −αp2 )l2

··· 1(ξm1 −αk)

··· 1(ξm1 −αk)

κnk

1 1(ξm2 −αp1 )

··· 1

(ξm2 −αp1 )l1

1(ξm2 −αp2 )

··· 1

(ξm2 −αp2 )l2

··· 1(ξm2 −αk)

··· 1(ξm2 −αk)

κnk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

,

and

cD =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× ··· dμ(ξm2−κnk+1) ··· dμ(ξm1 )

(ξ0−αk)0(ξ1−αp1 )1···(ξl1−αp1 )

l1 (ξl1+1−αp2 )1···(ξl1+l2−αp2 )

l2 ···(ξm2−κnk+1−αk)1···(ξm1−αk)κnk−1

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk−1

1 1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk−1

.........

.........

......

......

...1 1(ξl1

−αp1 )··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1

(ξl1−αk)κnk−1

1 1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1

(ξl1+1−αk)κnk−1

.........

.........

......

......

...1 1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1(ξl1+l2

−αk) ··· 1

(ξl1+l2−αk)κnk−1

.........

.........

......

......

...1 1(ξm1−1−αp1 )

··· 1

(ξm1−1−αp1 )l1

1(ξm1−1−αp2 )

··· 1

(ξm1−1−αp2 )l2

··· 1(ξm1−1−αk) ··· 1

(ξm1−1−αk)κnk−1

1 1(ξm1 −αp1 )

··· 1

(ξm1 −αp1 )l1

1(ξm1 −αp2 )

··· 1

(ξm1 −αp2 )l2

··· 1(ξm1 −αk)

··· 1


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

For n∈N and k=1, 2, . . . ,K, cN is studied first; then, cD. For n∈N and k=1, 2, . . . ,K, introduce the following notation (cf. Subsection 1.2; recall thatlm :=κnkkm

, m=1, 2, . . . , s−1, ls :=κnk, and∑0

j=1 ∗∗∗:=0):

ϕ0(z) :=s−1∏m=1

(z−αpm)lm(z−αk)

κnk =:(n−1)K+k∑

j=0

a j,0z j,


and, for r=1, 2, . . . , s, q(r)=∑r−1i=1 li+1,

∑r−1i=1 li+2, . . . ,

∑r−1i=1 li+lr, and m(r) =

1, 2, . . . , lr,

ϕq(r)(z) :=⎧⎨⎩ϕ0(z)(z−αpr )

−m(r)=:(n−1)K+k∑

j=0

a j,q(r)z j

⎫⎬⎭ ;

e.g., for r=1, the notation ϕq(1)(z) :={ϕ0(z)(z−αp1)−m(1)=:∑(n−1)K+k

j=0 a j,q(1)z j},q(1)=1, 2, . . . , l1, m(1)=1, 2, . . . , l1, encapsulates the l1 :=κnkk1

functions

ϕ1(z)= ϕ0(z)z−αp1

=:(n−1)K+k∑

j=0

a j,1z j,

ϕ2(z)= ϕ0(z)(z−αp1)

2=:(n−1)K+k∑

j=0

a j,2z j, . . .

. . . , ϕl1(z) =ϕ0(z)

(z−αp1)l1

=:(n−1)K+k∑

j=0

a j,l1 z j,

for r=2, the notation ϕq(2)(z) :={ϕ0(z)(z−αp2)−m(2)=:∑(n−1)K+k

j=0 a j,q(2)z j},q(2)= l1+1, l1+2, . . . , l1+l2, m(2)=1, 2, . . . , l2, encapsulates the l2 :=κnkk2

functions

ϕl1+1(z) = ϕ0(z)z−αp2

=:(n−1)K+k∑

j=0

a j,l1+1z j,

ϕl1+2(z)= ϕ0(z)(z−αp2)

2=:(n−1)K+k∑

j=0

a j,l1+2z j, . . .

. . . , ϕl1+l2(z)=ϕ0(z)

(z−αp2)l2

=:(n−1)K+k∑

j=0

a j,l1+l2 z j,

etc., and, for r=s (cf. Subsection 1.2; recall that αps:=αk and

∑s−1i=1 li=(n − 1)

·K + k−κnk), the notation ϕq(s)(z) :={ϕ0(z)(z−αk)−m(s)=:∑(n−1)K+k

j=0 a j,q(s)z j},q(s)=(n−1)K+k−κnk + 1, (n−1)K+k−κnk+2, . . . , (n− 1)K + k,m(s) = 1,2, . . . , ls, encapsulates the ls :=κnk functions

ϕ(n−1)K+k−κnk+1(z) = ϕ0(z)z−αk

=:(n−1)K+k∑

j=0

a j,(n−1)K+k−κnk+1z j,

ϕ(n−1)K+k−κnk+2(z) = ϕ0(z)(z−αk)

2=:(n−1)K+k∑

j=0

a j,(n−1)K+k−κnk+2z j, . . .

. . . , ϕ(n−1)K+k(z) = ϕ0(z)(z−αk)

κnk=:(n−1)K+k∑

j=0

a j,(n−1)K+kz j.


(Note: #{ϕ0(z)(z − αpr )−m(r)} = lr, r = 1, 2, . . . , s, and # ∪s

r=1 {ϕ0(z)(z −αpr )

−m(r)} =∑s

r=1 lr =(n−1)K+k.) One notes that, for n∈N and k= 1,2, . . . ,K, the l1+l2+· · ·+ls+1=(n−1)K+k+1 functions ϕ0(z), ϕ1(z), . . . ,ϕl1(z), ϕl1+1(z),. . . , ϕl1+l2(z),. . . , ϕ(n−1)K+k−κnk+1(z),. . . , ϕ(n−1)K+k(z) are linear-ly independent on R, that is, for z∈R,

∑(n−1)K+kj=0 c jϕ j(z)=0 ⇒ (via a

Vandermonde-type argument) c j=0, j=0, 1, . . . , (n−1)K+k (see the((n−1)K+k+1)× ((n−1)K+k+1) non-zero determinant D in (22) below).For n∈N and k=1, 2, . . . ,K, let S(n−1)K+k+1 denote the ((n−1)K+k+1)!permutations of {0, 1, . . . , (n−1)K+k}. Using the above notation and themulti-linearity property of the determinant, one studies, thus, for n∈N andk=1, 2, . . . ,K, cN (see, also, [70]):

cN =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · · dμ(ξl1+l2 ) · · ·

× ···dμ(ξm2−κnk+1) ···dμ(ξm2 )

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)

1···(ξm2−αk)κnk

× 1

ϕ0(ξ0)ϕ0(ξ1) · · ·ϕ0(ξl1 )ϕ0(ξl1+1) · · ·ϕ0(ξl1+l2 ) · · ·ϕ0(ξm2−κnk+1) · · ·ϕ0(ξm2 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ϕ0(ξ0) ϕ1(ξ0) ··· ϕl1 (ξ0) ϕl1+1(ξ0) ··· ϕl1+l2 (ξ0) ··· ϕm2−κnk+1(ξ0) ··· ϕ(n−1)K+k(ξ0)


.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1 ) ϕ1(ξl1 ) ··· ϕl1 (ξl1 ) ϕl1+1(ξl1 ) ··· ϕl1+l2 (ξl1 ) ··· ϕm2−κnk+1(ξl1 ) ··· ϕ(n−1)K+k(ξl1 )

ϕ0(ξl1+1) ϕ1(ξl1+1) ··· ϕl1 (ξl1+1) ϕl1+1(ξl1+1) ··· ϕl1+l2 (ξl1+1) ··· ϕm2−κnk+1(ξl1+1) ··· ϕ(n−1)K+k(ξl1+1)

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1+l2 )ϕ1(ξl1+l2 ) ··· ϕl1 (ξl1+l2 )ϕl1+1(ξl1+l2 ) ··· ϕl1+l2 (ξl1+l2 ) ··· ϕm2−κnk+1(ξl1+l2 ) ··· ϕ(n−1)K+k(ξl1+l2 )

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2 (ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k(ξm1 )


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=:G(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξm2−κnk+1,...,ξ(n−1)K+k)

= 1

(m2+1)!∑

σ∈Sm2+1

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξσ(0)) dμ(ξσ(1)) · · · dμ(ξσ(l1)) dμ(ξσ(l1+1)) · · ·

× ···dμ(ξσ(l1+l2)) ···dμ(ξσ(m2−κnk+1))···dμ(ξσ(m2))

(ξσ(0)−αk)0(ξσ(1)−αp1 )

1···(ξσ(l1)−αp1 )l1 (ξσ(l1+1)−αp2 )

1···(ξσ(l1+l2)−αp2 )l2 ···(ξσ(m2−κnk+1)−αk)

1···(ξσ(m2)−αk)κnk

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1)) · · ·ϕ0(ξσ(l1))ϕ0(ξσ(l1+1)) · · ·ϕ0(ξσ(l1+l2)) · · ·ϕ0(ξσ(m2−κnk+1)) · · ·ϕ0(ξσ(m2))

×G(ξσ(0), ξσ(1),. . . ,ξσ(l1), ξσ(l1+1),. . . ,ξσ(l1+l2),. . . ,ξσ(m2−κnk+1),. . . ,ξσ((n−1)K+k))

= 1

(m2+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0)dμ(ξ1) · · · dμ(ξl1 )dμ(ξl1+1) · · · dμ(ξl1+l2 ) · · · dμ(ξm2−κnk+1) · · ·


×· · · dμ(ξm2 )G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k)∑

σ∈Sm2+1

sgn(σσσσσσσσσ)

× 1(ξσ(0)−αk)

0(ξσ(1)−αp1 )1···(ξσ(l1)−αp1 )

l1 (ξσ(l1+1)−αp2 )1···(ξσ(l1+l2)−αp2 )

l2 ···(ξσ(m2−κnk+1)−αk)1···(ξσ(m2)−αk)

κnk

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1))· · ·ϕ0(ξσ(l1))ϕ0(ξσ(l1+1))· · ·ϕ0(ξσ(l1+l2))· · ·ϕ0(ξσ(m2−κnk+1))· · ·ϕ0(ξσ(m2))

= 1

(m2+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1)· · ·dμ(ξl1 ) dμ(ξl1+1) · · · dμ(ξl1+l2 )· · ·dμ(ξm2−κnk+1)· · ·

×· · · dμ(ξm2 )G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k)

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(ϕ0(ξ0))−1

(ξ0−αk)0(ϕ0(ξ0))

−1

(ξ0−αp1 )··· (ϕ0(ξ0))

−1

(ξ0−αp1 )l1

(ϕ0(ξ0))−1

(ξ0−αp2 )··· (ϕ0(ξ0))

−1

(ξ0−αp2 )l2

··· (ϕ0(ξ0))−1

(ξ0−αk) ··· (ϕ0(ξ0))−1

(ξ0−αk)κnk

(ϕ0(ξ1))−1

(ξ1−αk)0(ϕ0(ξ1))

−1

(ξ1−αp1 )··· (ϕ0(ξ1))

−1

(ξ1−αp1 )l1

(ϕ0(ξ1))−1

(ξ1−αp2 )··· (ϕ0(ξ1))

−1

(ξ1−αp2 )l2

··· (ϕ0(ξ1))−1

(ξ1−αk) ··· (ϕ0(ξ1))−1

(ξ1−αk)κnk

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.(ϕ0(ξl1

))−1

(ξl1−αk)0

(ϕ0(ξl1))−1

(ξl1−αp1 )

··· (ϕ0(ξl1))−1

(ξl1−αp1 )

l1

(ϕ0(ξl1))−1

(ξl1−αp2 )

··· (ϕ0(ξl1))−1

(ξl1−αp2 )

l2··· (ϕ0(ξl1

))−1

(ξl1−αk) ··· (ϕ0(ξl1

))−1

(ξl1−αk)κnk

(ϕ0(ξl1+1))−1

(ξl1+1−αk)0(ϕ0(ξl1+1))

−1

(ξl1+1−αp1 )··· (ϕ0(ξl1+1))

−1

(ξl1+1−αp1 )l1

(ϕ0(ξl1+1))−1

(ξl1+1−αp2 )··· (ϕ0(ξl1+1))

−1

(ξl1+1−αp2 )l2

··· (ϕ0(ξl1+1))−1

(ξl1+1−αk) ··· (ϕ0(ξl1+1))−1

(ξl1+1−αk)κnk

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.(ϕ0(ξl1+l2

))−1

(ξl1+l2−αk)0

(ϕ0(ξl1+l2))−1

(ξl1+l2−αp1 )

··· (ϕ0(ξl1+l2))−1

(ξl1+l2−αp1 )

l1

(ϕ0(ξl1+l2))−1

(ξl1+l2−αp2 )

··· (ϕ0(ξl1+l2))−1

(ξl1+l2−αp2 )

l2··· (ϕ0(ξl1+l2

))−1

(ξl1+l2−αk) ··· (ϕ0(ξl1+l2

))−1

(ξl1+l2−αk)κnk

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.(ϕ0(ξm1 ))

−1

(ξm1 −αk)0(ϕ0(ξm1 ))

−1

(ξm1 −αp1 )··· (ϕ0(ξm1 ))

−1

(ξm1 −αp1 )l1

(ϕ0(ξm1 ))−1

(ξm1 −αp2 )··· (ϕ0(ξm1 ))

−1

(ξm1 −αp2 )l2

··· (ϕ0(ξm1 ))−1

(ξm1 −αk) ··· (ϕ0(ξm1 ))−1

(ξm1 −αk)κnk

(ϕ0(ξm2 ))−1

(ξm2 −αk)0(ϕ0(ξm2 ))

−1

(ξm2 −αp1 )··· (ϕ0(ξm2 ))

−1

(ξm2 −αp1 )l1

(ϕ0(ξm2 ))−1

(ξm2 −αp2 )··· (ϕ0(ξm2 ))

−1

(ξm2 −αp2 )l2

··· (ϕ0(ξm2 ))−1

(ξm2 −αk) ··· (ϕ0(ξm2 ))−1

(ξm2 −αk)κnk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= 1

(m2+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1


× · · · dμ(ξm2−κnk+1) · · · dμ(ξm2 )G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k)

ϕ20 (ξ0)ϕ

20(ξ1) · · ·ϕ2

0 (ξl1 )ϕ20(ξl1+1) · · ·ϕ2

0(ξl1+l2 ) · · ·ϕ20(ξm2−κnk+1) · · ·ϕ2

0(ξm2 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1 ) ϕ1(ξl1 ) ··· ϕl1 (ξl1 ) ϕl1+1(ξl1 ) ··· ϕl1+l2 (ξl1 ) ··· ϕm2−κnk+1(ξl1 ) ··· ϕ(n−1)K+k(ξl1 )


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1+l2 ) ϕ1(ξl1+l2 ) ··· ϕl1 (ξl1+l2 ) ϕl1+1(ξl1+l2 ) ··· ϕl1+l2 (ξl1+l2 ) ··· ϕm2−κnk+1(ξl1+l2 ) ··· ϕ(n−1)K+k(ξl1+l2 )

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2 (ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k(ξm1 )


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=G(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξm2−κnk+1,...,ξ(n−1)K+k)

= 1

(m2+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1



×· · · dμ(ξm2−κnk+1) · · · dμ(ξm2 )

×(

G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k)


)2

;

but, noting the determinantal factorisation

G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k)

= D V1(ξ0, ξ1, . . . , ξ(n−1)K+k),

where

D :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a0,0 a1,0 ··· al1 ,0 al1+1,0 ··· al1+l2 ,0 ··· am2−κnk+1,0 ··· a(n−1)K+k,0a0,1 a1,1 ··· al1 ,1 al1+1,1 ··· al1+l2 ,1 ··· am2−κnk+1,1 ··· a(n−1)K+k,1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,l1 a1,l1 ··· al1 ,l1 al1+1,l1 ··· al1+l2 ,l1 ··· am2−κnk+1,l1 ··· a(n−1)K+k,l1a0,l1+1 a1,l1+1 ··· al1 ,l1+1 al1+1,l1+1 ··· al1+l2 ,l1+1 ··· am2−κnk+1,l1+1 ··· a(n−1)K+k,l1+1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,l1+l2 a1,l1+l2 ··· al1 ,l1+l2 al1+1,l1+l2 ··· al1+l2 ,l1+l2 ··· am2−κnk+1,l1+l2 ··· a(n−1)K+k,l1+l2

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,m1 a1,m1 ··· al1 ,m1al1+1,m1

··· al1+l2 ,m1··· am2−κnk+1,m1 ··· a(n−1)K+k,m1

a0,m2 a1,m2 ··· al1 ,m2al1+1,m2

··· al1+l2 ,m2··· am2−κnk+1,m2 ··· a(n−1)K+k,m2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

( �=0),

(22)

andV1(ξ0, ξ1, . . . , ξ(n−1)K+k) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 ··· 1 1 ··· 1 ··· 1 ··· 1ξ0 ξ1 ··· ξl1 ξl1+1 ··· ξl1+l2 ··· ξm2−κnk+1 ··· ξ(n−1)K+k

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

l10 ξ

l11 ··· ξ

l1l1

ξl1l1+1 ··· ξ l1

l1+l2··· ξ l1

m2−κnk+1 ··· ξ l1(n−1)K+k

ξl1+10 ξ

l1+11 ··· ξ l1+1

l1ξ

l1+1l1+1 ··· ξ l1+1

l1+l2··· ξ l1+1

m2−κnk+1 ··· ξ l1+1(n−1)K+k

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

l1+l20 ξ

l1+l21 ··· ξ l1+l2

l1ξ

l1+l2l1+1 ··· ξ l1+l2

l1+l2··· ξ l1+l2

m2−κnk+1 ··· ξ l1+l2(n−1)K+k

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

m10 ξ

m11 ··· ξ

m1l1

ξm1l1+1 ··· ξm1

l1+l2··· ξm1

m2−κnk+1 ··· ξm1(n−1)K+k

ξm20 ξ

m21 ··· ξ

m2l1

ξm2l1+1 ··· ξm2

l1+l2··· ξm2

m2−κnk+1 ··· ξm2(n−1)K+k

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= ∏(n−1)K+ki, j=0

j<i

(ξi−ξ j),

it follows that, for n∈N and k=1, 2, . . . ,K,

cN = 1

(m2+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1) · · ·dμ(ξl1) dμ(ξl1+1) · · · dμ(ξl1+l2) · · ·

×· · ·dμ(ξm2−κnk+1) · · ·dμ(ξm2)D

2 ∏(n−1)K+ki, j=0

j<i

(ξi−ξ j)2

(ϕ0(ξ0)ϕ0(ξ1)· · ·ϕ0(ξl1)ϕ0(ξl1+1)· · ·ϕ0(ξl1+l2) · · ·ϕ0(ξm2−κnk+1)· · ·ϕ0(ξ(n−1)K+k))2

= D2

((n−1)K+k+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1) dμ(ξl1+1) · · ·


× · · · dμ(ξl1+l2) · · ·dμ(ξ(n−1)K+k−κnk+1) · · · dμ(ξ(n−1)K+k)

×(n−1)K+k∏

i, j=0j<i

(ξi−ξ j)2

⎛⎝(n−1)K+k∏

m=0

ϕ0(ξm)

⎞⎠

−2

= D2

((n−1)K+k+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1) dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2) · · ·dμ(ξ(n−1)K+k−κnk+1) · · · dμ(ξ(n−1)K+k)

(n−1)K+k∏i, j=0

j<i

(ξi−ξ j)2

×⎛⎝(n−1)K+k∏

l=0

s−1∏q=1

(ξl−αpq )κnkkq (ξl−αk)

κnk

⎞⎠

−2

⇒

cN = D2

((n−1)K+k+1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(ξ0) dμ(ξ1) · · · dμ(ξ(n−1)K+k)

×(n−1)K+k∏

i, j=0j<i

(ξi−ξ j)2

⎛⎝(n−1)K+k∏

m=0

s−1∏q=1

(ξm−αpq )κnkkq (ξm−αk)

κnk

⎞⎠

−2

(>0). (23)

Now, cD is studied. For n∈N and k=1, 2, . . . ,K, introduce the followingnotation:

ϕ0(z) :=s−1∏m=1


κnk−1=:(n−1)K+k−1∑

j=0

a�j,0z j,

and, for r=1, 2, . . . , s, q(r)=∑r−1i=1 li+1,

∑r−1i=1 li+2, . . . ,

∑r−1i=1 li+lr−δrs, and

m(r)=1, 2, . . . , lr−δrs,

ϕq(r)(z) :=⎧⎨⎩ϕ0(z)(z−αpr )

−m(r)=:(n−1)K+k−1∑

j=0

a�j,q(r)zj

⎫⎬⎭ ;


e.g., for r=1, the notation ϕq(1)(z) :={ϕ0(z)(z−αp1 )−m(1)=:∑(n−1)K+k−1

j=0 a�j,q(1)zj},

q(1)=1, 2, . . . , l1, m(1)=1, 2, . . . , l1, encapsulates the l1 :=κnkk1functions

ϕ1(z) = ϕ0(z)z−αp1

=:(n−1)K+k−1∑

j=0

a�j,1z j,

ϕ2(z) = ϕ0(z)(z−αp1 )

2=:(n−1)K+k−1∑

j=0

a�j,2z j, . . .

. . . , ϕl1 (z) =ϕ0(z)

(z−αp1 )l1=:(n−1)K+k−1∑

j=0

a�j,l1 z j,

for r=2, the notation ϕq(2)(z) :={ϕ0(z)(z−αp2)−m(2)=:∑(n−1)K+k−1

j=0 a�j,q(2)zj},

q(2)= l1+1, l1+2, . . . , l1+l2, m(2)=1, 2, . . . , l2, encapsulates the l2 :=κnkk2

functions

ϕl1+1(z) = ϕ0(z)z−αp2

=:(n−1)K+k−1∑

j=0

a�j,l1+1z j,

ϕl1+2(z) = ϕ0(z)(z−αp2 )

2=:(n−1)K+k−1∑

j=0

a�j,l1+2z j, . . .

. . ., ϕl1+l2 (z) =ϕ0(z)

(z−αp2 )l2=:(n−1)K+k−1∑

j=0

a�j,l1+l2z j,

etc., and, for r=s, the notation ϕq(s)(z) :={ϕ0(z)(z−αk)−m(s)=:∑(n−1)K+k−1

j=0 a�j,q(s)zj}, q(s)=(n−1)K+k−κnk+1, (n−1)K+k−κnk+2, . . . ,

(n−1)K+k−1,m(s)=1, 2, . . . , ls−1, encapsulates the ls−1=κnk−1 functions

ϕ(n−1)K+k−κnk+1(z) = ϕ0(z)z−αk

=:(n−1)K+k−1∑

j=0

a�j,(n−1)K+k−κnk+1z j,

ϕ(n−1)K+k−κnk+2(z) = ϕ0(z)(z−αk)

2=:(n−1)K+k−1∑

j=0

a�j,(n−1)K+k−κnk+2z j, . . .

. . . , ϕ(n−1)K+k−1(z) = ϕ0(z)(z−αk)

κnk−1=:(n−1)K+k−1∑

j=0

a�j,(n−1)K+k−1z j.

(Note: #{ϕ0(z)(z−αpr )−m(r)}= lr−δrs, r=1, 2, . . . , s, and # ∪s

r=1 {ϕ0(z)(z−αpr )

−m(r)}=∑s

r=1 lr− 1= (n−1)K+k−1.) One notes that, for n∈N andk=1, 2, . . . ,K, the l1+l2+· · ·+ (ls−1)+1=(n−1)K+k functions ϕ0(z), ϕ1(z),. . . , ϕl1(z), ϕl1+1(z), . . . , ϕl1+l2(z), . . . , ϕ(n− 1)K + k−κnk + 1(z), . . . , ϕ(n− 1)K + k− 1(z)are linearly independent on R, that is, for z∈R,

∑(n−1)K+k−1j=0 c�jϕ j(z)= 0 ⇒


(via a Vandermonde-type argument) c�j=0, j=0, 1, . . . , (n−1)K+k−1 (seethe ((n−1)K+k)× ((n−1)K+k) non-zero determinant D

� in (24) below). Forn∈N and k=1, 2, . . . ,K, let S(n−1)K+k denote the ((n−1)K+k)! permutationsof {0, 1, . . . , (n−1)K+k−1}. Using the above notation and the multi-linearity property of the determinant, one proceeds to study, for n∈N andk=1, 2, . . . ,K, cD:

cD =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξl1 )dμ(ξl1+1) · · · dμ(ξl1+l2 ) · · ·

× ···dμ(ξm2−κnk+1) ···dμ(ξm1 )

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)

1···(ξm1−αk)κnk−1

× 1

ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξm1 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ϕ0(ξ0) ϕ1(ξ0) ··· ϕl1 (ξ0) ϕl1+1(ξ0) ··· ϕl1+l2 (ξ0) ··· ϕm2−κnk+1(ξ0) ··· ϕ(n−1)K+k−1(ξ0)


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1 ) ϕ1(ξl1 ) ··· ϕl1 (ξl1 ) ϕl1+1(ξl1 ) ··· ϕl1+l2 (ξl1 ) ··· ϕm2−κnk+1(ξl1 ) ··· ϕ(n−1)K+k−1(ξl1 )

ϕ0(ξl1+1) ϕ1(ξl1+1) ··· ϕl1 (ξl1+1) ϕl1+1(ξl1+1) ··· ϕl1+l2 (ξl1+1) ··· ϕm2−κnk+1(ξl1+1) ··· ϕ(n−1)K+k−1(ξl1+1)

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1+l2 ) ϕ1(ξl1+l2 ) ··· ϕl1 (ξl1+l2 ) ϕl1+1(ξl1+l2 ) ··· ϕl1+l2 (ξl1+l2 ) ··· ϕm2−κnk+1(ξl1+l2 ) ··· ϕ(n−1)K+k−1(ξl1+l2 )

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξm1−1) ϕ1(ξm1−1) ··· ϕl1 (ξm1−1) ϕl1+1(ξm1−1) ··· ϕl1+l2 (ξm1−1) ··· ϕm2−κnk+1(ξm1−1) ··· ϕ(n−1)K+k−1(ξm1−1)

ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2 (ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k−1(ξm1 )

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=: G(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξm2−κnk+1,...,ξ(n−1)K+k−1)

= 1

m2!∑

σ∈Sm2

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξσ(0)) dμ(ξσ(1)) · · · dμ(ξσ(l1))dμ(ξσ(l1+1)) · · ·

× ···dμ(ξσ(l1+l2 ))···dμ(ξσ(m2−κnk+1)) ···dμ(ξσ(m1 ))

(ξσ(0)−αk)0(ξσ(1)−αp1 )

1 ···(ξσ(l1 )−αp1 )l1 (ξσ(l1+1)−αp2 )

1 ···(ξσ(l1+l2 )−αp2 )l2 ···(ξσ(m2−κnk+1)−αk)

1 ···(ξσ(m1 )−αk)κnk−1

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1)) · · · ϕ0(ξσ(l1))ϕ0(ξσ(l1+1)) · · · ϕ0(ξσ(l1+l2)) · · · ϕ0(ξσ(m2−κnk+1)) · · · ϕ0(ξσ(m1))

× G(ξσ(0), ξσ(1), . . . , ξσ(l1), ξσ(l1+1), . . . , ξσ(l1+l2), . . . , ξσ(m2−κnk+1), . . . , ξσ((n−1)K+k−1))

= 1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× · · · dμ(ξm2−κnk+1) · · ·dμ(ξm1 ) G(ξ0, ξ1, . . . , ξl1, ξl1+1, . . . , ξl1+l2

, . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−1)

×∑

σ∈Sm2

sgn(σσσσσσσσσ) 1(ξσ(0)−αk)

0(ξσ(1)−αp1 )1 ···(ξσ(l1 )−αp1 )

l1 (ξσ(l1+1)−αp2 )1 ···(ξσ(l1+l2 )−αp2 )

l2 ···(ξσ(m2−κnk+1)−αk)1 ···(ξσ(m1 )−αk)

κnk−1

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1)) · · · ϕ0(ξσ(l1))ϕ0(ξσ(l1+1)) · · · ϕ0(ξσ(l1+l2)) · · · ϕ0(ξσ(m2−κnk+1)) · · · ϕ0(ξσ(m1))

= 1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× · · · dμ(ξm2−κnk+1) · · ·dμ(ξm1 ) G(ξ0, ξ1, . . . , ξl1, ξl1+1, . . . , ξl1+l2

, . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−1)


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

(ϕ0(ξ0))−1

(ξ0−αk)0(ϕ0(ξ0 ))

−1

(ξ0−αp1 )··· (ϕ0(ξ0))

−1

(ξ0−αp1 )l1

(ϕ0(ξ0))−1

(ξ0−αp2 )··· (ϕ0(ξ0))

−1

(ξ0−αp2 )l2

··· (ϕ0(ξ0))−1

(ξ0−αk) ··· (ϕ0(ξ0))−1

(ξ0−αk)κnk−1

(ϕ0(ξ1))−1

(ξ1−αk)0(ϕ0(ξ1 ))

−1

(ξ1−αp1 )··· (ϕ0(ξ1))

−1

(ξ1−αp1 )l1

(ϕ0(ξ1))−1

(ξ1−αp2 )··· (ϕ0(ξ1))

−1

(ξ1−αp2 )l2

··· (ϕ0(ξ1))−1

(ξ1−αk) ··· (ϕ0(ξ1))−1

(ξ1−αk)κnk−1

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.

(ϕ0(ξl1))−1

(ξl1−αk)0

(ϕ0(ξl1))−1

(ξl1−αp1 )

··· (ϕ0(ξl1))−1

(ξl1−αp1 )

l1

(ϕ0(ξl1))−1

(ξl1−αp2 )

··· (ϕ0(ξl1))−1

(ξl1−αp2 )

l2··· (ϕ0(ξl1

))−1

(ξl1−αk) ··· (ϕ0(ξl1

))−1


(ϕ0(ξl1+1))−1

(ξl1+1−αk)0(ϕ0(ξl1+1))

−1

(ξl1+1−αp1 )··· (ϕ0(ξl1+1))

−1

(ξl1+1−αp1 )l1

(ϕ0(ξl1+1))−1

(ξl1+1−αp2 )··· (ϕ0(ξl1+1))

−1

(ξl1+1−αp2 )l2

··· (ϕ0(ξl1+1))−1

(ξl1+1−αk) ··· (ϕ0(ξl1+1))−1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.

(ϕ0(ξl1+l2))−1

(ξl1+l2−αk)0

(ϕ0(ξl1+l2))−1

(ξl1+l2−αp1 )

··· (ϕ0(ξl1+l2))−1

(ξl1+l2−αp1 )

l1

(ϕ0(ξl1+l2))−1

(ξl1+l2−αp2 )

··· (ϕ0(ξl1+l2))−1

(ξl1+l2−αp2 )

l2··· (ϕ0(ξl1+l2

))−1

(ξl1+l2−αk) ··· (ϕ0(ξl1+l2

))−1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.

(ϕ0(ξm1−1))−1

(ξm1−1−αk)0(ϕ0(ξm1−1))

−1

(ξm1−1−αp1 )··· (ϕ0(ξm1−1))

−1

(ξm1−1−αp1 )l1

(ϕ0(ξm1−1))−1

(ξm1−1−αp2 )··· (ϕ0(ξm1−1))

−1

(ξm1−1−αp2 )l2

··· (ϕ0(ξm1−1))−1

(ξm1−1−αk) ··· (ϕ0(ξm1−1))−1

(ξm1−1−αk)κnk−1

(ϕ0(ξm1 ))−1

(ξm1 −αk)0(ϕ0(ξm1 ))

−1

(ξm1 −αp1 )··· (ϕ0(ξm1 ))

−1

(ξm1 −αp1 )l1

(ϕ0(ξm1 ))−1

(ξm1 −αp2 )··· (ϕ0(ξm1 ))

−1

(ξm1 −αp2 )l2

··· (ϕ0(ξm1 ))−1

(ξm1 −αk) ··· (ϕ0(ξm1 ))−1


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= 1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


×· · · dμ(ξm2−κnk+1) · · ·dμ(ξm1 ) G(ξ0, ξ1, . . . , ξl1

, ξl1+1, . . . , ξl1+l2, . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−1)

ϕ 20 (ξ0)ϕ

20 (ξ1) · · · ϕ 2

0 (ξl1)ϕ 2

0 (ξl1+1) · · · ϕ 20 (ξl1+l2

) · · · ϕ 20 (ξm2−κnk+1) · · · ϕ 2

0 (ξm1 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1 ) ϕ1(ξl1 ) ··· ϕl1 (ξl1 ) ϕl1+1(ξl1 ) ··· ϕl1+l2 (ξl1 ) ··· ϕm2−κnk+1(ξl1 ) ··· ϕ(n−1)K+k−1(ξl1 )


.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1+l2 ) ϕ1(ξl1+l2 ) ··· ϕl1 (ξl1+l2 ) ϕl1+1(ξl1+l2 ) ··· ϕl1+l2 (ξl1+l2 ) ··· ϕm2−κnk+1(ξl1+l2 ) ··· ϕ(n−1)K+k−1(ξl1+l2 )

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξm1−1) ϕ1(ξm1−1) ··· ϕl1 (ξm1−1) ϕl1+1(ξm1−1) ··· ϕl1+l2 (ξm1−1) ··· ϕm2−κnk+1(ξm1−1) ··· ϕ(n−1)K+k−1(ξm1−1)


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

= G(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξm2−κnk+1,...,ξ(n−1)K+k−1)

= 1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× · · · dμ(ξm2−κnk+1) · · ·dμ(ξm1)

×(

G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2, . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−1)

ϕ0(ξ0)ϕ0(ξ1)· · ·ϕ0(ξl1)ϕ0(ξl1+1)· · ·ϕ0(ξl1+l2) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξm1)

)2

;


but, noting the determinantal factorisationG(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−1)

= D�V2(ξ0, ξ1, . . . , ξ(n−1)K+k−1),

whereD� :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a�0,0 a�1,0 ··· a�l1 ,0a�l1+1,0 ··· a�l1+l2 ,0

··· a�m2−κnk+1,0 ··· a�(n−1)K+k−1,0

a�0,1 a�1,1 ··· a�l1 ,1a�l1+1,1 ··· a�l1+l2 ,1

··· a�m2−κnk+1,1 ··· a�(n−1)K+k−1,1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a�0,l1a�1,l1

··· a�l1 ,l1a�l1+1,l1

··· a�l1+l2 ,l1··· a�m2−κnk+1,l1

··· a�(n−1)K+k−1,l1

a�0,l1+1 a�1,l1+1 ··· a�l1 ,l1+1 a�l1+1,l1+1 ··· a�l1+l2 ,l1+1 ··· a�m2−κnk+1,l1+1 ··· a�(n−1)K+k−1,l1+1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a�0,l1+l2a�1,l1+l2

··· a�l1 ,l1+l2a�l1+1,l1+l2

··· a�l1+l2 ,l1+l2··· a�m2−κnk+1,l1+l2

··· a�(n−1)K+k−1,l1+l2

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a�0,m1−1 a�1,m1−1 ··· a�l1 ,m1−1 a�l1+1,m1−1 ··· a�l1+l2 ,m1−1 ··· a�m2−κnk+1,m1−1 ··· a�(n−1)K+k−1,m1−1

a�0,m1a�1,m1

··· a�l1 ,m1a�l1+1,m1

··· a�l1+l2 ,m1··· a�m2−κnk+1,m1

··· a�(n−1)K+k−1,m1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

( �=0),

(24)

andV2(ξ0, ξ1, . . . , ξ(n−1)K+k−1) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 ··· 1 1 ··· 1 ··· 1 ··· 1ξ0 ξ1 ··· ξl1 ξl1+1 ··· ξl1+l2 ··· ξm2−κnk+1 ··· ξ(n−1)K+k−1

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

l10 ξ

l11 ··· ξ

l1l1

ξl1l1+1 ··· ξ l1

l1+l2··· ξ l1

m2−κnk+1 ··· ξ l1(n−1)K+k−1

ξl1+10 ξ

l1+11 ··· ξ l1+1

l1ξ

l1+1l1+1 ··· ξ l1+1

l1+l2··· ξ l1+1

m2−κnk+1 ··· ξ l1+1(n−1)K+k−1

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

l1+l20 ξ

l1+l21 ··· ξ l1+l2

l1ξ

l1+l2l1+1 ··· ξ l1+l2

l1+l2··· ξ l1+l2

m2−κnk+1 ··· ξ l1+l2(n−1)K+k−1

.

.

..... . .

.

.

..... . .

.

.

.. . .

.

.

.. . .

.

.

.ξ

m1−10 ξ

m1−11 ··· ξm1−1

l1ξ

m1−1l1+1 ··· ξm1−1

l1+l2··· ξm1−1

m2−κnk+1 ··· ξm1−1(n−1)K+k−1

ξm10 ξ

m11 ··· ξ

m1l1

ξm1l1+1 ··· ξm1

l1+l2··· ξm1

m2−κnk+1 ··· ξm1(n−1)K+k−1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=∏(n−1)K+k−1i, j=0

j<i

(ξi−ξ j),


cD = 1

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


×· · · dμ(ξm2−κnk+1) · · ·dμ(ξm1 )(D

�)2∏(n−1)K+k−1

i, j=0j<i

(ξi−ξ j)2

(ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξ(n−1)K+k−1))2

= (D�)2

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2 ) · · ·dμ(ξ(n−1)K+k−κnk+1) · · · dμ(ξ(n−1)K+k−1)


×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2

((n−1)K+k−1∏

m=0

ϕ0(ξm)

)−2

= (D�)2

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξl1)dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2) · · · dμ(ξ(n−1)K+k−κnk+1) · · · dμ(ξ(n−1)K+k−1)

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2

⎛⎝(n−1)K+k−1∏

l=0

s−1∏q=1

(ξl−αpq)κnkkq (ξl−αk)

κnk−1

⎞⎠

−2

⇒

cD = (D�)2

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξ(n−1)K+k−1)

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2

⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1

(ξm−αpq)κnkkq (ξm−αk)

κnk−1

⎞⎠

−2

(>0).

(25)

Hence, for n∈N and k=1, 2, . . . ,K, (23) and (25) establish the existenceand the uniqueness of the monic FPC ORF, πππn

k : N × {1, 2, . . . ,K} × C \{α1, α2, . . . , αK}→C, (n,k, z) →πππn

k(z)=Y11(z)/(z−αk).There remains, still, the question of the existence and the uniqueness

of Y21 : N × {1, 2, . . . ,K} × C \ {α1, α2, . . . , αK}→C, which necessitates a de-tailed analysis of the elements of the second row of Y (z), that is, Y2m(z), m=1, 2; thus the gist of the subsequent calculations. Recalling that

∫Rξmw(ξ)dξ <

∞ and∫

R(ξ−α j)

−(m+1)w(ξ)dξ <∞, m∈Z+0 , j=1, 2, . . . ,K, it follows via

the expansion 1z1−z2

=∑li=0

zi2

zi+11+ zl+1

2

zl+11 (z1−z2)

, l∈Z+0 , the integral representation

(cf. (14)) Y22(z)=∫

R

(z−αk)Y21(ξ)w(ξ)

(ξ−αk)(ξ−z)dξ2π i , z∈C \ R, and the second of each of the

asymptotic conditions (11), (12), and (13) that, for n∈N and k=1, 2, . . . ,K,∫

R

(Y21(ξ)

ξ−αk

)w(ξ)

(ξ−αk)pdξ=0, p=0, 1, . . . ,κnk−2, (26)

∫R

(Y21(ξ)

ξ−αk

)w(ξ)

(ξ−αk)κnk−1dξ=2π i, (27)

∫R

(Y21(ξ)

ξ−αk

)w(ξ)

(ξ−αpq)r

dξ=0, q=1, 2, . . . , s−1, r=1, 2, . . . ,κnkkq. (28)


Note: for n=1 and k=1, 2, . . . ,K, depending on the distribution of thereal poles α1, α2, . . . , αK, it can happen that the corresponding κ1k<2, inwhich case, (26) is vacuous (of course, for n�2, κnk �2, k=1, 2, . . . ,K);moreover, if, for n∈N and k=1, 2, . . . ,K, the set {αk′ ; k′ ∈{1, 2, . . . ,K}, αk′ �=αk}=∅, then the corresponding (28) are vacuous (actually, this can onlyoccur if n=1). Via the above ordered disjoint partition for the repeated realpole sequence {α1, α2, . . . , αK} ∪ · · · ∪ {α1, α2, . . . , αK} ∪ {α1, α2, . . . , αk} and(26)–(28), one writes, in the indicated order,

Y21(z)z−αk

=s−1∑q=1

lq∑r=1

νq,r(n,k)(z−αi(q)kq

)r+

ls∑m=1

νs,m(n,k)(z−αi(s)ks

)m−1

=s−1∑q=1

lq=κnkkq∑r=1

νq,r(n,k)(z−αpq)

r+

ls=κnk∑m=1

νs,m(n,k)(z−αk)m−1

=l1=κnkk1∑

r=1

ν1,r(n,k)(z−αp1)

r+· · ·+

ls−1=κnkks−1∑r=1

νs−1,r(n,k)(z−αps−1)

r

+ls=κnk∑

m=1

νs,m(n,k)(z−αk)m−1

, νs,ls(n,k) �=0.

Substituting the latter partial fraction expansion for Y21(z)/(z−αk) into(26)–(28), one arrives at, for n∈N and k=1, 2, . . . ,K, the orthogonalityconditions

∫R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1

νm,q(n, k)(ξ−αpm)

q +ls=κnk∑

q=1

νs,q(n, k)(ξ−αk)

q−1

⎞⎠ w(ξ)

(ξ−αk)r dξ=0,

r=0, 1, . . . ,κnk−2, (29)

∫R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1

νm,q(n, k)(ξ−αpm)

q +ls=κnk∑

q=1


q−1

⎞⎠ w(ξ)

(ξ−αk)κnk−1

dξ=2π i, (30)

∫R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1

νm,q(n, k)(ξ − αpm)

q +ls=κnk∑

q=1


q−1

⎞⎠ w(ξ)

(ξ−αpi )j dξ=0,

i =1, 2, . . . , s−1, j=1, 2, . . . , li. (31)

Incidentally, for n∈N and k=1, 2, . . . ,K, via the orthogonality conditions (29)and (31), condition (30) can be manipulated thus:

2π i =∫

R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1


q+

ls=κnk∑q=1

νs,q(n,k)(ξ−αk)q−1

⎞⎠ νs,ls(n,k)(ξ−αk)κnk−1


× w(ξ)

νs,ls(n,k)dξ =

∫R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1


q+

ls=κnk∑q=1


⎞⎠

×⎛⎝s−1∑

m=1

lm=κnkkm∑q=1


q+

ls−1=κnk−1∑q=1


+ νs,ls(n,k)(ξ−αk)κnk−1

⎞⎠

× w(ξ)

νs,ls(n,k)dξ =

∫R

⎛⎝s−1∑

m=1

lm=κnkkm∑q=1


q+

ls=κnk∑q=1


⎞⎠

︸︷︷︸=Y21(ξ)/(ξ−αk)

×⎛⎝s−1∑

m=1

lm=κnkkm∑q=1


q+

ls=κnk∑q=1


⎞⎠

︸︷︷︸=Y21(ξ)/(ξ−αk)

w(ξ)

νs,ls(n,k)dξ

=∫

R

(Y21(ξ)

ξ−αk

)2w(ξ)

νs,ls(n,k)dξ ;

hence, for n∈N and k=1, 2, . . . ,K, one arrives at the interesting ‘normalisa-tion formula’:

∫R

(Y21(ξ)

ξ−αk

)2

w(ξ)dξ=2π i νs,ls(n,k).

For n∈N and k=1, 2, . . . ,K, the orthogonality conditions (29)–(31)give rise to a total of (cf. Subsection 1.2; recall that

∑s

r=1 lr =∑s−1r=1 lr+ ls=∑s−1

r=1 κnkkr+ κnk=(n−1)K+k)(n−1)K+k linear inhomogeneous algebraic

equations for the (n−1)K+k real unknowns ν1,1(n,k), . . . , ν1,l1(n,k),ν2,1(n,k), . . . , ν2,l2(n,k), . . . , νs,1(n,k), . . . , νs,ls(n,k), namely:

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp1 )

··· ∫R


(ξ0−αp1 )l1

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp2 )

··· ∫R


(ξ0−αp2 )l2

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )l1

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )l2

......

......

......∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

l2

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )l1

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )l2

......

......

......∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

l1

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2

......

......

......∫

R


(ξm1 −αp1 )··· ∫

R


(ξm1 −αp1 )l1

∫R


(ξm1 −αp2 )··· ∫

R


(ξm1 −αp2 )l2


··· ··· ··· ∫R


(ξ0−αk)0··· ∫

R


(ξ0−αk)κnk−1

··· ··· ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)0··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)κnk−1

.........

......

...

··· ··· ··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk)0

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)


··· ··· ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk)0··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)


.........

......

...

··· ··· ··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk)0

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)


.........

......

...

··· ··· ··· ∫R


(ξm1 −αk)0··· ∫

R



⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

×

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ν1,1(n,k)...

ν1,l1(n,k)ν2,1(n,k)...

ν2,l2(n,k)...

νs,1(n,k)...

νs,ls(n,k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

0...

00...

0...

02π i

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (32)

where m1 :=(n−1)K+k−1. For n∈N and k=1, 2, . . . ,K, the linear sys-tem (32) of (n−1)K+k inhomogeneous algebraic equations for the (n−1)K+k real unknowns ν1,1(n,k), . . . , ν1,l1(n,k), ν2,1(n,k), . . . , ν2,l2(n,k), . . . ,νs,1(n,k), . . . , νs,ls(n,k) admits a unique solution if, and only if, the determi-nant of the coefficient matrix, denoted N �(n,k), is non-zero: this fact will nowbe established. Via the multi-linearity property of the determinant, one showsthat, for n∈N and k=1, 2, . . . ,K, N �(n,k) is given by

N �(n,k) =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× ···dμ(ξm2−κnk+1)···dμ(ξm1 )

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)



×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk )0

··· 1

(ξ0−αk )κnk−1

1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk )0

··· 1

(ξ1−αk )κnk−1

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1−αp1 )

··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk )0

··· 1

(ξl1−αk )κnk−1

1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk )0

··· 1

(ξl1+1−αk )κnk−1

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1+l2−αp1 )

··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk )0

··· 1

(ξl1+l2−αk )κnk−1

.

.

.. . .

.

.

.

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξm1 −αp1 )··· 1

(ξm1 −αp1 )l1

1(ξm1 −αp2 )

··· 1

(ξm1 −αp2 )l2

··· 1(ξm1 −αk )0

··· 1

(ξm1 −αk )κnk−1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

,

where m2 :=(n−1)K+k. Recalling, for n∈N and k=1, 2, . . . ,K, the (n−1)K+k linearly independent (on R) functions ϕ0(z), ϕ1(z), . . . , ϕl1(z), ϕl1+1(z), . . . ,ϕl1+l2(z), . . . , ϕ(n−1)K+k−κnk+1(z), . . . , ϕ(n−1)K+k−1(z) introduced above for theanalysis of cD, one shows that N �(n,k) can be written as

N �(n,k) =∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× · · · dμ(ξm2−κnk+1) · · · dμ(ξm1 )

× (−1)(n−1)K+k−κnk (ϕ0(ξ0)ϕ0(ξ1)···ϕ0(ξl1 )ϕ0(ξl1+1)···ϕ0(ξl1+l2 )···ϕ0(ξm2−κnk+1)···ϕ0(ξm1 ))−1

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



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.. . .

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.. . .

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.ϕ0(ξm1−1) ϕ1(ξm1−1) ··· ϕl1 (ξm1−1) ϕl1+1(ξm1−1) ··· ϕl1+l2 (ξm1−1) ··· ϕm2−κnk+1(ξm1−1) ··· ϕ(n−1)K+k−1(ξm1−1)


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

Modulo the factor (−1)(n−1)K+k−κnk , this is the same determinantal expressionencountered while studying cD; therefore, for n∈N and k=1, 2, . . . ,K,

N �(n,k) = (−1)(n−1)K+k−κnk (D�)2

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0) dμ(ξ1) · · · dμ(ξ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1


κnk−1

⎞⎠−2

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2 ( �=0), (33)


whence follows, for n∈N and k=1, 2, . . . ,K, the existence and the uniquenessof Y21(z)/(z−αk). ��

Corollary 2.1 Let V: R\{α1, α2, . . . , αK}→R satisfy conditions (3)–(5). LetY :N × {1, 2, . . . ,K} × C \ R→SL2(C) be the unique solution of RHPFPC withintegral representation given in Lemma 2.1, where, in particular, πππn

k : N×{1, 2, . . . ,K} × C \ {α1, α2, . . . , αK} → C, (n,k, z) → πππn

k(z) := (μn,κnk(n,k))−1

·φnk(z) = Y11(z)/(z−αk) is the monic FPC ORF defined in Subsection 1.2, withμn,κnk(n,k) the associated norming constant, and φn

k(z) the corresponding FPCorthonormal rational function. Then, for n∈N and k=1, 2, . . . ,K,

μn,κnk(n,k)=√(n−1)K+k+1

√ς(n,k)λ(n,k)

,

where

ς(n,k) := (D�)2∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0) dμ(ξ1) · · ·dμ(ξ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1


κnk−1

⎞⎠

−2

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2,

λ(n,k) := D2∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k+1

dμ(τ0) dμ(τ1) · · · dμ(τ(n−1)K+k)

×⎛⎝(n−1)K+k∏

m=0

s−1∏q=1

(τm−αpq )κnkkq (τm−αk)

κnk

⎞⎠

−2

×(n−1)K+k∏

i, j=0j<i

(τi−τ j)2,

with dμ(z)=exp(−N V(z))dz, N∈N,

D=(−1)(n−1)K+k det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

A (1)

A (2)

.

.

.

A (s−1)

A (s)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,


where (with the convention∑0

m=1 ∗∗∗:=0), for r=1, . . . , s−1, s,

i(r) = 1+∑r−1

m=1lm, 2+

∑r−1

m=1lm, . . . , lr+

∑r−1

m=1lm,

q(i(r), r) = i(r)−∑r−1

m=1lm,

(A (r))i(r) j(r) =

⎧⎪⎪⎨⎪⎪⎩

(−1)q(i(r),r)∏q(i(r),r)−1m=0 (lr−m)

(∂

∂αpr

)q(i(r),r)a j(r)−1(α), j(r)=1, 2, . . . , (n−1)K+k−q(i(r), r)+1,

0, j(r)=(n−1)K+k−q(i(r), r)+2, . . . , (n−1)K+k,

am1(α) :=

∑ip=0,1,...,l p

p∈{1,2,...,s−1}is=0,1,...,κnk∑s

m=1 im=(n−1)K+k−m1

(−1)(n−1)K+k−m1s−1∏j=1

l j!i j!(l j−i j)!

κnk!is !(κnk−is )!

×s−1∏m=1

(αpm )im (αk)

is ,

and

D�=(−1)(n−1)K+k+1 det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

A �(1)

A �(2)...

A �(s−1)

A �(s)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where, for r=1, . . . , s−1, s,

i(r) = 1+∑r−1

m=1lm, 2+

∑r−1

m=1lm, . . . , lr−δrs+

∑r−1

m=1lm,

q(i(r), r)= i(r)−∑r−1

m=1lm,

(A �(r))i(r) j(r) =

⎧⎪⎪⎨⎪⎪⎩

(−1)q(i(r),r)∏q(i(r),r)−1m=0 (lr−δrs−m)

(∂

∂αpr

)q(i(r),r)a�j(r)−1(α), j(r)=1, 2, . . . , (n−1)K+k−q(i(r), r),

0, j(r)=(n−1)K+k−q(i(r), r)+1, . . . , (n−1)K+k−1,

a�m1(α) :=

∑ip=0,1,...,l p

p∈{1,2,...,s−1}is=0,1,...,κnk−1∑s

m=1 im=(n−1)K+k−m1−1

(−1)(n−1)K+k−m1−1s−1∏j=1


× (κnk−1)!is !(κnk−1−is )!

s−1∏m=1

(αpm )im (αk)

is.


Proof Recall from the proof of Lemma 2.1 that, for n∈N and k=1, 2, . . . ,K,(μn,κnk(n,k))

−2= cN/cD, where cN is given in (23) (with the determinant D

given in (22)), and cD is given in (25) (with the determinant D� given in

(24)); hence, substituting (23) and (25) into the formula for (μn,κnk(n,k))−2

and taking the positive square root of both sides of the resulting quotient, onearrives at the formula for μn,κnk(n,k) stated in the Corollary. ��

Lemma 2.2 Let V: R \ {α1, α2, . . . , αK}→R satisfy conditions (3)–(5). Let Y :N × {1, 2, . . . ,K} × C \ R→SL2(C) be the unique solution of RHPFPC withintegral representation given in Lemma 2.1, where, in particular, πππn

k : N×{1, 2, . . . ,K} × C \ {α1, α2, . . . , αK} → C, (n,k, z) → πππn

k(z) := (μn,κnk(n,k))−1

·φnk(z) = Y11(z)/(z−αk) is the monic FPC ORF defined in Subsection 1.2,

with μn,κnk(n,k) given in Corollary 2.1, and φnk(z) is the corresponding FPC

orthonormal rational function. Then, for n∈N and k=1, 2, . . . ,K, πππnk(z) and

Y21(z)/(z−αk) have, respectively, the following integral representations:

πππnk(z)=

Y11(z)z−αk

= D

D�

⎛⎝s−1∏

q=1

(z−αpq)κnkkq (z−αk)

κnk

⎞⎠

−1

ϒN(n,k; z)ϒD(n,k)

,

where D and D� are given in Corollary 2.1,

ϒN(n,k; z) :=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1


κnk

⎞⎠−2

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−1∏

l=0

(ξl−αk)(z−ξl),

ϒD(n,k) :=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(τ0)dμ(τ1) · · · dμ(τ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1

(τm−αpq)κnkkq (τm−αk)

κnk−1

⎞⎠

−2

×(n−1)K+k−1∏

i, j=0j<i

(τi−τ j)2,


with dμ(z)=exp(−N V(z))dz, N∈N; and

Y21(z)z−αk

=2π i((n−1)K+k)D

D�

⎛⎝s−1∏

q=1


κnk−1

⎞⎠

−1

N(n,k; z)D(n,k)

,

where

D =(−1)(n−1)K+k det

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

A (1)

A (2)...

A (s−1)

A (s)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠,

where (with the convention∑0

m=1 ∗∗∗:=0), for r=1, . . . , s−1, s,

i(r) = 1+�r−1m=1lm, 2+�r−1

m=1lm, . . . , lr−2δrs+�r−1m=1lm,

q(i(r), r)= i(r)−�r−1m=1lm,

(A (r))i(r) j(r)=

⎧⎪⎨⎪⎩

(−1)q(i(r),r)∏q(i(r),r)−1m=0 (lr−2δrs−m)

(∂

∂αpr

)q(i(r),r)a j(r)−1(α), j(r)=1,2, . . . , (n−1)K+k−q(i(r),r)−1,

0, j(r)=(n−1)K+k−q(i(r), r), . . . , (n−1)K+k−2,

a m1(α) :=

∑ip=0,1,...,l p

p∈{1,2,...,s−1}is=0,1,...,κnk−2∑s

m=1 im=(n−1)K+k−m1−2

(−1)(n−1)K+k−m1s−1∏j=1


× (κnk−2)!is !(κnk−2−is )!

s−1∏m=1

(αpm )im (αis

k ),

and

ΛN(n,k; z) :=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1

dμ(ξ0)dμ(ξ1) · · · dμ(ξ(n−1)K+k−2)

×⎛⎝(n−1)K+k−2∏

m=0

s−1∏q=1


κnk−1

⎞⎠

−2

×(n−1)K+k−2∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−2∏

l=0

(ξl−αk)(z−ξl),


ΛD(n,k) :=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(τ0)dμ(τ1) · · · dμ(τ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1

(τm−αpq)κnkkq (τm−αk)

κnk−1

⎞⎠

−2

×(n−1)K+k−1∏

i, j=0j<i

(τi−τ j)2.

Proof For n∈N and k=1, 2, . . . ,K, recall, from the proof of Lemma 2.1,the ordered disjoint partition for {α1, α2, . . . , αK} ∪ · · · ∪ {α1, α2, . . . , αK} ∪{α1, α2, . . . , αk} and the corresponding formula for the monic FPC ORF,

πππnk(z)=

Y11(z)z−αk

= φ0(n,k)+s−1∑m=1

lm=κnkkm∑q=1


q

+ls−1=κnk−1∑

q=1

νs,q(n,k)(z−αk)q

+ 1

(z−αk)κnk,

where the (n−1)K+k+1 real coefficients φ0(n,k), ν1,1(n,k), . . . , ν1,l1(n,k),ν2,1(n,k), . . . , ν2, l2(n,k), . . . , νs, 1(n,k), . . . , νs,ls−1(n,k), μn,κnk(n,k), withμn,κnk(n,k) the ‘norming constant’ (cf. Corollary 2.1), satisfy the linearinhomogeneous algebraic system (21) given in the proof of Lemma 2.1. Viathe multi-linearity property of the determinant and an application of Cramer’sRule to system (21), one arrives at, for n∈N and k=1, 2, . . . ,K, the following(ordered) determinantal representation for the monic FPC ORF:

πππnk(z)=

Y11(z)z−αk

= Ξnk (z)

cD,


where cD is given in the proof of Lemma 2.1, (25), and, with m1 :=(n−1)K+k−1,

Ξnk (z) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∫R

dμ(ξ0)(ξ0−αk)0

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp1 )

··· ∫R


(ξ0−αp1 )l1

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp2 )

··· ···

∫R

dμ(ξ1)(ξ1−αp1 )

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )l1

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )··· ···

.

.

....

. . ....

.

.

.. . .. . .

∫R

dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

··· ···

∫R

dμ(ξl1+1)

(ξl1+1−αp2 )

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )l1

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )··· ···

.

.

....

. . ....

.

.

.. . .. . .

∫R

dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

l1

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

··· ···

.

.

....

. . ....

.

.

.. . .. . .

∫R

dμ(ξm1 )

(ξm1−αk)κnk−1

∫R

(ξm1−αk)−κnk+1 dμ(ξm1 )(ξm1−αp1 )

··· ∫R

(ξm1−αk)−κnk+1 dμ(ξm1 )

(ξm1−αp1 )l1

∫R

(ξm1−αk)−κnk+1 dμ(ξm1 )(ξm1−αp2 )

··· ···

1 1z−αp1

··· 1

(z−αp1 )l1

1z−αp2

··· ···

··· ∫R


(ξ0−αp2 )l2

··· ··· ∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αk) ··· ∫

R

(ξ0−αk)0 dμ(ξ0)(ξ0−αk)κnk

··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )l2

··· ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk) ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)κnk

. . ....

. . .. . .

.

.

.. . .

.

.

.

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

l2··· ··· ∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk) ··· ∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk)κnk

··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )l2

··· ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk) ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk)κnk

. . ....

. . .. . .

.

.

.. . .

.

.

.

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2··· ··· ∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk) ··· ∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk)κnk

. . ....

. . .. . .

.

.

.. . .

.

.

.

··· ∫R


(ξm1 −αp2 )l2

··· ··· ∫R

(ξm1 −αk)−κnk+1 dμ(ξm1 )(ξm1 −αk) ··· ∫

R


(ξm1 −αk)κnk

··· 1

(z−αp2 )l2

··· ··· 1z−αk ··· 1

(z−αk)κnk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× ··· dμ(ξm2−κnk+1) ··· dμ(ξm1 )

(ξ0−αk)0(ξ1−αp1 )1···(ξl1−αp1 )

l1 (ξl1+1−αp2 )1···(ξl1+l2−αp2 )

l2 ···(ξm2−κnk+1−αk)1···(ξm1−αk)κnk−1


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk

1 1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk

......

......

......

......

......

...1 1

(ξl1−αp1 )

··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1

(ξl1−αk)κnk

1 1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1

(ξl1+1−αk)κnk

......

......

......

......

......

...1 1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk) ··· 1

(ξl1+l2−αk)κnk

......

......

......

......

......

...1 1(ξm1 −αp1 )

··· 1

(ξm1 −αp1 )l1

1(ξm1 −αp2 )

··· 1

(ξm1 −αp2 )l2

··· 1(ξm1 −αk)

··· 1(ξm1 −αk)

κnk

1 1z−αp1

··· 1

(z−αp1 )l1

1z−αp2

··· 1

(z−αp2 )l2

··· 1z−αk ··· 1

(z−αk)κnk

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

,

where dμ(z) is given in the Lemma. For n∈N and k=1, 2, . . . ,K, recalling,from the proof of Lemma 2.1, the (n−1)K+k+1 linearly independent (onR) functions ϕ0(z) :=∏s−1

m=1(z−αpm)lm(z−αk)

κnk =:∑(n−1)K+kj=0 a j,0z j, ϕq(r1)(z) :=

{ϕ0(z)(z−αpr1)−m(r1)=:∑(n−1)K+k

j=0 a j,q(r1)zj}, r1=1, 2, . . . , s, q(r1)=∑r1−1

i=1 li+1,∑r1−1i=1 li+2, . . . ,

∑r1−1i=1 li+lr1 , m(r1)=1, 2, . . . , lr1 , and the (n−1)K+k linearly

independent (on R) functions ϕ0(z) :=∏s−1m=1(z−αpm)

lm(z−αk)κnk−1=:∑(n−1)K+k−1

j=0 a�j,0z j, ϕq(r2)(z):={ϕ0(z)(z−αpr2)−m(r2)=:∑(n−1)K+k−1

j=0 a�j,q(r2)z j}, r2=

1, 2, . . . , s, q(r2)=∑r2−1i=1 li+1,

∑r2−1i=1 li+2, . . . ,

∑r2−1i=1 li+lr2−δr2s,m(r2)= 1, 2,

. . . , lr2−δr2s, one proceeds, via the latter determinantal expression, with theanalysis of Ξn

k (z) (with m2 :=(n−1)K+k):

Ξnk (z)=

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · ·dμ(ξl1)dμ(ξl1+1) · · ·dμ(ξl1+l2) · · ·

× ···dμ(ξm2−κnk+1) ···dμ(ξm1 )

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)


× (ϕ0(z))−1

ϕ0(ξ0)ϕ0(ξ1) · · ·ϕ0(ξl1)ϕ0(ξl1+1) · · ·ϕ0(ξl1+l2) · · ·ϕ0(ξm2−κnk+1) · · ·ϕ0(ξm1)

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ϕ0(ξ0) ϕ1(ξ0) ··· ϕl1(ξ0) ϕl1+1(ξ0) ··· ϕl1+l2

(ξ0) ··· ϕm2−κnk+1(ξ0) ··· ϕ(n−1)K+k(ξ0)

ϕ0(ξ1) ϕ1(ξ1) ··· ϕl1(ξ1) ϕl1+1(ξ1) ··· ϕl1+l2

(ξ1) ··· ϕm2−κnk+1(ξ1) ··· ϕ(n−1)K+k(ξ1)

.

.

.

.

.

.

...

.

.

.

.

.

.

...

.

.

.

...

.

.

.

...

.

.

.ϕ0(ξl1

) ϕ1(ξl1) ··· ϕl1

(ξl1) ϕl1+1(ξl1

) ··· ϕl1+l2(ξl1) ··· ϕm2−κnk+1(ξl1

) ··· ϕ(n−1)K+k(ξl1)

ϕ0(ξl1+1) ϕ1(ξl1+1) ··· ϕl1 (ξl1+1) ϕl1+1(ξl1+1) ··· ϕl1+l2(ξl1+1) ··· ϕm2−κnk+1(ξl1+1) ··· ϕ(n−1)K+k(ξl1+1)

.

.

.

.

.

.

...

.

.

.

.

.

.

...

.

.

.

...

.

.

.

...

.

.

.ϕ0(ξl1+l2

) ϕ1(ξl1+l2) ··· ϕl1 (ξl1+l2

) ϕl1+1(ξl1+l2) ··· ϕl1+l2

(ξl1+l2) ··· ϕm2−κnk+1(ξl1+l2

) ··· ϕ(n−1)K+k(ξl1+l2)

.

.

.

.

.

.

...

.

.

.

.

.

.

...

.

.

.

...

.

.

.

...

.

.

.ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2

(ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k(ξm1 )

ϕ0(z) ϕ1(z) ··· ϕl1(z) ϕl1+1(z) ··· ϕl1+l2

(z) ··· ϕm2−κnk+1(z) ··· ϕ(n−1)K+k(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=:G (ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2,...,ξ(n−1)K+k−1;z)


= (ϕ0(z))−1

m2!∑

σ∈Sm2

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


× ···dμ(ξσ(l1+l2))···dμ(ξσ(m2−κnk+1)) ···dμ(ξσ(m1))

(ξσ(0)−αk)0(ξσ(1)−αp1 )

1···(ξσ(l1)−αp1 )l1 (ξσ(l1+1)−αp2 )


1···(ξσ(m1)−αk)κnk−1

× 1


× G (ξσ(0), ξσ(1), . . . , ξσ(l1), ξσ(l1+1), . . . , ξσ(l1+l2), . . . , ξσ((n−1)K+k−1); z)

= (ϕ0(z))−1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · · dμ(ξl1+l2 ) . . .

× · · · dμ(ξm2−κnk+1) · · · dμ(ξm1 )G (ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξ(n−1)K+k−1; z)

×∑

σ∈Sm2


0(ξσ(1)−αp1)1···(ξσ(l1)−αp1)

l1(ξσ(l1+1)−αp2)1···(ξσ(l1+l2)−αp2)

l2···(ξσ(m2−κnk+1)−αk)1···(ξσ(m1)−αk)

κnk−1

× 1


= (ϕ0(z))−1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k




×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1(ξ0−αk)0

1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk−1

1(ξ1−αk)0

1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk−1

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1−αk)0

1(ξl1

−αp1 )··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1


1(ξl1+1−αk)0

1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1+l2−αk)0

1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξm1 −αk)01

(ξm1 −αp1 )··· 1

(ξm1 −αp1 )l1

1(ξm1 −αp2 )

··· 1

(ξm1 −αp2 )l2

··· 1(ξm1 −αk) ··· 1


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= (ϕ0(z))−1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k





× 1


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2 (ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k−1(ξm1 )

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸=D�

∏(n−1)K+k−1i, j=0

j<i

(ξi−ξ j) (cf. proof of Lemma 2.1)

= D�(ϕ0(z))−1

m2!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k


×· · · dμ(ξm2−κnk+1) · · · dμ(ξm1 )

∏(n−1)K+k−1i, j=0

j<i

(ξi − ξ j)


× 1


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1 ) ϕ1(ξl1 ) ··· ϕl1 (ξl1 ) ϕl1+1(ξl1 ) ··· ϕl1+l2 (ξl1 ) ··· ϕm2−κnk+1(ξl1 ) ··· ϕ(n−1)K+k(ξl1 )


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξl1+l2 ) ϕ1(ξl1+l2 ) ··· ϕl1 (ξl1+l2 ) ϕl1+1(ξl1+l2 ) ··· ϕl1+l2 (ξl1+l2 ) ··· ϕm2−κnk+1(ξl1+l2 ) ··· ϕ(n−1)K+k(ξl1+l2 )

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξm1 ) ϕ1(ξm1 ) ··· ϕl1 (ξm1 ) ϕl1+1(ξm1 ) ··· ϕl1+l2 (ξm1 ) ··· ϕm2−κnk+1(ξm1 ) ··· ϕ(n−1)K+k(ξm1 )

ϕ0(z) ϕ1(z) ··· ϕl1 (z) ϕl1+1(z) ··· ϕl1+l2 (z) ··· ϕm2−κnk+1(z) ··· ϕ(n−1)K+k(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=D∏(n−1)K+k−1

i, j=0j<i

(ξi−ξ j)∏(n−1)K+k−1

m=0 (z−ξm) (cf. Proof of Lemma 2.1)

= DD�(ϕ0(z))−1

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2 ) · · · dμ(ξ(n−1)K+k−1)

× (ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξ(n−1)K+k−1))−1

ϕ0(ξ0)ϕ0(ξ1) · · ·ϕ0(ξl1 )ϕ0(ξl1+1) · · ·ϕ0(ξl1+l2 ) · · ·ϕ0(ξm2−κnk+1) · · ·ϕ0(ξ(n−1)K+k−1)

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−1∏

m=0

(z−ξm);


but, recalling that, for n∈N and k=1, 2, . . . ,K, ϕ0(z)=ϕ0(z)/(z−αk), onearrives at

Ξnk (z) =

DD�(ϕ0(z))−1

((n−1)K+k)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξl1)dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2) · · · dμ(ξ(n−1)K+k−1)

((n−1)K+k−1∏

l=0

ϕ0(ξl)

)−2

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−1∏

m=0

(ξm−αk)(z−ξm) ⇒

Ξnk (z) =

DD�

((n−1)K+k)!

⎛⎝s−1∏

q=1


κnk

⎞⎠

−1

×∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k

dμ(ξ0)dμ(ξ1) · · · dμ(ξ(n−1)K+k−1)

×⎛⎝(n−1)K+k−1∏

m=0

s−1∏q=1


κnk

⎞⎠

−2

×(n−1)K+k−1∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−1∏

l=0

(ξl−αk)(z−ξl).

Recalling that, for n∈N and k=1, 2, . . . ,K, πππnk(z)=Y11(z)/(z−αk)=Ξn

k (z)/cD, where Ξn

k (z) is given directly above, and cD is given in the proof ofLemma 2.1, (25), one arrives at the integral representation for πππn

k(z) statedin the Lemma.

The determinantal representation for Y21(z)/(z−αk) is now studied.For n∈N and k=1, 2, . . . ,K, recall the ordered disjoint partition for{α1, α2, . . . , αK} ∪ · · · ∪ {α1, α2, . . . , αK} ∪ {α1, α2, . . . , αk} introduced in theproof of Lemma 2.1. For this ordered disjoint partition, introduce, for n∈N

and k=1, 2, . . . ,K, the following notation (cf. Subsection 1.2; recall that lm :=κnkkm

, m=1, 2, . . . , s−1, and ls :=κnk):

ϕ0(z) :=s−1∏m=1


κnk−2=:(n−1)K+k−2∑

j=0

a j,0z j,


and, for r=1, 2, . . . , s, q(r)=∑r−1i=1 li+1,

∑r−1i=1 li+2, . . . ,

∑r−1i=1 li+lr−2δrs, and

m(r)=1, 2, . . . , lr−2δrs,

ϕq(r)(z) :=⎧⎨⎩ϕ0(z)(z−αpr )

−m(r)=:(n−1)K+k−2∑

j=0

a j,q(r)z j

⎫⎬⎭ ;

e.g., for r=1, the notation ϕq(1)(z) :={ϕ0(z)(z−αp1)−m(1)=: ∑(n−1)K+k−2

j=0

a j,q(1)z j}, q(1) = 1, 2, . . . , l1, m(1) = 1, 2, . . . , l1, encapsulates the l1=κnkk1

functions

ϕ1(z) = ϕ0(z)z−αp1

=:(n−1)K+k−2∑

j=0

a j,1z j,

ϕ2(z) = ϕ0(z)(z−αp1)

2=:(n−1)K+k−2∑

j=0

a j,2z j, . . .

. . . , ϕl1(z) =ϕ0(z)(z−αp1)

l1=:(n−1)K+k−2∑

j=0

a j,l1 z j,

for r=2, the notation ϕq(2)(z) :={ϕ0(z)(z−αp2)−m(2)=:∑(n−1)K+k−2

j=0 a j,q(2)z j},q(2)= l1+1, l1+2, . . . , l1+l2, m(2)=1, 2, . . . , l2, encapsulates the l2=κnkk2

functions

ϕl1+1(z) = ϕ0(z)z−αp2

=:(n−1)K+k−2∑

j=0

a j,l1+1z j,

ϕl1+2(z) = ϕ0(z)(z−αp2)

2=:(n−1)K+k−2∑

j=0

a j,l1+2z j, . . .

. . . , ϕl1+l2(z) =ϕ0(z)(z−αp2)

l2=:(n−1)K+k−2∑

j=0

a j,l1+l2 z j,

etc., and, for r=s (cf. Subsection 1.2; recall that αps:=αk and

∑s−1i=1 li=(n−1)

·K+k−κnk), the notation ϕq(s)(z) :={ϕ0(z)(z−αk)−m(s)=:∑(n−1)K+k−2

j=0 a j,q(s)z j},


q(s) = (n−1)K+k−κnk+1, (n−1)K+k−κnk+2, . . . , (n−1)K+k−2, m(s) =1, 2, . . . , ls−2, encapsulates the ls−2=κnk−2 functions

ϕ(n−1)K+k−κnk+1(z) = ϕ0(z)z − αk

=:(n−1)K+k−2∑

j=0

a j,(n−1)K+k−κnk+1z j,

ϕ(n−1)K+k−κnk+2(z) = ϕ0(z)(z − αk)2

=:(n−1)K+k−2∑

j=0

a j,(n−1)K+k−κnk+2z j, . . .

. . . , ϕ(n−1)K+k−2(z) = ϕ0(z)(z − αk)κnk−2

=:(n−1)K+k−2∑

j=0

a j,(n−1)K+k−2z j.

(Note: #{ϕ0(z)(z−αpr )−m(r)}= lr−2δrs, r=1, 2, . . . , s, and # ∪s

r=1 {ϕ0(z)(z−αpr )

−m(r)}=∑s

r=1 lr− 2=(n−1)K+k−2.) One notes that, for n∈N and k=1, 2,. . . ,K, the 1+l1+l2+· · ·+(ls−2)= (n−1)K+k−1 functions ϕ0(z), ϕ1(z), . . . ,ϕl1(z), ϕl1+1(z), . . . , ϕl1+l2(z), . . . , ϕ(n−1)K+k−κnk+1(z), . . . , ϕ(n−1)K+k−2(z) are lin-early independent on R, that is, for z∈R,

∑(n−1)K+k−2j=0 c j ϕ j(z)=0 ⇒ (via

a Vandermonde-type argument) c j=0, j=0, 1, . . . , (n−1)K+k−2 (see the((n−1)K+k−1)× ((n−1)K+k−1) non-zero determinant D

in (34) below).For n∈N and k=1, 2, . . . ,K, corresponding to the ordered disjoint partionabove, recall the formula for Y21(z)/(z−αk) given in the proof of Lemma 2.1:

Y21(z)z−αk

=s−1∑m=1

lm=κnkkm∑q=1


q+

ls=κnk∑q=1

νs,q(n,k)(z−αk)q−1

, νs,ls(n,k) �=0,

where the (n − 1)K+k real coefficients ν1,1(n,k), . . . , ν1,l1(n,k), ν2,1(n,k), . . . ,ν2,l2(n,k), . . . , νs,1(n,k), . . . , νs,ls(n,k), with νs,ls(n,k) �=0, satisfy the linearinhomogeneous algebraic system of equations (32) given in the proof ofLemma 2.1. Via the multi-linearity property of the determinant and anapplication of Cramer’s Rule to system (32), one arrives at, for n∈N

and k=1, 2, . . . ,K, the following (ordered) determinantal representation forY21(z)/(z−αk):

1

2π iY21(z)z−αk

= Ξnk (z)

N �(n,k),


where N �(n,k) is given in the proof of Lemma 2.1, and, with n1 :=(n−1)K+k−2,

Ξnk (z) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp1 )

··· ∫R


(ξ0−αp1 )l1

∫R

(ξ0−αk)0 dμ(ξ0)(ξ0−αp2 )

···

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp1 )l1

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )···

.

.

.

...

.

.

.

.

.

.

...

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp1 )

l1

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

···

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp1 )l1

∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )···

.

.

.

...

.

.

.

.

.

.

...

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp1 )

l1

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

···

.

.

.

...

.

.

.

.

.

.

...

∫R

(ξn1−αk)−κnk+2 dμ(ξn1 )(ξn1−αp1 )

··· ∫R

(ξn1−αk)−κnk+2 dμ(ξn1 )

(ξn1−αp1 )l1

∫R

(ξn1−αk)−κnk+2 dμ(ξn1 )(ξn1−αp2 )

···

1z−αp1

··· 1(z−αp1 )

l11

z−αp2···

∫R


(ξ0−αp2 )l2

··· ··· ··· ∫R


(ξ0−αk)0··· ∫

R


(ξ0−αk)κnk−1

∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αp2 )l2

··· ··· ··· ∫R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)0··· ∫

R

(ξ1−αp1 )−1 dμ(ξ1)

(ξ1−αk)κnk−1

.

.

.. . .. . .. . .

.

.

.. . .

.

.

.

∫R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αp2 )

l2··· ··· ··· ∫

R

(ξl1−αp1 )

−l1 dμ(ξl1)

(ξl1−αk)0

··· ∫R

(ξl1−αp1 )

−l1 dμ(ξl1)


∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αp2 )l2

··· ··· ··· ∫R

(ξl1+1−αp2 )−1 dμ(ξl1+1)

(ξl1+1−αk)0··· ∫

R

(ξl1+1−αp2 )−1 dμ(ξl1+1)


.

.

.. . .. . .. . .

.

.

.. . .

.

.

.

∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αp2 )

l2··· ··· ··· ∫

R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)

(ξl1+l2−αk)0

··· ∫R

(ξl1+l2−αp2 )

−l2 dμ(ξl1+l2)


.

.

..... . .. . .

.

.

.. . .

.

.

.∫R

(ξn1 −αk)−κnk+2 dμ(ξn1 )

(ξn1 −αp1 )l2

··· ··· ··· ∫R


(ξn1 −αk)0··· ∫

R


(ξn1 −αk)κnk−1

1

(z−αp2 )l2

··· ··· ··· 1(z−αk)0

··· 1

(z−αk)κnk−1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1


× ··· dμ(ξm2−κnk+1) ··· dμ(ξn1 )(−1)(n−1)K+k−κnk

(ξ0−αk)0(ξ1−αp1 )1···(ξl1−αp1 )

l1 (ξl1+1−αp2 )1···(ξl1+l2−αp2 )

l2 ···(ξm2−κnk+1−αk)1···(ξn1−αk)κnk−2


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk−1

1 1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk−1

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1 1

(ξl1−αp1 )

··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1


1 1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1 1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1 1(ξn1 −αp1 )

··· 1

(ξn1 −αp1 )l1

1(ξn1 −αp2 )

··· 1

(ξn1 −αp2 )l2

··· 1(ξn1 −αk) ··· 1


1 1z−αp1

··· 1

(z−αp1 )l1

1z−αp2

··· 1

(z−αp2 )l2

··· 1z−αk ··· 1

(z−αk)κnk−1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

.

For n∈N and k=1, 2, . . . ,K, recalling the (n−1)K+k linearly independent(on R) functions ϕ0(z) :=∏s−1

m=1(z−αpm)lm(z−αk)

κnk−1=:∑(n−1)K+k−1j=0 a�j,0z j,

ϕq(r1)(z) :={ϕ0(z)(z−αpr1)−m(r1)=:∑(n−1)K+k−1

j=0 a�j,q(r1)z j}, r1=1, 2, . . . , s,q(r1) =∑r1−1

i=1 li+1,∑r1−1

i=1 li+2, . . . ,∑r1−1

i=1 li+lr1−δr1s, m(r1)=1, 2, . . . , lr1−δr1s, andthe (n−1)K+k−1 linearly independent (on R) functions ϕ0(z) :=∏s−1

m=1(z−αpm)

lm(z−αk)κnk−2 =: ∑(n−1)K+k−2

j=0 a j,0z j, ϕq(r2)(z) := {ϕ0(z)(z − αpr2)−m(r2) =:∑(n−1)K+k−2

j=0 a j,q(r2)zj }, r2 = 1, 2, . . . , s,q(r2) = ∑ r2−1

i=1 li + 1,∑r2−1

i=1 li + 2, . . . ,∑r2−1i=1 li + lr2 − 2δr2s, m(r2) = 1, 2, . . . , lr2−2δr2s, one proceeds, via the latter

determinantal expression, with the analysis of Ξnk (z) :=(−1)(n−1)K+k−κnkΞn

k (z):

Ξnk (z) =

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1


× ···dμ(ξm2−κnk+1) ···dμ(ξn1 )

(ξ0−αk)0(ξ1−αp1 )

1···(ξl1−αp1 )l1 (ξl1+1−αp2 )

1···(ξl1+l2−αp2 )l2 ···(ξm2−κnk+1−αk)

1···(ξn1−αk)κnk−2

× (ϕ0(z))−1

ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξn1 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

ϕ0(ξ0) ϕ1(ξ0) ··· ϕl1 (ξ0) ϕl1+1(ξ0) ··· ϕl1+l2 (ξ0) ··· ϕm2−κnk+1(ξ0) ··· ϕ(n−1)K+k−1(ξ0)ϕ0(ξ1) ϕ1(ξ1) ··· ϕl1 (ξ1) ϕl1+1(ξ1) ··· ϕl1+l2 (ξ1) ··· ϕm2−κnk+1(ξ1) ··· ϕ(n−1)K+k−1(ξ1)

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1) ϕ1(ξl1) ··· ϕl1 (ξl1) ϕl1+1(ξl1) ··· ϕl1+l2 (ξl1) ··· ϕm2−κnk+1(ξl1) ··· ϕ(n−1)K+k−1(ξl1)ϕ0(ξl1+1) ϕ1(ξl1+1) ··· ϕl1 (ξl1+1) ϕl1+1(ξl1+1) ··· ϕl1+l2 (ξl1+1) ··· ϕm2−κnk+1(ξl1+1) ··· ϕ(n−1)K+k−1(ξl1+1)

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξl1+l2) ϕ1(ξl1+l2) ··· ϕl1 (ξl1+l2) ϕl1+1(ξl1+l2) ··· ϕl1+l2 (ξl1+l2) ··· ϕm2−κnk+1(ξl1+l2) ··· ϕ(n−1)K+k−1(ξl1+l2)

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

ϕ0(ξn1) ϕ1(ξn1) ··· ϕl1 (ξn1) ϕl1+1(ξn1) ··· ϕl1+l2 (ξn1) ··· ϕm2−κnk+1(ξn1) ··· ϕ(n−1)K+k−1(ξn1)ϕ0(z) ϕ1(z) ··· ϕl1 (z) ϕl1+1(z) ··· ϕl1+l2 (z) ··· ϕm2−κnk+1(z) ··· ϕ(n−1)K+k−1(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸=:G∨(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξ(n−1)K+k−2;z)

= (ϕ0(z))−1

m1!∑

σ∈Sm1

∫R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1



× ···dμ(ξσ(l1+l2)) ···dμ(ξσ(m2−κnk+1)) ···dμ(ξσ(n1))

(ξσ(0)−αk)0(ξσ(1)−αp1 )

1···(ξσ(l1)−αp1 )l1 (ξσ(l1+1)−αp2 )


1···(ξσ(n1)−αk)κnk−2

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1)) · · · ϕ0(ξσ(l1))ϕ0(ξσ(l1+1)) · · · ϕ0(ξσ(l1+l2)) · · · ϕ0(ξσ(m2−κnk+1)) · · · ϕ0(ξσ(n1))

× G∨(ξσ(0), ξσ(1), . . . , ξσ(l1), ξσ(l1+1), . . . , ξσ(l1+l2), . . . , ξσ((n−1)K+k−2); z)

= (ϕ0(z))−1

m1!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1


× · · ·dμ(ξm2−κnk+1) · · · dμ(ξn1 )G∨(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξ(n−1)K+k−2; z)

×∑

σ∈Sm1


0(ξσ(1)−αp1 )1 ···(ξσ(l1 )−αp1 )

l1 (ξσ(l1+1)−αp2 )1 ···(ξσ(l1+l2 )−αp2 )

l2 ···(ξσ(m2−κnk+1)−αk)1 ···(ξσ(n1 )−αk)

κnk−2

× 1

ϕ0(ξσ(0))ϕ0(ξσ(1)) · · · ϕ0(ξσ(l1))ϕ0(ξσ(l1+1)) · · · ϕ0(ξσ(l1+l2)) · · · ϕ0(ξσ(m2−κnk+1)) · · · ϕ0(ξσ(n1))

= (ϕ0(z))−1

m1!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1


× · · ·dμ(ξm2−κnk+1) · · · dμ(ξn1 )G∨(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξ(n−1)K+k−2; z)


×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1(ξ0−αk)0

1(ξ0−αp1 )

··· 1

(ξ0−αp1 )l1

1(ξ0−αp2 )

··· 1

(ξ0−αp2 )l2

··· 1(ξ0−αk) ··· 1

(ξ0−αk)κnk−2

1(ξ1−αk)0

1(ξ1−αp1 )

··· 1

(ξ1−αp1 )l1

1(ξ1−αp2 )

··· 1

(ξ1−αp2 )l2

··· 1(ξ1−αk) ··· 1

(ξ1−αk)κnk−2

.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1−αk)0

1(ξl1

−αp1 )··· 1

(ξl1−αp1 )

l11

(ξl1−αp2 )

··· 1

(ξl1−αp2 )

l2··· 1

(ξl1−αk) ··· 1


1(ξl1+1−αk)0

1(ξl1+1−αp1 )

··· 1

(ξl1+1−αp1 )l1

1(ξl1+1−αp2 )

··· 1

(ξl1+1−αp2 )l2

··· 1(ξl1+1−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξl1+l2−αk)0

1(ξl1+l2

−αp1 )··· 1

(ξl1+l2−αp1 )

l11

(ξl1+l2−αp2 )

··· 1

(ξl1+l2−αp2 )

l2··· 1

(ξl1+l2−αk) ··· 1


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.1

(ξn1 −αk)01

(ξn1 −αp1 )··· 1

(ξn1 −αp1 )l1

1(ξn1 −αp2 )

··· 1

(ξn1 −αp2 )l2

··· 1(ξn1 −αk) ··· 1


∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= (ϕ0(z))−1

m1!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1


× · · · dμ(ξm2−κnk+1) · · · dμ(ξn1 )G∨(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2 , . . . , ξ(n−1)K+k−2; z)


× 1



×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξn1 ) ϕ1(ξn1 ) ··· ϕl1 (ξn1 ) ϕl1+1(ξn1 ) ··· ϕl1+l2 (ξn1 ) ··· ϕm2−κnk+1(ξn1 ) ··· ϕ(n−1)K+k−2(ξn1 )

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=: G(ξ0,ξ1,...,ξl1 ,ξl1+1,...,ξl1+l2 ,...,ξm2−κnk+1,...,ξ(n−1)K+k−2)

;

but, noting the determinantal factorisation

G(ξ0, ξ1, . . . , ξl1 , ξl1+1, . . . , ξl1+l2, . . . , ξm2−κnk+1, . . . , ξ(n−1)K+k−2)

= D V3(ξ0, ξ1, . . . , ξ(n−1)K+k−2),

where

D :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

a0,0 a1,0 ··· al1 ,0 al1+1,0 ··· al1+l2 ,0 ··· am2−κnk+1,0 ··· a(n−1)K+k−2,0

a0,1 a1,1 ··· al1 ,1 al1+1,1 ··· al1+l2 ,1 ··· am2−κnk+1,1 ··· a(n−1)K+k−2,1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,l1 a1,l1 ··· al1 ,l1 al1+1,l1 ··· al1+l2 ,l1 ··· am2−κnk+1,l1 ··· a(n−1)K+k−2,l1a0,l1+1 a1,l1+1 ··· al1 ,l1+1 al1+1,l1+1 ··· al1+l2 ,l1+1 ··· am2−κnk+1,l1+1 ··· a(n−1)K+k−2,l1+1

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,l1+l2 a1,l1+l2 ··· al1 ,l1+l2 al1+1,l1+l2 ··· al1+l2 ,l1+l2 ··· am2−κnk+1,l1+l2 ··· a(n−1)K+k−2,l1+l2

.

.

..... . .

.

.

....

. . ....

. . ....

. . ....

a0,n1−1 a1,n1−1 ··· al1 ,n1−1 al1+1,n1−1 ··· al1+l2 ,n1−1 ··· am2−κnk+1,n1−1 ··· a(n−1)K+k−2,n1−1

a0,n1 a1,n1 ··· al1 ,n1al1+1,n1

··· al1+l2 ,n1··· am2−κnk+1,n1 ··· a(n−1)K+k−2,n1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

( �=0),

(34)

and

V3(ξ0, ξ1, . . . , ξ(n−1)K+k−2) :=∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 ··· 1 1 ··· 1 ··· 1 ··· 1ξ0 ξ1 ··· ξl1 ξl1+1 ··· ξl1+l2 ··· ξm2−κnk+1 ··· ξ(n−1)K+k−2

........................

......

...ξ

l10 ξ

l11 ··· ξ

l1l1ξ

l1l1+1 ··· ξ l1

l1+l2··· ξ l1

m2−κnk+1 ··· ξ l1(n−1)K+k−2

ξl1+10 ξ

l1+11 ··· ξ l1+1

l1ξ

l1+1l1+1 ··· ξ l1+1

l1+l2··· ξ l1+1

m2−κnk+1 ··· ξ l1+1(n−1)K+k−2

........................

......

...ξ

l1+l20 ξ

l1+l21 ··· ξ l1+l2

l1ξ

l1+l2l1+1 ··· ξ l1+l2

l1+l2··· ξ l1+l2

m2−κnk+1 ··· ξ l1+l2(n−1)K+k−2

........................

......

...ξ

n1−10 ξ

n1−11 ··· ξn1−1

l1ξ

n1−1l1+1 ··· ξn1−1

l1+l2··· ξn1−1

m2−κnk+1 ··· ξn1−1(n−1)K+k−2

ξn10 ξ

n11 ··· ξ

n1l1

ξn1l1+1 ··· ξn1

l1+l2··· ξn1

m2−κnk+1 ··· ξn1(n−1)K+k−2

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

=(n−1)K+k−2∏

i, j=0j<i

(ξi−ξ j),



Ξnk (z) =

D (ϕ0(z))−1

m1!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · ·

×· · · dμ(ξl1+l2 ) · · · dμ(ξm2−κnk+1) · · · dμ(ξn1 )

∏(n−1)K+k−2i, j=0

j<i

(ξi − ξ j)

ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1)· · ·ϕ0(ξl1+l2 )· · ·ϕ0(ξm2−κnk+1)· · ·ϕ0(ξn1 )

× 1

ϕ0(ξ0)ϕ0(ξ1)· · ·ϕ0(ξl1 )ϕ0(ξl1+1)· · ·ϕ0(ξl1+l2 )· · ·ϕ0(ξm2−κnk+1)· · ·ϕ0(ξn1 )

×

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.



.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.


.

.

....

. . ....

.

.

.. . .

.

.

.. . .

.

.

.. . .

.

.

.ϕ0(ξn1 ) ϕ1(ξn1 ) ··· ϕl1 (ξn1 ) ϕl1+1(ξn1 ) ··· ϕl1+l2 (ξn1 ) ··· ϕm2−κnk+1(ξn1 ) ··· ϕ(n−1)K+k−1(ξn1 )

ϕ0(z) ϕ1(z) ··· ϕl1 (z) ϕl1+1(z) ··· ϕl1+l2 (z) ··· ϕm2−κnk+1(z) ··· ϕ(n−1)K+k−1(z)

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣︸︷︷︸

=D�∏(n−1)K+k−2

i, j=0j<i

(ξi−ξ j)∏(n−1)K+k−2

m=0 (z−ξm) (cf. Proof of Lemma 2.1)

= D D�(ϕ0(z))−1

((n−1)K+k−1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1

dμ(ξ0) dμ(ξ1) · · · dμ(ξl1 ) dμ(ξl1+1) · · ·

× · · · dμ(ξl1+l2 ) · · · dμ(ξ(n−1)K+k−2)

× (ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξ(n−1)K+k−2))−1

ϕ0(ξ0)ϕ0(ξ1) · · · ϕ0(ξl1 )ϕ0(ξl1+1) · · · ϕ0(ξl1+l2 ) · · · ϕ0(ξm2−κnk+1) · · · ϕ0(ξ(n−1)K+k−2)

×(n−1)K+k−2∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−2∏

m=0

(z−ξm);

but, recalling that, for n∈N and k=1, 2, . . . ,K, ϕ0(z)= ϕ0(z)/(z−αk) andΞn

k (z) :=(−1)(n−1)K+k−κnkΞnk (z), one arrives at

Ξnk (z) =

(−1)(n−1)K+k−κnkD D�(ϕ0(z))−1

((n−1)K+k−1)!∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1

dμ(ξ0)dμ(ξ1) · · ·dμ(ξl1)

× dμ(ξl1+1) · · · dμ(ξl1+l2) · · · dμ(ξ(n−1)K+k−2)

((n−1)K+k−2∏

l=0

ϕ0(ξl)

)−2


×(n−1)K+k−2∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−2∏

m=0

(ξm−αk)(z−ξm)⇒

Ξnk (z) =

(−1)(n−1)K+k−κnkD D�

((n−1)K+k−1)!

⎛⎝s−1∏

q=1


κnk−1

⎞⎠

−1

×∫

R

∫R

· · ·∫

R︸︷︷︸(n−1)K+k−1

dμ(ξ0)dμ(ξ1) · · · dμ(ξ(n−1)K+k−2)

×⎛⎝(n−1)K+k−2∏

m=0

s−1∏q=1


κnk−1

⎞⎠

−2

×(n−1)K+k−2∏

i, j=0j<i

(ξi−ξ j)2(n−1)K+k−2∏

l=0

(ξl−αk)(z−ξl).

Recalling that, for n ∈ N and k = 1, 2, . . . ,K, Y21(z)/(z − αk) = 2π i Ξnk (z)/

N �(n,k), where Ξnk (z) is given directly above, and N �(n,k) is given in

the proof of Lemma 2.1, (33), one arrives at the integral representation forY21(z)/(z−αk) stated in the Lemma. ��

Remark 2.3 For n∈N and k=1, 2, . . . ,K, the integral representation forthe FPC orthonormal rational function φn

k : N × {1, 2, . . . ,K} × C \ {α1,

α2, . . . , αK}→C is obtained via (cf. Subsection 1.2) φnk(z)=μn,κnk(n,k)πππ

nk(z),

where the ‘norming constant’, μn,κnk : N × {1, 2, . . . ,K}→R+, is given inCorollary 2.1, and the integral representation for the monic FPC ORF,πππn

k : N × {1, 2, . . . ,K} × C \ {α1, α2, . . . , αK}→C, is given in Lemma 2.2. �

3 The Monic FPC ORF Family of Variational Problemsand Transformed RHPs

In this section, the analysis of the family of K (different) variational problemsassociated with the monic FPC ORFs is undertaken; more precisely, the cor-responding family of equilibrium measures and generalised weighted Feketesets (see Lemmata 3.1–3.3 and Lemmata 3.5–3.8), as well as the correspondingfamily of complex potentials, or ‘g-functions’ (see Lemma 3.4), are studied.Furthermore, Lemma RHPFPC is reformulated as a family of K equivalentmatrix RHPs on R (see Lemma 3.4).

One begins by establishing the existence of the family of equilibriummeasures.


Lemma 3.1 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8), and set � (z) :=exp(−V(z)). For n∈N and k=1, 2, . . . ,K, withassociated measure6 μEQ (=μEQ(n,k)) ∈M1(R), define the weighted (n- andk-dependent) logarithmic energy functional

IV : N × {1, 2, . . . ,K} × M1(R)→R, (n,k, μEQ) →IV

[n,k;μEQ

] :=IV

[μEQ

]

=∫∫

R2ln

⎛⎝|s−t| 1

n

( |s−t||s−αk||t−αk|

)κnk−1n

s−1∏q=1

( |s−t||s−αpq ||t−αpq |

)κnkkqn

× � (s)� (t)

⎞⎠

−1

dμEQ(s)dμEQ(t),

and consider the family of minimisation problems

EV(n,k) :=EV = inf{IV

[μEQ

] ; μEQ ∈M1(R)}.

Then, for n∈N and k=1, 2, . . . ,K: (1) EV is finite; (2) the infimum is at-tained, that is, there exists μV (=μV(n,k)) ∈M1(R) such that IV[μV]=EV,and μV has finite weighted logarithmic energy, that is, −∞<IV[μV]<+∞;and (3) ({x∈R; � (x)>0} ⊃) J :=supp(μV) is a proper compact subset ofR \ {α1, α2, . . . , αK}.

Sketch of Proof For n∈N and k=1, 2, . . . ,K, let μEQ (=μEQ(n,k)) ∈M1(R),and set, as in the Lemma, � (z) :=exp(−V(z)), where (cf. Subsection 1.3)V: R \ {α1, α2, . . . , αK}→R satisfies conditions (6)–(8). For n∈N and k=1, 2, . . . ,K, from the definition of IV : N × {1, 2, . . . ,K} × M1(R)→R given inthe Lemma, one shows that

IV[μEQ]=∫∫

R2KV(s, t)dμ

EQ(s)dμEQ(t),

where the symmetric kernel KV : N × {1, 2, . . . ,K} × R2→R, (n,k, s, t) →

KV(n,k; s, t) :=KV(s, t) is given by

KV(s, t) = KV(t, s) :=1

nln(|s−t|−1

)+(

κnk−1

n

)ln

(∣∣∣∣ 1

t−αk− 1

s−αk

∣∣∣∣−1)

+s−1∑q=1

κnkkq

nln

(∣∣∣∣ 1

t−αpq

− 1

s−αpq

∣∣∣∣−1)+V(s)+V(t).

6The measure, etc., vary according to the parameters n (∈N) and k (=1, 2, . . . ,K), and thus, strictlyspeaking, all the introduced variables in the proof, too, depend on n and k, which would necessitatethe introduction of additional, superfluous notation(s) to encode these n- and k-dependencies;however, for simplicity of notation, unless where absolutely necessary, such cumbersome n- and k-dependencies will not be introduced, and the reader should be cognizant of this fact: this commentapplies, mutatis mutandis, throughout the remainder of the paper.


(Of course, the definition of IV[μEQ] only makes sense provided both integralsexist and are finite.) Recall the following inequalities (see, e.g., Chapter 6of [56]): for s, t∈R,

|s−t|�(1+s2)1/2(1+t2)1/2,

| (t−αk)−1−(s−αk)

−1|�(1+(t−αk)−2)1/2 (

1+(s−αk)−2)1/2,

| (t−αpq

)−1−(s−αpq

)−1 |�(1+(t−αpq)−2)1/2 (

1+(s−αpq)−2)1/2,

whence

ln(|s−t|−1) �− 12 ln

(1+s2

)− 12 ln

(1+t2

),

ln(|(t−αk)

−1−(s−αk)−1|−1

)�− 12 ln

(1+(t−αk)

−2)− 12 ln

(1+(s−αk)

−2),

ln(| (t−αpq

)−1−(s−αpq

)−1 |−1)

�− 12 ln

(1+(t−αpq

)−2)− 1

2 ln(

1+(s−αpq

)−2);

thus, for n∈N and k=1, 2, . . . ,K,

KV(s, t) �1

2

⎛⎝2V(s)− 1

nln(s2+1

)−(

κnk−1

n

)ln((s−αk)

−2+1)

−s−1∑q=1

κnkkq

nln((

s−αpq

)−2+1)⎞⎠+ 1

2

⎛⎝2V(t)− 1

nln(t2+1)

−(

κnk−1

n

)ln((t−αk)

−2+1)−s−1∑q=1

κnkkq

nln((

t−αpq

)−2+1)⎞⎠ .

Recalling conditions (6)–(8) for the external field V: R \ {α1, α2, . . . , αK}→R,in particular, there exists some arbitrarily fixed, sufficiently small positive realnumber δ∞ (=δ∞(n,k)) such that, for x∈O∞ :={x∈R; |x|>δ−1∞ }, V(x)�(1+c∞) ln(x2+1), where c∞ (=c∞(n,k)) is some bounded, positive real number,and, for q=1, 2, . . . , s, there exist arbitrarily fixed, sufficiently small positivereal numbers δq (= δq(n,k)) such that, for x∈Oδq(αpq) :={x∈R; |x−αpq |<δq}, V(x)�(1+cq) ln((x−αpq)

−2+1), where cq (=cq(n,k)), q=1, 2, . . . , s, arebounded, positive real numbers, it follows that, for n∈N and k=1, 2, . . . ,K,

2V(x)− 1

nln(x2+1

)−(

κnk−1

n

)ln((x−αk)

−2+1)

−s−1∑q=1

κnkkq

nln((x−αpq)

−2+1)�CV(n,k) :=CV>−∞,


whence KV(s, t)�CV (>−∞), which shows that, for n∈N and k=1, 2, . . . ,K,KV(s, t) is bounded from below (on R

2); hence, for n∈N and k=1, 2, . . . ,K,

IV[μEQ]�∫∫

R2CV dμEQ(s)dμEQ(t)=CV

∫R

dμEQ(s)∫

R

dμEQ(t)�CV (>−∞).

It follows from the above inequality and the definition of EV stated in theLemma that, for n∈N and k=1, 2, . . . ,K, for all μEQ∈M1(R), EV �CV>−∞,which shows that, for n∈N and k=1, 2, . . . ,K, EV is bounded from below. Letε (=ε(n,k)) be an arbitrarily fixed, sufficiently small positive real number, andset �ε :={x∈R; � (x)�ε}; then �ε is compact, and �0 :=∪∞

j=1�1/j=∪∞j=1{x∈

R; � (x)>1/j}={x∈R; � (x)>0}. Since, for V: R \ {α1, α2, . . . , αK}→R satis-fying conditions (6)–(8), � is an admissible weight [60], it follows that, forn∈N and k=1, 2, . . . ,K, there exits j∗ (= j∗(n,k)) ∈N such that cap(�1/j∗)=exp(− inf{IV[μEQ]; μEQ ∈M1(�1/j∗)})>0, which, in turn, means that, for n∈N

and k=1, 2, . . . ,K, there exists a probability measure μEQj∗ (=μEQ

j∗ (n,k)), say,with supp(μ j∗) ⊆ �1/j∗ such that

∫∫�1/j∗ ×�1/j∗

ln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

⎞⎠

−1

dμEQj∗ (s)dμ

EQj∗ (t)<+∞,

with �1/j∗ ×�1/j∗ ⊆ R2. For x∈supp(μEQ

j∗ ) ⊆ �1/j∗ , it follows that � (x)�1/j∗

(bounded from below), whence

∫∫�1/j∗ ×�1/j∗

ln(� (s)� (t))−1 dμEQj∗ (s)dμ

EQj∗ (t)�2 ln( j∗)<+∞,

which implies that, for n∈N and k=1, 2, . . . ,K,

IV[μEQj∗ ] =

∫∫�1/j∗ ×�1/j∗

ln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

� (s)� (t)

⎞⎠

−1

dμEQj∗ (s)dμ

EQj∗ (t)<+∞;

thus, for n∈N and k=1, 2, . . . ,K, −∞<EV := inf{IV[μEQ]; μEQ ∈M1(R)}<+∞. This establishes the first claim of the Lemma.


For n∈N and k=1, 2, . . . ,K, choose a sequence of probability measures{μEQ

m =μEQm (n,k)}∞m=1 in M1(R) such that IV[μEQ

m ]� EV+ 1m . From the analysis

above, it follows that, for n∈N and k=1, 2, . . . ,K,

IV[μEQm ] =

∫∫R2

KV(s, t)dμEQm (s)dμ

EQm (t) �

∫∫R2

(1

2

(2V(s)− 1

nln(s2+1)

−(

κnk−1

n

)ln((s−αk)

−2+1)−s−1∑q=1

κnkkq

nln((s−αpq)

−2+1)

⎞⎠

+1

2

(2V(t)− 1

nln(t2+1)−

(κnk−1

n

)ln((t−αk)

−2+1)

−s−1∑q=1

κnkkq

nln((t−αpq)

−2+1)

⎞⎠⎞⎠dμEQ

m (s)dμEQm (t).

For n∈N and k=1, 2, . . . ,K, set

ψV(z) := 2V(z)− 1

nln(z2+1)−

(κnk−1

n

)ln((z−αk)

−2+1)

−s−1∑q=1

κnkkq

nln((z−αpq)

−2+1). (35)

Then, for n∈N and k=1, 2, . . . ,K, IV [μEQm ]�∫∫

R2 (12 ψV(s)+ 1

2 ψV(t))dμEQm (s)dμ

EQm (t) �∫

RψV(ξ)dμ

EQm (ξ)⇒EV+ 1

m �IV[μEQm ]�∫

RψV(ξ)dμ

EQm (ξ). For n∈N and k=

1, 2, . . . ,K, recalling that, for x∈O∞, ∃ c∞(=c∞(n,k)) >0 suchthat V(x)� (1+c∞) ln(x2+1), and, for q=1, 2, . . . , s, ∃ cq (=cq(n,k)) >0 suchthat, for x∈Oδq(αpq), V(x)�(1+cq) ln((x−αpq)

−2+ 1), it follows that,for any b (=b(n,k)) >0, there exists (some) M (=M(n,k)) >1, forwhich O 1

M(αpi) ∩ O 1

M(αpj)=∅, i �= j∈{1, 2, . . . , s}, such that ψV(z)>b ∀

z∈DM :={|x|� M} ∪ (∪sq=1O 1

M(αpq)), which implies that, upon writing

R=(R \ DM) ∪ DM, with (R \ DM) ∩ DM=∅,

EV+1

m�∫(R\DM)∪DM

ψV(ξ)dμEQm (ξ)=

∫DM

ψV(ξ)︸︷︷︸> b

dμEQm (ξ)

+∫

R\DM

ψV(ξ)︸︷︷︸�−|CV |

dμEQm (ξ) � b

∫DM

dμEQm (ξ)

− |CV |∫

R\DM

dμEQm (ξ)

︸︷︷︸∈ [0,1]

�b∫

DM

dμEQm (ξ)−|CV |;


thus, for n∈N and k=1, 2, . . . ,K, b∫DM

dμEQm (ξ)� EV+|CV |+ 1

m , whence

lim supm→∞

∫DM

dμEQm (ξ)� lim sup

m→∞

(b−1

(EV+|CV |+m−1

)).

Recall that, for n∈N and k=1, 2, . . . ,K, EV is bounded above. By theArchimedean property, it follows that, for n∈N and k=1, 2, . . . ,K, ∀ ε (=ε(n,k)) >0, ∃ N (= N(n,k)) ∈N such that, ∀ m (=m(n,k)) � N, m−1<ε; thus,choosing b (=b(n,k)) =[ε−1(EV+|CV |+ε)], where [∗∗∗] denotes the largestinteger less than or equal to ∗∗∗, with ε (=ε(n,k)) some arbitrarily fixed, suffi-ciently small positive real number, it follows that, for n∈N and k=1, 2, . . . ,K,with m>b (∈N),

lim supm→∞

∫DM

dμEQm (ξ)�ε,

which implies that the sequence of probability measures {μEQm }∞m=1 in M1(R)

is tight [61] (that is, for n∈N and k=1, 2, . . . ,K, given ε (=ε(n,k)) >0,∃ M (=M(n,k)) >1, for which O 1

M(αpi) ∩ O 1

M(αpj)=∅, i �= j∈{1, 2, . . . , s},

such that lim supm→∞ μEQm (DM)�ε, where μEQ

m (DM) :=∫DM

dμEQm (ξ)). Since, for

n∈N and k=1, 2, . . . ,K, the sequence of probability measures {μEQm }∞m=1 in

M1(R) is tight, by a Helly selection theorem [60], there exists, for n∈N

and k=1, 2, . . . ,K, a (weak-∗ convergent) subsequence of probability mea-sures {μEQ

m j=μEQ

m j(n,k)}∞j=1 (with j= j(n,k)) in M1(R) converging (weakly) to

a probability measure μEQ (=μEQ(n,k)) ∈M1(R), symbolically μEQm j

∗→μEQ asj→∞.7 One now shows that, if μEQ

m∗→μEQ, then lim infm→∞ IV[μEQ

m ]�IV[μEQ].Since � (resp., V) is upper semi-continuous (resp., lower semi-continuous),there exists, for n∈N and k=1, 2, . . . ,K, a decreasing (resp., an increasing)sequence {� m = � m(n,k)}∞m=1 (resp., {Vm = Vm(n,k)}∞m=1) of continuousfunctions on R such that � m+1(·)�� m(·) (resp., Vm+1(·)� Vm(·))8, and, point-wise, � m(x)↘� (x) (resp., Vm(x)↗ V(x)) as m→∞ for every x∈R. Not-ing that, for n∈N and k=1, 2, . . . ,K, IV[μEQ

m ]=∫∫R2 KV(s, t)dμ

EQm (s)dμ

EQm (t)�∫∫

R2 KVm(s, t)dμEQ

m (s)dμEQm (t), it follows that, for any � (= �(n,k))∈R, IV[μEQ

m ]�∫∫R2 g(s, t)dμEQ

m (s)dμEQm (t), where g: N×{1, 2, . . . ,K}×R

2→R, (n,k, s, t) →g(n,k, s, t) :=g(s, t)=g(t, s)=min{�,KVm

(s, t)} is, by virtue of conditions (6)–(8) on the external field V: R \ {α1, α2, . . . , αK}→R, bounded and con-tinuous on R

2. Let ε (=ε(n,k)) >0 be given, and choose M0(=M0(n,k)) >1, for which O 1

M0(αpi) ∩ O 1

M0(αpj)=∅, i �= j∈{1, 2, . . . , s}, such

that lim supm→∞∫DM0

dμEQm (ξ)�ε. For n∈N and k=1, 2, . . . ,K, with M0>1,

7A sequence of probability measures {μEQm }m∈N in M1(D) is said to converge weakly as m→∞

to μEQ ∈M1(D), symbolically μEQm

∗→μEQ, if μEQm ( f ) :=∫D f (λ) dμEQ

m (λ)→∫D f (λ) dμEQ(λ)=:

μEQ( f ) as m→∞ for all f ∈CCC0b(D), where CCC0

b(D) denotes the set of all bounded, continuousfunctions on D with compact support8Adding a suitable constant, if necessary, which does not change μEQ

m , or the regularity of V: R \{α1, α2, . . . , αK}→R, one may assume that V �0 and Vm �0, m∈N.


let (the test function) CCC0b(R)�hM : N × {1, 2, . . . ,K} × R→[0, 1], (n,k, x) →

hM(n,k; x) :=hM(x) be such that: (i) 0�hM(x)�1, x∈R; (ii) hM(x)=1, x∈R \DM0 , where DM0 :={|x|� M0} ∪ (∪s

q=1O 1M0(αpq)); and (iii) hM(x)=0, x∈DM0+1.

Note the splitting (decomposition) [56]

∫∫R2

g(t, s)dμEQm (t)dμ

EQm (s)= Ia+ Ib + Ic,

where

Ia :=∫∫

R2g(t, s)(1−hM(s))dμEQ

m (t)dμEQm (s),

Ib :=∫∫

R2g(t, s)hM(s)(1−hM(t))dμEQ

m (t)dμEQm (s),

Ic :=∫∫

R2g(t, s)hM(t)hM(s)dμEQ

m (t)dμEQm (s).

One shows that

|Ia| �∫∫

R2|g(t, s)|(1−hM(s))dμEQ

m (t)dμEQm (s)� sup

(t,s)∈R2

|g(t, s)|∫

R

dμEQm (t)

×⎛⎝∫

DM0+1

(1−hM(s)︸︷︷︸= 0

)dμEQm (s)+

∫R\DM0

(1−hM(s)︸︷︷︸= 1

)dμEQm (s)

⎞⎠ ,

whence

lim supm→∞

|Ia|� sup(t,s)∈R2

|g(t, s)| lim supm→∞

∫R\DM0+1

dμEQm (ξ)�ε sup

(t,s)∈R2

|g(t, s)|;

similarly,

lim supm→∞

|Ib |�ε sup(t,s)∈R2

|g(t, s)|.

Since, for n∈N and k=1, 2, . . . ,K, g(t, s) is bounded and continuous on R2,

that is, g∈CCCb(R2), there exists, by an appropriate generalisation of the Stone–

Weierstrass theorem for the single-variable case, a sequence of polynomials,with n- and k-dependent coefficients (suppressed for notational simplicity),{pm(t, s)= pm(n,k; t, s)}∞m=1 in R[t, s] (the algebra of polynomials in the in-determinates t and s with coefficients in R) such that sup(t,s)∈R2 |pm(t, s)−


g(t, s)|→0 as m→∞, that is, there exists a (symmetric) polynomial, p(s, t)=p(t, s), say, in R[s, t], with representation p(t, s)=∑i�i0

∑j� j0 γijt is j, where

γij=γij(n,k), with |γij(n,k)|<+∞, such that |g(t, s)− p(t, s)|�ε (=ε(n,k));thus, |hM(t)hM(s)g(t, s)−hM(t)hM(s)p(t, s)|�ε for t, s∈R. Split Ic as follows:Ic= Iαc + Iβc , where

Iαc :=∫∫

R2hM(s)hM(t)p(t, s)dμEQ

m (t)dμEQm (s),

Iβc :=∫∫

R2hM(s)hM(t)(g(t, s)− p(t, s))dμEQ

m (t)dμEQm (s).

One now shows that

|Iβc | �∫∫

R2hM(s)hM(t) |g(t, s)− p(t, s)|︸︷︷︸

� ε

dμEQm (t)dμ

EQm (s)

� ε∫

R

hM(s)dμEQm (s)

∫R

hM(t)dμEQm (t)

� ε

⎛⎝∫

DM0+1

hM(s)︸︷︷︸= 0

dμEQm (s)+

∫R\DM0

hM(s)︸︷︷︸= 1

dμEQm (s)

⎞⎠

2

� ε(∫

R\DM0

dμEQm (s)

)2

�ε(∫

R

dμEQm (s)

)2

�ε,

whence lim supm→∞ |Iβc |�ε, and

Iαc =∫∫

R2hM(s)hM(t)

∑i�i0

∑j� j0

γijtis j dμEQm (t) dμ

EQm (s)

=∑i�i0

∑j� j0

γij

(∫R

hM(t)ti dμEQm (t)

)(∫R

hM(s)s j dμEQm (s)

)

→∑i�i0

∑j� j0

γij

(∫R

hM(t)ti dμEQ(t))(∫

R

hM(s)s j dμEQ(s))(since μEQ

m∗→μEQ as m→∞)

=∫∫

R2

⎛⎝∑

i�i0

∑j� j0

γijtis j

⎞⎠ hM(t)hM(s) dμEQ(t) dμEQ(s) ⇒

Iαc =∫∫

R2p(t, s)hM(t)hM(s) dμEQ(t) dμEQ(s).


Furthermore, recalling that |hM(t)hM(s)g(t, s)−hM(t)hM(s)p(t, s)|�ε,

Iαc �∫∫

R2g(t, s)hM(t)hM(s) dμEQ(t) dμEQ(s)

+ ε⎛⎜⎝∫

D M0+1

hM(s)︸︷︷︸= 0

dμEQ(s)+∫

R\D M0

hM(s)︸︷︷︸= 1

dμEQ(s)

⎞⎟⎠

2

�∫∫

R2g(t, s)hM(t)hM(s) dμEQ(t) dμEQ(s)+ε

(∫R\D M0

dμEQ(s)

)2

�∫∫

R2g(t, s)hM(t)hM(s) dμEQ(t) dμEQ(s)+ε

(∫R

dμEQ(s))2

=∫∫

R2g(t, s)hM(t)hM(s) dμEQ(t) dμEQ(s)+ε ⇒

Iαc �∫∫

R2g(t, s)|1+(hM(t)−1)||1+(hM(s)−1)|dμEQ(t) dμEQ(s)+ε

�∫∫

R2g(t, s) dμEQ(t) dμEQ(s)+2

∫∫R2

g(t, s)|hM(s)−1|dμEQ(t) dμEQ(s)+ε

+∫∫

R2g(t, s)|hM(t)−1||hM(s)−1|dμEQ(t) dμEQ(s)

�∫∫

R2g(t, s) dμEQ(t) dμEQ(s)+ε+2 sup

(t,s)∈R2|g(t, s)|

∫R

dμEQ(t)︸︷︷︸

= 1

×⎛⎜⎝∫

D M0+1

| hM(s)︸︷︷︸= 0

−1|dμEQ(s)+∫

R\D M0

| hM(s)︸︷︷︸= 1

−1|dμEQ(s)

⎞⎟⎠

+ sup(t,s)∈R2

|g(t, s)|⎛⎜⎝∫

D M0+1

| hM(s)︸︷︷︸= 0

−1|dμEQ(s)+∫

R\D M0

| hM(s)︸︷︷︸= 1

−1|dμEQ(s)

⎞⎟⎠

2

�∫∫

R2g(t, s) dμEQ(t) dμEQ(s)+ 2 sup

(t,s)∈R2|g(t, s)|

∫D M0+1

dμEQ(s)

︸︷︷︸� ε

+ sup(t,s)∈R2

|g(t, s)|

⎛⎜⎜⎜⎜⎝∫

D M0+1

dμEQ(s)

︸︷︷︸� ε

⎞⎟⎟⎟⎟⎠

2

+ε

=∫∫

R2g(t, s) dμEQ(t) dμEQ(s)+ε

(1+2 sup

(t,s)∈R2|g(t, s)|

)+O(ε2),


whereupon, neglecting the O(ε2) term, and setting κ (=κ (n,k)) := 1+2 sup(t,s)∈R2 |g(t, s)|, one obtains

Iαc �∫∫

R2g(t, s)dμEQ(t)dμEQ(s)+κ ε.

Hence, assembling the above-derived bounds for Ia, Ib , Iβc , and Iαc , one arrivesat, for n∈N and k=1, 2, . . . ,K, upon setting ε (=ε (n,k)) :=2κ ε,∫∫

R2g(t, s)dμEQ

m (t)dμEQm (s)−

∫∫R2

g(t, s)dμEQ(t)dμEQ(s)�ε ;

thus,∫∫R2

g(t, s)dμEQm (t)dμ

EQm (s)→

∫∫R2

g(t, s)dμEQ(t)dμEQ(s) as m→∞.

Recalling that g(t, s) :=min{�,KVm(t, s)}, (�,m)∈R×N, it follows that, for n∈N

and k=1, 2, . . . ,K,

lim infm→∞ IV[μEQ

m ]�∫∫

R2min

{�,KVm

(t, s)}

dμEQ(t)dμEQ(s) :

letting �↑+∞ and m→∞, and noting that min{�,KVm(t, s)}→KV(t, s), after

an application of the monotone convergence theorem, one arrives at, for n∈N

and k=1, 2, . . . ,K,

lim infm→∞ IV[μEQ

m ]�∫∫

R2KV(t, s)dμ

EQ(t)dμEQ(s)=IV[μEQ].

Since, from the analysis above, it was shown that, for n∈N and k=1, 2, . . . ,K,there exists a weakly (weak-∗) convergent subsequence of probability mea-sures {μEQ

m j}∞j=1 (⊂ M1(R)) of {μEQ

m }∞m=1 (⊂ M1(R)) with a weak-∗ limit μEQ ∈M1(R), that is, μEQ

m j

∗→μEQ as j→∞, upon recalling that IV[μEQm ]� EV+ 1

m ,m∈N, it follows that, for n∈N and k=1, 2, . . . ,K, in the limit as m→∞,IV[μEQ]� EV = inf{IV[μEQ]; μEQ ∈M1(R)}; from the latter two inequalities, itfollows, thus, that for n∈N and k=1, 2, . . . ,K, there existsμEQ (=μEQ(n,k)) :=μV ∈M1(R), the equilibrium measure, such that IV[μV]= inf{IV[μEQ]; μEQ ∈M1(R)}, that is, for n∈N and k=1, 2, . . . ,K, the infimum is attained. Thisestablishes the second claim of the Lemma (the unicity of the equilibriummeasure is proven in Lemma 3.3 below).

It remains, finally, to establish the third claim of the Lemma, that is, toshow that, for n∈N and k=1, 2, . . . ,K, ({x∈R; � (x)>0} ⊃) J :=supp(μV) ⊂R \ {α1, α2, . . . , αK} is compact. The following argument is valid for anyμEQ (=μEQ(n,k)) ∈M1(R) achieving the above minimum; in particular, forμEQ =μV . Without loss of generality, therefore, for n∈N and k=1, 2, . . . ,K,let M1(R)�μEQ

� (=μEQ� (n,k)) be such that IV[μEQ

� ]=EV , and let D (=D(n,k))be any proper, measurable subset of R for which μEQ

� (D)=∫

D dμEQ� (ξ)>0. As

in [61], set, for n∈N and k=1, 2, . . . ,K,

με� (n,k; z) :=με� (z)=(1+εμEQ� (D))

−1(μEQ

� (z)+ε(μEQ� �D)(z)

), ε∈(−1, 1),


where μEQ� �D denotes the restriction of μEQ

� to D. (Note that∫

Rdμε� (ξ)=1; in

particular, με� ∈M1(R), ε∈(−1, 1).) Using the fact that KV(s, t)=KV(t, s), oneshows that, for n∈N and k=1, 2, . . . ,K,

IV[με� ] =∫∫

R2KV(s, t)dμ

ε� (s)dμ

ε� (t) = (1+εμEQ

� (D))−2

×(

IV[μEQ� ]+2ε

∫∫R2

KV(s, t)dμEQ� (s)d(μ

EQ� �D)(t)

+ ε2∫∫

R2KV(s, t)d(μ

EQ� �D)(s)d(μEQ

� �D)(t))

(note that all the above integrals are finite due to the argument at the beginningof the proof). By the minimal property of μEQ

� ∈M1(R), it follows that [61], forn∈N and k=1, 2, . . . ,K, ∂εIV[με� ]=0, which implies that

∫∫R2(KV(s, t)−IV[μEQ

� ])dμEQ� (s)d(μ

EQ� �D)(t)=0;

but, recalling that, for n∈N and k=1, 2, . . . ,K, KV(t, s)� 12 (ψV(s)+ψV(t)),

where ψV(z) is defined in (35), it follows, from the latter minimisation con-dition, that

∫∫R2

IV[μEQ� ]dμEQ

� (s)d(μEQ� �D)(t) �

∫∫R2

(1

2ψV(s)+

1

2ψV(t)

)

× dμEQ� (s)d(μ

EQ� �D)(t)⇒

0 �∫∫

R2

(1

2ψV(s)+

1

2ψV(t)−IV

[μEQ

�

])dμEQ

� (s)d(μEQ� �D)(t),

whence, for n∈N and k=1, 2, . . . ,K,

0�∫

R

(ψV(t)+

(∫R

ψV(s)dμEQ� (s)

)−2IV

[μEQ

�

])d(μEQ

� �D)(t).

But, for n ∈ N and k = 1, 2, . . . ,K, from the growth conditions on V: R\{α1, α2, . . . , αK} → R, that is, ∃ c∞ (= c∞(n,k)) > 0 such that, for x ∈ O∞,V(x) � (1 + c∞) ln(x2 + 1), and, for q = 1, 2, . . . , s, ∃ cq (= cq(n,k)) > 0 suchthat, for x ∈ Oδq(αpq), V(x) � (1 + cq) ln((x − αpq)

−2 + 1), it follows that,for n ∈ N and k = 1, 2, . . . ,K, there exists (some) TM (= TM(n,k)) > 1(take, say, TM = max{δ∞,maxq=1,2,...,s{δ−1

q }}), with O 1TM(αpi) ∩ O 1

TM(αpj) = ∅,

i �= j ∈ {1, 2, . . . , s}, such that, for τ ∈ {|x| � TM} ∪ (∪sq=1O 1

TM(αpq)), ψV(τ )−

2IV[μEQ� ] + ∫

RψV(ξ)dμ

EQ� (ξ) � 1; hence, if D is such that D ⊂ {|x| � TM} ∪


(∪sq=1O 1

TM(αpq)), with TM > 1, it follows from the above calculations that, for

n ∈ N and k = 1, 2, . . . ,K,

0�∫

R

(ψV(t)+

(∫R

ψV(s)dμEQ� (s)

)−2IV

[μEQ

�

])d(μEQ

� �D)(t)�1,

which is a contradiction; hence, for n ∈ N and k = 1, 2, . . . ,K, supp(μEQ� ) ⊆

R \ ({|x| � TM} ∪ (∪sq=1O 1

TM(αpq))), for TM > 1; in particular, J := supp(μV) ⊆

R \ ({|x| � TM} ∪ (∪sq=1O 1

TM(αpq))), which establishes the compactness of the

support of the equilibrium measure μV ∈ M1(R): it is a straightforward conse-quence that supp(μV) ∩ {α1, α2, . . . , αK} = ∅. Since V: R \ {α1, α2, . . . , αK} →R is real analytic on R \ {α1, α2, . . . , αK} and supp(μV) ∩ {α1, α2, . . . , αK} =∅, in which case, for x ∈ supp(μV), infx∈J V(x) =: m (= m(n,k)) � V(x) �M (= M(n,k)) := supx∈J V(x), whence, for n ∈ N and k = 1, 2, . . . ,K, −∞ <∫

J V(ξ)dμV(ξ) < +∞, it follows, too, that for n ∈ N and k = 1, 2, . . . ,K, theweighted logarithmic energy of μV ∈ M1(R) is finite, that is (recalling that� (x) := exp(−V(x))),

−∞<IV[μV] =∫∫

J × Jln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

� (s)� (t)

⎞⎠

−1

dμV(s)dμV(t)<+∞,

which gives rise to the corollary that, for n∈N and k=1, 2, . . . ,K, the logarith-mic energy of μV ∈M1(R) is finite, that is,

−∞<IV[μV]=∫∫

J × Jln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

⎞⎠

−1

dμV(s)dμV(t)<+∞.

Furthermore, it is a straightforward consequence of the facts establishedthat, for n∈N and k=1, 2, . . . ,K, J :=supp(μV) (a proper compact subset ofR \ {α1, α2, . . . , αK}) has positive logarithmic capacity, that is, for n∈N andk=1, 2, . . . ,K, cap(J)=exp(−EV)>0. This establishes the third claim of theLemma; and thus concludes the proof. ��

Prior to establishing, for n∈N and k=1, 2, . . . ,K, the unicity of the equilib-rium measure μV (=μV(n,k)) ∈M1(R), whose support, supp(μV), as proven inLemma 3.1, is a proper compact subset of R \ {α1, α2, . . . , αK}, the variationalinequality of Lemma 3.2 is requisite.


Lemma 3.2 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8), and set � (z) :=exp(−V(z)). For n∈N and k=1, 2, . . . ,K, letμ

EQ

j (=μEQ

j (n,k)) ∈M1(R), j=1, 2, be non-negative, finite-moment measureson R supported on distinct compact sets, that is,

∫supp(μEQ

j )ξm dμEQ

j (ξ)<∞ and∫supp(μEQ

j )(ξ−αi)

−m dμEQ

j (ξ)<∞, m∈N, i=1, 2, . . . ,K, j=1, 2, and supp(μEQ

1 ) ∩supp(μEQ

2 )=∅. For n∈N and k=1, 2, . . . ,K, let μEQ (=μEQ(n,k)) :=μEQ

1 −μEQ

2be the (unique) Jordan decomposition of the finite-moment signed measure onR with compact support and mean zero, that is,

∫supp(μEQ)

dμEQ(ξ)=0. Suppose,furthermore, that for n∈N and k=1, 2, . . . ,K, the measures μEQ

j , j=1, 2, havefinite logarithmic energy:

−∞<∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× dμEQ

j (s)dμEQ

j (t)<+∞, j=1, 2.

Then, for n∈N and k=1, 2, . . . ,K,

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× dμEQ(s)dμEQ(t)

=∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

)−1

dμEQ(s)dμEQ(t)

=((n−1)K+k

n

)∫ +∞

0ξ−1|μ(ξ)|2 dξ�0,

where μ(λ) :=∫R

eiλξ dμEQ(ξ), and equality holds if, and only if, μEQ=0.

Remark 3.1 Lemma 3.2 says that, for n ∈ N and k = 1, 2, . . . ,K, uponwriting dμEQ(s)dμEQ(t) = d(μEQ

1 − μEQ2 )(s)d(μ

EQ1 − μEQ

2 )(t) = dμEQ1 (s)dμ

EQ1 (t) +

dμEQ2 (s)dμ

EQ2 (t)− dμEQ

1 (s)dμEQ2 (t)− dμEQ

2 (s)dμEQ1 (t), and taking note of the

symmetry of the (n- and k-dependent) integrand under the interchange s↔ t,

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn


× � (s)� (t)

)−1 (dμEQ

1 (s)dμEQ1 (t)+ dμEQ

2 (s)dμEQ2 (t)

)

�∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1 (dμEQ

1 (s)dμEQ2 (t)+dμEQ

2 (s)dμEQ1 (t)

)

= 2∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

)−1

dμEQ1 (s)dμ

EQ2 (t)

= 2∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

)−1

dμEQ2 (s)dμ

EQ1 (t);

hence, if, for n ∈ N and k = 1, 2, . . . ,K, ln(|s − t| 1

n( |s−t||s−αk||t−αk|

)κnk−1n∏s−1

q=1( |s−t||s−αpq ||t−αpq |

)κnkkqn)

is integrable with respect to the product measures dμEQ1 (λ)

·dμEQ1 (τ ) and dμEQ

2 (λ)dμEQ2 (τ ), which is the supposition of the Lemma,

then it is integrable with respect to the ‘mixed’ product measures dμEQ1 (λ)·dμEQ

2 (τ ) and dμEQ2 (λ)dμ

EQ1 (τ ). �

Proof of Lemma 3.2 Recall the following identity [56] (see pg. 147, (6.44)): forλ∈R and (any) ε>0,

ln(λ2+ε2)= ln(ε2)+2 Im

(∫ +∞

0(iν)−1(eiλν−1)e−εν dν

);

thus, for n∈N and k=1, 2, . . . ,K, with r∈{s, t},1

2n

∫∫R2

ln((s−t)2 + ε2)dμEQ(s)dμEQ(t) = 1

2nln(ε2)

∫∫R2

dμEQ(s)dμEQ(t)

+1

n

∫∫R2

Im

(∫ +∞

0(iν)−1(ei(s−t)ν−1)e−εν dν

)dμEQ(s)dμEQ(t),


1

2

(κnk−1

n

)∫∫R2

ln((s−t)2 + ε2)dμEQ(s)dμEQ(t) = 1

2

(κnk−1

n

)ln(ε2)

×∫∫

R2dμEQ(s)dμEQ(t)+

(κnk−1

n

)∫∫R2

Im

(∫ +∞


)

×dμEQ(s)dμEQ(t),

κnkkq

2n

∫∫R2

ln((s−t)2 + ε2)dμEQ(s)dμEQ(t)= κnkkq

2nln(ε2)

∫∫R2

dμEQ(s)dμEQ(t)

+κnkkq

n

∫∫R2

Im

(∫ +∞


)dμEQ(s)dμEQ(t),

1

2

(κnk−1

n

)∫∫R2

ln((r−αk)2 + ε2)dμEQ(s)dμEQ(t) = 1

2

(κnk−1

n

)ln(ε2)

×∫∫

R2dμEQ(s)dμEQ(t)+

(κnk−1

n

)∫∫R2

Im

(∫ +∞

0(iν)−1(ei(r−αk)ν−1)e−εν dν

)

×dμEQ(s)dμEQ(t),

κnkkq

2n

∫∫R2

ln((r−αpq)2+ε2)dμEQ(s)dμEQ(t)= κnkkq

2nln(ε2)

∫∫R2

dμEQ(s)dμEQ(t)

+ κnkkq

n

∫∫R2

Im

(∫ +∞

0(iν)−1(ei(r−αpq )ν−1)e−εν dν

)dμEQ(s)dμEQ(t).

Noting that, for n∈N and k=1, 2, . . . ,K,∫∫

R2 dμEQ(s)dμEQ(t)=(∫R

dμEQ(ξ))2 =

0, after some re-arrangement, the above simplifies to, with r ∈ {s, t},

1

2n

∫∫R2

ln((s−t)2+ε2)dμEQ(s)dμEQ(t)

= 1

nIm

(∫ +∞

0e−εν

(∫∫R2(iν)−1(ei(s−t)ν−1)dμEQ(s)dμEQ(t)

)dν),

1

2

(κnk−1

n

)∫∫R2


=(

κnk−1

n

)Im

(∫ +∞

0e−εν


)dν),

κnkkq

2n

∫∫R2



= κnkkq

nIm

(∫ +∞

0e−εν


)dν),

1

2

(κnk−1

n

)∫∫R2

ln((r−αk)2+ε2)dμEQ(s)dμEQ(t)

=(

κnk−1

n

)Im

(∫ +∞

0e−εν

(∫∫R2(iν)−1(ei(r−αk)ν−1)dμEQ(s)dμEQ(t)

)dν),

κnkkq

2n

∫∫R2

ln((r−αpq)2+ε2)dμEQ(s)dμEQ(t)

= κnkkq

nIm

(∫ +∞

0e−εν

(∫∫R2(iν)−1(ei(r−αpq )ν−1)dμEQ(s)dμEQ(t)

)dν).

Noting that, for n∈N and k=1, 2, . . . ,K,∫∫

R2(iν)−1(ei(s−t)ν−1)dμEQ(s)dμEQ(t) = 1

iν

∫∫R2

ei(s−t)ν dμEQ(s)dμEQ(t)

− 1

iν

∫∫R2

dμEQ(s)dμEQ(t)︸︷︷︸

= 0

= 1

iν

∫R

eisν dμEQ(s)∫

R

e−itν dμEQ(t),

and setting μ(λ) (= μ(n,k; λ)) :=∫R

eiλξ dμEQ(ξ), one gets that∫∫

R2(iν)−1(ei(s−t)ν−1)dμEQ(s)dμEQ(t)=(iν)−1|μ(ν)|2;

similarly, one shows that, for n∈N and k=1, 2, . . . ,K, with r∈{s, t},∫∫

R2(iν)−1(ei(r−αk)ν−1)dμEQ(s)dμEQ(t) =

∫∫R2(iν)−1(ei(r−αpq )ν−1)

× dμEQ(s)dμEQ(t)=0.

Hence, for n∈N and k=1, 2, . . . ,K, with r∈{s, t},1

2n

∫∫R2


= 1

nIm

(∫ +∞

0(iν)−1|μ(ν)|2e−εν dν

),

1

2

(κnk−1

n

)∫∫R2



=(

κnk−1

n

)Im

(∫ +∞


),

κnkkq

2n

∫∫R2


= κnkkq

nIm

(∫ +∞


),

1

2

(κnk−1

n

)∫∫R2

ln((r−αk)2+ε2)dμEQ(s)dμEQ(t)

= κnkkq

2n

∫∫R2

ln((r−αpq)2+ε2)dμEQ(s)dμEQ(t)=0.

Noting that μ(0)=∫R

dμEQ(ξ)=0, a Taylor expansion about λ=0 showsthat μ(λ)=λ→0 μ

′(0)λ+O(λ2), where μ′(0) :=∂λμ(λ)|λ=0; thus, λ−1|μ(λ)|2=λ→0

|μ′(0)|2λ+O(λ2), which means that there is no singularity in the integrandat λ=0 (in fact, λ−1|μ(λ)|2 is real analytic in an open neighbourhood of theorigin), whence, for n∈N and k=1, 2, . . . ,K,∫∫

R2ln((s−t)2+ε2)−

12n dμEQ(s)dμEQ(t) = 1

n

∫ +∞

0ξ−1|μ(ξ)|2e−εξ dξ,

∫∫R2

ln((s−t)2+ε2)−(κnk−1)

2n dμEQ(s)dμEQ(t)=(

κnk−1

n

)∫ +∞

0ξ−1|μ(ξ)|2e−εξ dξ,

∫∫R2

ln((s−t)2+ε2)−κnkkq

2n dμEQ(s)dμEQ(t) = κnkkq

n

∫ +∞

0ξ−1|μ(ξ)|2e−εξ dξ.

Adding the above, and recalling that∫∫

R2 ln((r−r)2+ε2)dμEQ(s)dμEQ(t)=0,r∈{s, t}, r∈{αk, αpq}, and κnk+∑s−1

q=1 κnkkq=(n−1)K+k, for n∈N and k=

1, 2, . . . ,K,

∫∫R2

ln

⎛⎝((s−t)2+ε2

) 12n

((s−t)2+ε2

((s−αk)2+ε2)((t−αk)2+ε2)

)κnk−1n

×s−1∏q=1

((s−t)2+ε2

((s−αpq)2+ε2)((t−αpq)

2+ε2)

)κnkkqn

⎞⎠

−1

dμEQ(s)dμEQ(t)

=((n−1)K+k

n

)∫ +∞

0ξ−1|μ(ξ)|2e−εξ dξ.

Now, using the fact that ln((s−t)2+ε2)−1 (resp., ln((r−r)2+ε2), r∈{s, t},r∈{αk, αpq}) is (resp., are) bounded from below (resp., above) uniformly with


respect to ε and that the measures have compact support, letting ε↓0 andinvoking the monotone and dominated convergence theorems, one arrives at,for n∈N and k=1, 2, . . . ,K,

∫∫R2

ln

⎛⎝((s−t)2+ε2

) 12n(

(s−t)2+ε2((s−αk)

2+ε2)((t−αk)2+ε2)

)κnk−12n

×s−1∏q=1

((s−t)2+ε2

((s−αpq )2+ε2)((t−αpq )

2+ε2)

)κnkkq2n

⎞⎟⎠

−1

dμEQ(s)dμEQ(t)

=ε↓0

∫∫R2

ln

⎛⎜⎝|s−t| 1

n


)κnk−1n

s−1∏q=1

(|s−t|

|s−αpq ||t−αpq |

)κnkkqn

⎞⎟⎠

−1

× dμEQ(s)dμEQ(t) =((n−1)K+k

n

)∫ +∞

0ξ−1|μ(ξ)|2 dξ�0,

where, trivially, equality holds if, and only if, μEQ = 0. Furthermore, recallingthat, by assumption,

∫R

dμEQ(ξ)=0, it follows that∫∫

R2 ln(� (s)� (t))−1

·dμEQ(s)dμEQ(t)=0; hence, letting ε↓0 and using, again, monotone and domi-nated convergence, one arrives at, for n∈N and k=1, 2, . . . ,K,

∫∫R2

ln

⎛⎝((s−t)2+ε2

) 12n(

(s−t)2+ε2((s−αk)

2+ε2)((t−αk)2+ε2)

)κnk−12n

×s−1∏q=1

((s−t)2+ε2

((s−αpq )2+ε2)((t−αpq )

2+ε2)

)κnkkq2n

⎞⎟⎠

−1

dμEQ(s)dμEQ(t)

=∫∫

R2ln

⎛⎝((s−t)2+ε2

) 12n(

(s−t)2+ε2((s−αk)

2+ε2)((t−αk)2+ε2)

)κnk−12n

×s−1∏q=1

((s−t)2+ε2

((s−αpq )2+ε2)((t−αpq )

2+ε2)

)κnkkq2n

� (s)� (t)

⎞⎟⎠

−1

× dμEQ(s)dμEQ(t)

=ε↓0

∫∫R2

ln

⎛⎜⎝|s−t| 1

n


)κnk−1n

s−1∏q=1

(|s−t|

|s−αpq ||t−αpq |

)κnkkqn

× � (s)� (t))−1 dμEQ(s)dμEQ(t)

=((n−1)K+k

n

)∫ +∞

0ξ−1|μ(ξ)|2 dξ�0, (36)

where, again, and trivially, equality holds if, and only if, μEQ =0. ��


With the variational inequality of Lemma 3.2 at hand, the unicity of theequilibrium measure μV (∈M1(R)) will now be established.

Lemma 3.3 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8), and set � (z) :=exp(−V(z)). For n∈N and k=1, 2, . . . ,K, defineIV : N × {1, 2, . . . ,K} × M1(R)→R as in Lemma 3.1, and consider the associ-ated minimisation problem EV = inf{IV[μEQ]; μEQ ∈M1(R)}. Then, for n∈N andk=1, 2, . . . ,K, there exists unique μV ∈M1(R) such that IV[μV]=EV.

Proof It was shown in Lemma 3.1 that, for n∈N and k=1, 2, . . . ,K, thereexists μV (=μV(n,k)) ∈M1(R), the equilibrium measure, such that IV[μV]=EV (=EV(n,k)), with ({x∈R; � (x)>0} ⊃) supp(μV)=: J a proper compactsubset of R \ {α1, α2, . . . , αK}, in particular, for (some) TM (=TM(n,k)) >1,supp(μV) ⊂ R \ ({|x|�TM} ∪ (∪s

q=1O 1TM(αpq))); therefore, it remains to estab-

lish the uniqueness of the equilibrium measure. For n∈N and k=1, 2, . . . ,K,let μV (= μV(n,k)) ∈M1(R) be a second probability measure for whichIV[μV]=EV =IV[μV]: the argument of Lemma 3.1 shows that, for n∈N andk=1, 2, . . . ,K, ({x∈R; � (x)>0} ⊃) supp(μV)=: J is a proper compact subsetof R \ {α1, α2, . . . , αK}, and −∞<IV[μV]<+∞. Define, for n∈N and k=1, 2, . . . ,K, the finite-moment signed measure (on R) μ� (=μ�(n,k)) := μV−μV , where μV, μV ∈M1(R), and J ∩ J=∅ (the measures are supported ondistinct proper compact subsets of R \ {α1, α2, . . . , αK}); thus, from Lemma 3.1(with the change μ→μ�), that is,

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× dμ�(s)dμ�(t)

=∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t))−1 dμ�(s)dμ�(t)�0,

it follows that, for n∈N and k=1, 2, . . . ,K, via the symmetry of the associatedintegrand under the interchange s↔ t,

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1 (dμV(s)dμV(t)+ dμV(s)dμV(t)

)


�∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1 (dμV(s)dμV(t)+dμV(s)dμV(t)

)

= 2∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1

dμV(s)dμV(t)

= 2∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1

dμV(s)dμV(t).

The above shows that (cf. Lemma 3.1), for n∈N and k=1, 2, . . . ,K, since both

IV[μV] and IV[μV] are finite, ln(|s−t| 1


)κnk−1n∏s−1

q=1


)κnkkqn)

is integrable with respect to both ‘product’ measures dμV(s)dμV(t) anddμV(s)dμV(t). From an argument on pg. 149 of [56], it follows that, for

n∈N and k=1, 2, . . . ,K, ln(|s−t| 1


)κnk−1n∏s−1

q=1


)κnkkqn)

isintegrable with respect to the one-parameter family of ‘product’ measuresdμEQλ (s)dμ

EQλ (t), where μEQ

λ (z) :=μV(z)+λ(μV−μV)(z), (z, λ)∈R × [0, 1]. Forn∈N and k=1, 2, . . . ,K, set Fμ(λ) :=IV[μEQ

λ ], that is,

Fμ :N×{1, 2, . . . ,K}×M1(R)×[0, 1]→R, (n,k, μEQλ ) →Fμ(n,k;μEQ

λ ) :=Fμ(λ)

=∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1

dμEQλ (s)dμ

EQλ (t). (37)

Noting that dμEQλ (s)dμ

EQλ (t) = dμV(s)dμV(t) + λdμV(s)d(μV − μV)(t) +

λdμV(t)d(μV − μV)(s)+ λ2d(μV − μV)(s)d(μV − μV)(t), it follows from the


above definition of Fμ(λ), Lemma 3.1, and the symmetry of the integrandunder the interchange s↔ t that, for n∈N and k=1, 2, . . . ,K,

Fμ(λ) = IV[μV]+2λ∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

� (s)� (t)

⎞⎠

−1

dμV(s)d(μV−μV)(t)

+ λ2∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1

d(μV−μV)(s)d(μV−μV)(t).

Sinceμ�∈M1(R) is a finite-moment signed measure on R with compact supportand mean zero, that is,

∫R

dμ�(ξ)=∫R

d(μV−μV)(ξ)=0, it follows from theexpression above and the variational inequality of Lemma 3.2 that, for n∈N

and k=1, 2, . . . ,K, Fμ(λ) is a twice-differentiable, real-valued function of λon [0, 1], and, with d2

λ :=d2/dλ2,

1

2d2λFμ(λ) =

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

� (s)� (t)

⎞⎠

−1

dμ�(s)dμ�(t)�0,

that is, Fμ(λ) is a real-valued convex9 function of λ on [0, 1]; thus, for n∈N andk=1, 2, . . . ,K, with [0, 1]�λ,

IV[μV] � Fμ(λ)=IV[μEQλ ]=Fμ(λ+(1−λ)0)�λFμ(1)+(1−λ)Fμ(0)

= λIV[μV]+(1−λ)IV[μV]=λIV[μV]+(1−λ)IV[μV] ⇒IV[μV] � IV[μEQ

λ ]�IV[μV],

9If f is twice differentiable on (a,b), then f ′′(x)�0 on (a,b) is both a necessary and sufficientcondition that f be convex on (a,b).


whence IV[μEQλ ]=IV[μV]=EV (=EV(n,k)). Since IV[μEQ

λ ]=Fμ(λ)=EV , itfollows, in particular, that for n∈N and k=1, 2, . . . ,K, d2

λFμ(λ)|λ=0=0, inwhich case

0 =∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

× � (s)� (t)

⎞⎠

−1

d(μV−μV)(s)d(μV−μV)(t)

=∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× d(μV−μV)(s)d(μV−μV)(t)+ 2∫

R

V(s)d(μV−μV)(s)

×∫

R

d(μV−μV)(t)︸︷︷︸= 0

⇒

0 =∫∫

R2ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× d(μV−μV)(s)d(μV−dμV)(t);but, in Lemma 3.2, it was shown that, for n∈N and k=1, 2, . . . ,K,

∫∫R2

ln

⎛⎝|s−t| 1

n


)κnk−1n

s−1∏q=1


)κnkkqn

⎞⎠

−1

× d(μV−μV)(s)d(μV−dμV)(t)

=((n−1)K+k

n

)∫ +∞

0τ−1|(μV−μV)(τ )|2 dτ�0,

whence, for n∈N and k=1, 2, . . . ,K,∫ +∞

0τ−1|(μV−μV)(τ )|2 dτ=0 ⇒

μV(τ )= μV(τ ), τ�0. Noting that μV(−τ)=∫

Re−iτξ dμV(ξ)= μV(τ ) and

μV(−τ)=∫

Re−iτξ dμV(ξ)= μV(τ ), it follows from μV(τ )= μV(τ ), τ�0, via

a complex-conjugation argument, that μV(−τ)= μV(−τ), τ�0; hence, forn∈N and k=1, 2, . . . ,K, μV(τ )= μV(τ ), τ ∈R. The latter relation shows that,


for n∈N and k=1, 2, . . . ,K,∫

Reiτξ d(μV−μV)(ξ)=0 ⇒ μV =μV ; thus the

uniqueness of the equilibrium measure. ��

For n∈N and k=1, 2, . . . ,K, RHPFPC, that is, (Y (n,k; z) :=Y (z), I+e−nV(z)σ+,R), is now reformulated as a family of K equivalent, auxiliary RHPs.

Notational Remark 3.1 For completeness, the integrand appearing in the de-finition of g(z) in Lemma 3.4 below is defined as follows: for n∈N and k=1, 2, . . . ,K,

ln

⎛⎝(z−s)

1n

((z−s)

(z−αk)(s−αk)

)κnk−1n

s−1∏q=1

((z−s)

(z−αpq)(s−αpq)

)κnkkqn

⎞⎠

:= 1

nln(z−s)+

(κnk−1

n

)(ln(z−s)−ln(z−αk)−ln(s−αk))

+s−1∑q=1

κnkkq

n

(ln(z−s)−ln(z−αpq)−ln(s−αpq)

),

where, for x<0, ln x := ln|x|+iπ .

Lemma 3.4 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8). For the associated equilibrium measure μV ∈M1(R), set J :=supp(μV), where J is a proper compact subset of R \ {α1, α2, . . . , αK}, and,for n∈N and k=1, 2, . . . ,K, let Y : N × {1, 2, . . . ,K} × C \ R→SL2(C) be theunique solution of RHPFPC. For n∈N and k=1, 2, . . . ,K, set

Y(n,k; z) :=Y(z)=e−n�2 ad(σ3)Y (z)e−n(g(z)−P0(n,k))σ3 ,

where � : N × {1, 2, . . . ,K}→R, the variational constant, is given in Lemma 3.8below,

g : N × {1, 2, . . . ,K} × C \(−∞,max

{max

q=1,2,...,s{αpq},max{J}

})→C,

(n,k, z) →g(n,k; z) :=g(z) =∫

Jln

⎛⎝(z−s)

1n

((z−s)

(z−αk)(s−αk)

)κnk−1n

×s−1∏q=1

((z−s)

(z−αpq)(s−αpq)

)κnkkqn

⎞⎠dμV(s),

and

P0(n,k)={

P+0 , z∈C+,

P−0 , z∈C−,


with

P±0 :=

∫J

ln

⎛⎝|s−αk| 1

n

s−1∏q=1

( |s−αk||s−αpq ||αpq−αk|

)κnkkqn

⎞⎠dμV(s)

− iπ(

κnk−1

n

)∫J∩R<αk

dμV(s) − iπs−1∑q=1

κnkkq

n

∫J∩R<αpq

dμV(s)

± iπ((n−1)K+k

n

)∫J∩R>αk

dμV(s)∓iπ∑

q∈�2(k)

κnkkq

n,

where �2(k) :={ j∈{1, 2, . . . , s−1}; αpj>αk}. Then, for n∈N and k=1, 2,. . . ,K, Y : N × {1, 2, . . . ,K} × C \ R→SL2(C) solves the following RHP: (i)Y(z) is holomorphic for z∈C \ R; (ii) the boundary values Y±(z) := limε↓0 Y(z±iε) satisfy the jump condition

Y+(z)=Y−(z)V(z), z∈R,

where, with g±(z) := limε↓0 g(z±iε),

V: N × {1, 2, . . . ,K} × R→GL2(C),

z →V(z) :=(

e−n(g+(z)−g−(z)+P−0 −P+

0 ) en(g+(z)+g−(z)−P−0 −P+

0 −V(z)−�)

0 en(g+(z)−g−(z)+P−0 −P+

0 )

);

(iii)

Y(z) =C± � z→αk

I+O(z−αk);

(iv)

Y(z) =C± � z→∞

O(1);

(v) for q∈{1, 2, . . . , s−1},Y(z) =

C± � z→αpq

O(1).

Proof For (arbitrary) z1, z2∈C±, and for n∈N and k=1, 2, . . . ,K, note fromthe definition of g(z) stated in the Lemma that g(z2)−g(z1)= iπ

∫ z2

z1F(ξ)dξ ,

where, with D :=C \ (supp(μV) ∪ {α1, α2, . . . , αK}),F: N × {1, 2, . . . ,K} × D→C, (n,k, z) →F(n,k; z) :=F(z)

=− 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)+((n−1)K+k

n

)∫J

dμV(ξ)

ξ−z

⎞⎠ , (38)


and, since supp(μV) ∩ {α1, α2, . . . , αK}=∅ and μV ∈M1(R), in particular,∫J(ξ− αk)

−m dμV(ξ)<∞, m∈N,

F(z)+ 1

iπ(κnk−1)

n(z−αk)=

D� z→αk

O(1)

(e.g., 0< |z−αk|%min{infξ∈J{||ξ |−αk|},min j=1,2,...,s−1{|αpj−αk|}}), and, forq∈{1, 2, . . . , s−1},

F(z)+ 1

iπ

κnkkq

n(z−αpq)=

D� z→αpq

O(1)

(e.g., 0< |z−αpq |%min{inf ξ∈Jj=1,2,...,s−1

{||ξ |−αpj|},min j=1,2,...,s−1{|αpj−αk|}}); thus,

|g(z2)−g(z1)|�π |z2−z1| supz∈C± |F(z)|. Thus, for n∈N and k=1, 2, . . . ,K,from the definition of g : C \ D→C, with D :=(−∞,max{maxq=1,2,...,s{αpq},max{supp(μV)}}), given in the Lemma: (1) for ξ ∈ J (J ∩ {α1, α2, . . . , αK}=∅), z∈C \ D, with 0< |z−αk|%min{infξ∈J{||ξ |−αk|},min j=1,2,...,s−1{|αpj−αk|}},and μV ∈M1(R), in particular,

∫J(ξ−αk)

−m dμV(ξ)<∞, m∈N, it follows

from the expansions 1(z−αk)−(ξ−αk)

=−∑lj=0

(z−αk)j

(ξ−αk) j+1 + (z−αk)l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 , and

ln(1−∗∗∗)=|∗∗∗|→0−∑∞j=1

∗∗∗ j

j , and a careful analysis of the branch cuts, that

g(z) =C± � z→αk

−(

κnk−1

n

)ln(z−αk) +

∫J

ln(|ξ−αk| 1

n

×s−1∏q=1

( |ξ−αk||ξ−αpq ||αpq−αk|

)κnkkqn

⎞⎠dμV(ξ)

− iπ(

κnk−1

n

)∫J∩R<αk

dμV(τ )−iπs−1∑q=1

κnkkq

n

∫J∩R<αpq

dμV(τ )

± iπ((n−1)K+k

n

)∫J∩R>αk

dμV(τ )∓iπ∑

q∈�2(k)

κnkkq

n

+∞∑

m=1

1

m

⎛⎝s−1∑

q=1

κnkkq

n(αpq−αk)m−((n−1)K+k

n

)∫J(ξ−αk)

−m

× dμV(ξ)

⎞⎠ (z−αk)

m, (39)

where �2(k) is defined in the Lemma; (2) for ξ ∈ J, z ∈ D, with|z| & max{supξ∈J{|ξ |},maxq=1,2,...,s{|αpq |}}, and μV ∈ M1(R), in particular,∫

J dμV(ξ) = 1 and∫

J ξm dμV(ξ)<∞, m∈N, it follows from the expansions


1ξ−z = −∑l

j=0ξ j

z j+1 + ξ l+1

zl+1(ξ−z) , l∈Z+0 , and ln(z −∗∗∗) =|z|→∞ ln(z)−∑∞

j=11j (

∗∗∗z )

j,and a careful analysis of the branch cuts, that

g(z) =C± � z→∞

1

nln(z)−

∫J

ln

⎛⎝|ξ−αk|

κnk−1n

s−1∏q=1

|ξ−αpq |κnkkq

n

⎞⎠dμV(ξ)

−iπ(

κnk−1

n

)∫J∩R<αk

dμV(τ )− iπs−1∑q=1

κnkkq

n

∫J∩R<αpq

dμV(τ )

+∞∑

m=1

1

m

⎛⎝(

κnk−1

n

)(αk)

m+s−1∑q=1

κnkkq

n(αpq)

m

−((n−1)K+k

n

)∫Jξm dμV(ξ)

⎞⎠ z−m; (40)

and (3) for q∈{1, 2, . . . , s−1}, ξ ∈ J, z∈D, with 0< |z−αpq |%min{inf ξ∈J

j=1,2,...,s−1

{||ξ |−αpj|},min j=1,2,...,s−1{|αpj−αk|}}, and μV ∈M1(R), in par-

ticular,∫

J(ξ− αpq)−mdμV(ξ)<∞, m∈N, it follows from the expansions

1(z−αpq )−(ξ−αpq )

=−∑lj=0

(z−αpq )j

(ξ−αpq )j+1 + (z−αpq )

l+1

(ξ−αpq )l+1(z−ξ) , l∈Z

+0 , and ln(1−∗∗∗)=|∗∗∗|→0

−∑∞j=1

∗∗∗ j

j , and a careful analysis of the branch cuts, that

g(z) =C± � z→αpq

− κnkkq

nln(z−αpq)+

∫J

ln

⎛⎜⎜⎝

|ξ−αpq |κnk

n

(|ξ−αk||αpq−αk|)κnk−1n

×s−1∏j=1j�=q

( |ξ−αpq ||ξ−αpj||αpq−αpj|

)κnkk jn

⎞⎟⎟⎠dμV(ξ)

− iπ(

κnk−1

n

)∫J∩R<αk

dμV(τ )−iπκnkkq

n

∫J∩R<αpq

dμV(τ )

−iπs−1∑j=1j�=q

κnkk j

n

∫J∩R<αp j

dμV(τ )∓ iπ(

κnk−1

n

)ε(k,q)

∓iπ∑

j∈�2(k,q)

κnkk j

n±iπ

((n−1)K+k

n

)∫J∩R>αpq

dμV(τ )


+∞∑

m=1

1

m

⎛⎜⎜⎝ (κnk−1)

n(αk−αpq)m+

s−1∑j=1j�=q

κnkk j

n(αpj−αpq)m

−((n−1)K+k

n

)∫J(ξ−αpq)

−m dμV(ξ)

⎞⎟⎟⎠ (z−αpq)

m, (41)

where

ε(k,q)={

1, αpq<αk,0, αpq>αk,

and �2(k,q) :={ j∈{1, 2, . . . , s−1}; αpj>αpq}. Using the definition of Y(z) (interms of Y (z)) stated in the Lemma, and recalling that Y : N × {1, 2, . . . ,K} ×C \ R→SL2(C) solves, uniquely, RHPFPC, via the above asymptotic expan-sions for g(z), one arrives at the RHP (Y(n,k; z) :=Y(z),V(z),R) stated in theLemma. ��

In order to prove the following Lemma 3.5, which is requisite for theproof of Lemma 3.6 (see below), one follows closely the idea of the proof ofTheorem 6.6.2 of Deift [56].

Lemma 3.5 Let V: R\{α1, α2, . . . , αK}→R satisfy conditions (6)–(8). For n∈N

and k=1, 2, . . . ,K, let

dV(n−1)K+k := 1

((n−1)K+k)((n−1)K+k−1)

× inf{x1,x2,...,x(n−1)K+k}⊂R

(n−1)K+k∑i, j=1

i �= j

⎛⎝ln

⎛⎝|xi−x j| 1

n

( |xi−x j||xi−αk||x j−αk|

)κnk−1n

×s−1∏q=1

( |xi−x j||xi−αpq ||x j−αpq |

)κnkkqn

⎞⎠−1

+2((n−1)K+k−1)(n−1)K+k∑

j=1

V(x j)

⎞⎟⎠ .

Then, for k=1, 2, . . . ,K, limn→∞ dV(n−1)K+k exists,

limn→∞ dV

(n−1)K+k=EV = inf{IV[μEQ]; μEQ ∈M1(R)

}=IV[μV],

and limn→∞ exp(−dV(n−1)K+k)=exp(−EV)>0 and finite.


For n∈N and k=1, 2, . . . ,K, let x∗1, x∗2, . . . , x

∗(n−1)K+k denote the generalised

weighted Fekete points, that is,

dV(n−1)K+k = 1

((n−1)K+k)((n−1)K+k−1)

(n−1)K+k∑i, j=1

i �= j

(ln(|x∗i −x∗j |

1n

×( |x∗i −x∗j ||x∗i −αk||x∗j−αk|

)κnk−1n s−1∏

q=1

( |x∗i −x∗j ||x∗i −αpq ||x∗j−αpq |

)κnkkqn

⎞⎟⎠

−1

+ 2((n−1)K+k−1)(n−1)K+k∑

j=1

V(x∗j)

⎞⎠ .

For n∈N and k=1, 2, . . . ,K, with {x∗1, x∗2, . . . , x∗(n−1)K+k} a generalised weighted((n−1)K+k)-Fekete set, denote by

λ(n−1)K+k := 1

((n−1)K+k)

(n−1)K+k∑j=1

δx∗j ,

where δx∗j , j=1, 2, . . . , (n−1)K+k, are the Dirac delta (atomic) masses concen-trated at x∗j , the normalised counting measure. Then, in the weak-∗ topology ofmeasures, λ(n−1)K+k converges (weakly) to μV as n→∞.

Sketch of Proof For n∈N and k=1, 2, . . . ,K, with V: R \ {α1, α2, . . . , αK}→R

satisfying conditions (6)–(8), set, with N :=(n−1)K+k,

δVN = sup

{x1,x2,...,xN }⊂R

⎛⎝∏

i< j

(f(xi, x j))2 e−2V(xi) e−2V(x j)

⎞⎠

1N (N −1)

= sup{x1,x2,...,xN }⊂R

⎛⎝∏

i< j

(f(xi, x j))2 e−2(N −1)

∑Nj=1 V(x j)

⎞⎠

1N (N −1)

,

where∏

i< j(∗∗∗) :=∏N −1

i=1

∏Nj=i+1(∗∗∗), and

f : N × {1, 2, . . . ,K} × R2→R+, (n,k, x, y) → f(n,k; x, y) := f(x, y)= f(y, x)

= |x−y| 1n

( |x−y||x−αk||y−αk|

)κnk−1n

s−1∏q=1

( |x−y||x−αpq ||y−αpq |

)κnkkqn

.


(Note: n→∞⇒N →∞). One begins by showing that, for n∈N and k=1, 2, . . . ,K, δV

N is finite, and (the supremum) is attained at some (or many)finite point set(s) {x∗1, x∗2, . . . , x∗N } ⊂ R, with x∗j =x∗j(n,k), j=1, 2, . . . ,N (ac-tually, as will be shown below, {x∗1, x∗2, . . . , x∗N } ⊂ R \ {α1, α2, . . . , αK}). Recall,from the (sketch of the) proof of Lemma 3.1, the following inequality: fors, t∈R, |s−t|�(1+s2)1/2(1+t2)1/2. Using this inequality, one shows that, forn∈N and k=1, 2, . . . ,K,

∏i �= j

|xi−x j|�∏i �= j

(1+x2

i

)1/2(1+x2

j

)1/2

=⎛⎝N∏

j=1

(1+x2

j

)⎞⎠N −1

,

∏i �= j


)κnk−1

�∏i �= j

((1+ 1

(xi−αk)2

)1/2

×(

1+ 1

(x j−αk)2

)1/2)κnk−1

=⎛⎝N∏

j=1

(1+ 1

(x j−αk)2

)κnk−1⎞⎠

N −1

,

∏i �= j

s−1∏q=1


)κnkkq

�∏i �= j

s−1∏q=1

((1+ 1

(xi−αpq)2

)1/2

×(

1+ 1

(x j−αpq)2

)1/2)κnkkq

=⎛⎝N∏

j=1

s−1∏q=1

(1+ 1

(x j−αpq)2

)κnkkq

⎞⎠

N −1

;

hence, using the fact that f(x, y)= f(y, x), via the above inequalities, one arrivesat, for n∈N and k=1, 2, . . . ,K:

∏i< j

(f(xi, x j))2 e−2V(xi) e−2V(x j) �

N∏j=1

⎛⎝(1+x2

j

) 1n

(1+ 1

(x j−αk)2

)κnk−1n

×s−1∏q=1

(1+ 1

(x j−αpq)2

)κnkkqn

e−2V(x j)

⎞⎠

N −1

.


Recalling that V: R \ {αi, α2, . . . , αK}→R satisfies conditions (6)–(8), oneshows that, for n∈N and k=1, 2, . . . ,K: (i) for x j∈R \ {α1, α2, . . . , αK}, j=1, 2, . . . ,N ,

0�(1+x2

j

) 1n

(1+ 1

(x j−αk)2

)κnk−1n

s−1∏q=1

(1+ 1

(x j−αpq)2

)κnkkqn

e−2V(x j)

�c(n,k)<+∞;

(ii) for x j∈O∞ :={x∈R; |x|>δ−1∞ }, j=1, 2, . . . ,N , where δ∞ (=δ∞(n,k)) isan arbitrarily fixed, sufficiently small positive real number, there exists c∞(=c∞(n,k)) >0 and bounded such that V(x j)�(1+c∞) ln(x2

j+1), it follows

that (1+x2j)

1/ne−2V(x j)→0 as |x j|→+∞, j=1, 2, . . . ,N ; hence, for x j∈O∞,j=1, 2, . . . ,N ,

0�(1+x2

j

)1n

(1+ 1

(x j−αk)2

)κnk−1n

s−1∏q=1

(1+ 1

(x j−αpq)2

)κnkkqn

e−2V(x j)

�c(n,k)<+∞;

and (iii) for x j∈Oδk :={x∈R; |x−αk|<δk}, j=1, 2, . . . ,N , k=1, 2, . . . ,K,where δk (= δk(n,k)) are arbitrarily fixed, sufficiently small positive realnumbers, there exists ci (=ci(n,k)) >0 and bounded such that V(x j)�(1+ck) ln((x j−αk)

−2+1), it follows that (1+(x j−αk)−2)

κnk−1n exp(−2V(x j))→0 as

x j→αk, j=1, 2, . . . ,N , k=1, 2, . . . ,K; hence, for x j∈Oδk , j=1, 2, . . . ,N ,k=1, 2, . . . ,K,

0�(1+x2

j

)1n

(1+ 1

(x j−αk)2

)κnk−1n

s−1∏q=1

(1+ 1

(x j−αpq)2

)κnkkqn

e−2V(x j)

�c(n,k)<+∞.

Hence, for n∈N and k=1, 2, . . . ,K, with xm∈R, m∈{1, 2, . . . ,N },∏i< j

(f(xi, x j))2e−2V(xi) e−2V(x j) =

∏i �= j

f(xi, x j)e−2(N −1)∑N

j=1 V(x j)

� (c(n,k))(N −1)2

⎛⎝(1+x2

m)1n

(1+ 1

(xm−αk)2

)κnk−1n


×s−1∏q=1

(1+ 1

(xm−αpq)2

)κnkkqn

e−2V(xm)

⎞⎠

N −1

→ 0 as |xm|→+∞ and as xm→αk.

Hence, for n∈N and k=1, 2, . . . ,K, δVN is finite, and (the supremum) is

attained at some (or many) finite point set(s) {x∗1, x∗2, . . . , x∗N } ⊂ R, wherex∗j =x∗j(n,k), j=1, 2, . . . ,N (actually, {x∗j}Nj=1 ⊂ R \ {α1, α2, . . . , αK}). Forn∈N and k=1, 2, . . . ,K, one agrees to call, with some abuse of nomenclature,a maximising set, that is, a set {x∗1, x∗2, . . . , x∗N } for which

δVN =

⎛⎝∏

i< j

(f(x∗i , x∗j))

2 e−2V(x∗i ) e−2V(x∗j )

⎞⎠

1N (N −1)

=⎛⎝∏

i �= j

f(x∗i , x∗j)e

−2(N −1)∑N

j=1 V(x∗j )

⎞⎠

1N (N −1)

,

a generalised weighted N -Fekete set, and the N points x∗1, x∗2, . . . , x

∗N will be

called general weighted Fekete points. For n∈N and k=1, 2, . . . ,K, define

K VN (x1, x2, . . . , xN ) :=

N∑i, j=1

i �= j

KV(xi, x j),

where, from the (sketch of the) proof of Lemma 3.1,

KV(s, t)=KV(t, s) :=1

nln(|s−t|−1)+

(κnk−1

n

)ln

(∣∣∣∣ 1

t−αk− 1

s−αk

∣∣∣∣−1)

+s−1∑q=1

κnkkq

nln

(∣∣∣∣ 1

t−αpq

− 1

s−αpq

∣∣∣∣−1)+V(s)+V(t).

An algebraic calculation shows that, for n∈N and k=1, 2, . . . ,K,

K VN (x1, x2, . . . , xN ) =

N∑i, j=1

i �= j

⎛⎝1

nln

(1

|xi−x j|)+(

κnk−1

n

)ln

( |xi−αk||x j−αk||xi−x j|

)

+s−1∑q=1

κnkkq

nln

( |xi−αpq ||x j−αpq ||xi − x j|

)⎞⎠+ 2(N −1)

N∑j=1

V(x j).

For n∈N and k=1, 2, . . . ,K, set

dVN := 1

N (N −1)inf

{x1,x2,...,xN }⊂R

K VN (x1, x2, . . . , xN ).


One shows, via the definition of K VN (x1, x2, . . . , xN ), that, for n∈N and k=

1, 2, . . . ,K,

e−K VN (x1,x2,...,xN ) =

∏i< j

⎛⎝|xi−x j| 1

n


)κnk−1n

×s−1∏q=1


)κnkkqn

⎞⎠

2

e−2(N −1)∑N

j=1 V(x j)

=∏i< j

(f(xi, x j))2e−2(N −1)

∑Nj=1 V(x j),

whence

e−1

N (N −1)KV

N (x1,x2,...,xN )=⎛⎝∏

i< j

(f(xi, x j))2e−2(N −1)

∑Nj=1 V(x j)

⎞⎠

1N (N −1)

.

Using the fact that − sup(−∗∗∗)= inf(∗∗∗), one arrives at, for n∈N and k=1, 2, . . . ,K,

e−dVN = sup

{x1,x2,...,xN }⊂R

⎛⎝∏

i< j

(f(xi, x j))2e−2(N −1)

∑Nj=1 V(x j)

⎞⎠

1N (N −1)

=δVN .

The above calculations also show that, for n∈N and k=1, 2, . . . ,K,

e−K VN (x1,x2,...,xN )�

⎛⎝N∏

j=1

(1+x2

j

) 1n (

1+(x j−αk)−2)κnk−1

n

×s−1∏q=1

(1+(x j−αpq)

−2)κnkkq

n e−2V(x j)

⎞⎠

N −1

,

whence

e−dVN � sup

{x1,x2,...,xN }⊂R

⎛⎝N∏

j=1

⎛⎝(1+x2

j

) 1n (

1+(x j−αk)−2)κnk−1

n

×s−1∏q=1

(1+(x j−αpq)

−2)κnkkqn e−2V(x j)

⎞⎠

N −1⎞⎟⎠

1N (N −1)

<+∞;

hence, for n∈N and k=1, 2, . . . ,K, dVN >−∞, that is, (N (N −1))−1

· inf{x1,x2,...,xN }⊂R K VN (x1, x2, . . . , xN )>−∞.


At the beginning of the proof it was shown that, for n∈N and k=1, 2, . . . ,K,a generalised weighted N -Fekete set for K V

N (x1, x2, . . . , xN ) exists, that is,a set {x∗1, x∗2, . . . , x∗N } (⊂ R \ {α1, α2, . . . , αK}) such that dV

N =(N (N −1))−1

·K VN (x

∗1, x

∗2, . . . , x

∗N ) (the points x∗j , j=1, 2, . . . ,N , are distinct). For n∈N

and k=1, 2, . . . ,K, define the normalised counting measure for the general-ised weighted N -Fekete set as follows:

λN := 1

N

N∑j=1

δx∗j ,

where δx∗j , j=1, 2, . . . ,N , are the Dirac delta (atomic) masses concentrated

at x∗j (note that, for n∈N and k=1, 2, . . . ,K,∫

RdλN (ξ)= 1

N

∫R

∑Nj=1 δ(ξ−

x∗j)dξ=1). One now shows that, in the weak-∗ topology of measures, fork=1, 2, . . . ,K, as n→∞ (⇒N →∞) λN converges (weakly) to μV (∈M1(R)), that is, for k=1, 2, . . . ,K, λN ⇀μV as n→∞. One proceeds viaa contradiction argument, namely, one assumes that, for k=1, 2, . . . ,K and(some) g∈CCC0

b(R) (the space of real-valued, bounded, continuous functions onR with compact support),

∫R

g(ξ)dλN (ξ)�∫

Rg(ξ)dμV(ξ) as n→∞, that is,

∃ ε>0 and a subsequence Nk→∞ such that, for all k∈N, |∫R

g(ξ)dλNk(ξ)−∫

Rg(ξ)dμV(ξ)|�ε. One first shows, though, that for n∈N and k=1, 2, . . . ,K,

the sequence of probability measures {λN }N ∈N is tight (cf. (sketch of the)proof of Lemma 3.1). Since, for n∈N and k=1, 2, . . . ,K,

1

N (N −1)inf{x1,x2,...,xN }⊂R

K VN (x1, x2, . . . , xN ) �

1

N (N −1)K V

N (x1, x2, . . . , xN )

� 1

N (N −1)sup

{x1,x2,...,xN }⊂R

K VN (x1, x2, . . . , xN ),

upon recalling the definition of dVN , one notes that (N (N −1))−1K V

N (x1,

x2, . . . , xN )�dVN : integrating both sides of this latter inequality with respect

to the ‘product measure’ (cf. Remark 3.1) dμV(x1)dμV(x2) · · · dμV(xN ), onearrives at, for n∈N and k=1, 2, . . . ,K, after a straightforward calculation:

∫∫R2

ln

⎛⎝ 1

|s−τ | 1n

( |s−αk||τ−αk||s−τ |

)κnk−1n

s−1∏q=1

( |s−αpq ||τ−αpq ||s−τ |

)κnkkqn

× eV(s) eV(τ ))

dμV(s)dμV(τ ) =∫∫

R2KV(s, τ )dμV(s)dμV(τ )

= IV[μV]=EV �dVN (>−∞).


Via this inequality and the definition of ψV(z) given in (35) (cf. (sketch of the)proof of Lemma 3.1), one proceeds thus; for n∈N and k=1, 2, . . . ,K:

EV �dVN = 1

N (N −1)inf

{x1,x2,...,xN }⊂R

K VN (x1, x2, . . . , xN )

= 1

N (N −1)K V

N (x∗1, x

∗2, . . . , x

∗N ) =

1

N (N −1)

N∑i, j=1

i �= j

KV(x∗i , x

∗j)

� 1

2

1

N (N −1)

N∑i, j=1

i �= j

(ψV(x

∗i )+ψV(x

∗j))

︸︷︷︸= 2(N −1)

∑Nj=1 ψV(x

∗j )

= 1

N

N∑j=1

ψV(x∗j)

=∫

R

ψV(ξ)dλN (ξ)⇒

EV = inf{IV[μEQ]; μEQ∈M1(R)

} = IV[μV]�dVN �

∫R

ψV(ξ)dλN (ξ).

For n∈N and k=1, 2, . . . ,K, writing DM :={|x|� M} ∪ (∪sq=1O 1

M(αpq)), for

some suitably chosen M (=M(n,k)) >1, one proceeds as in the (sketch ofthe) proof of Lemma 3.1 to show that

∫R

ψV(ξ)dλN (ξ)=∫

R\DM

ψV(ξ)︸︷︷︸�−|CV |

dλN (ξ)+∫

DM

ψV(ξ)︸︷︷︸> b

dλN (ξ),

whence follows the tightness of the sequence of probability measures{λN }N ∈N in M1(R), that is, for (some) sufficiently small ε (=ε(n,k)) >0,lim supN →∞

∫DM

dλN (ξ)�ε. For n∈N and k=1, 2, . . . ,K, with L∈R,

dVN = 1

N (N −1)inf

{x1,x2,...,xN }⊂R

K VN (x1, x2, . . . , xN )

= 1

N (N −1)K V

N (x∗1, x

∗2, . . . , x

∗N ) =

1

N (N −1)

N∑i, j=1

i �= j

KV(x∗i , x

∗j)

� 1

N (N −1)

N∑i, j=1

i �= j

min{

L,KV(x∗i , x

∗j)}= N 2

N (N −1)


×∫∫

R2min

{L,KV(s, τ )

} 1

N

N∑i=1

δ(s−x∗i )ds

︸︷︷︸= dλN (s)

1

N

N∑j=1

δ(τ−x∗j)dτ

︸︷︷︸= dλN (τ )

⇒

dVN � N 2

N (N −1)

∫∫R2

min{

L,KV(s, τ )}

dλN (s)dλN (τ ). (42)

Now, and as a—direct—consequence of tightness, for n∈N and k=1, 2, . . . ,K,as the (sub) sequence of probability measures {λNk

}∞k=1

in M1(R) is also tight,there exists, by a Helly selection theorem, a further weak-∗ convergent sub-sequence of probability measures {λNk j

}∞j=1 in M1(R) converging weakly to

a probability measure λ∈M1(R), that is, λNk j⇀λ as j→∞. Proceeding, verba-

tim, as in the (sketch of the) proof of Lemma 3.1, one shows, via inequality(42), that, as j→∞ (using N 2

k j(Nk j(Nk j

−1))−1→1) N 2k j(Nk j(Nk j

−1))−1

· ∫∫R2 min{L,KV(s, τ )}dλNk j

(s)dλNk j(τ )→∫∫

R2 min{L,KV(s, τ )}dλ(s)dλ(τ)⇒

lim infj→∞

dVNk j

�∫∫

R2min

{L,KV(s, τ )

}dλ(s)dλ(τ ) :

letting L↑+∞ and using monotone convergence (cf. (sketch of the) proof ofLemma 3.1), one arrives at

lim infj→∞

dVNk j

�∫∫

R2KV(s, τ )dλ(s)dλ(τ )=IV[λ] (>−∞);

but,

EV � lim supj→∞

dVNk j

� lim infj→∞

dVNk j

�IV[λ]

� EV = inf{IV[μEQ]; μEQ∈M1(R)

}=IV[μV] ⇒IV[λ] = IV[μV],

which, by the unicity of the equilibrium measure (cf. Lemma 3.3), impliesthat λ=μV , in particular, for g∈CCC0

b(R),∫

Rg(ξ)dλNk j

(ξ)→∫R

g(ξ)dμV(ξ) as

j→∞, which is a contradiction; hence, λN ⇀μV as n→∞ (⇒N →∞). Since,from the above calculation, lim sup j→∞ dV

Nk jand lim inf j→∞ dV

Nk jexist, it follows

that lim j→∞ dVNk j

exists, and equals EV = inf{IV[μEQ]; μEQ ∈M1(R)}=IV[μV]; in

fact, limN →∞ dVN =EV =IV[μV]. Observe from the calculations above that

EV �∫

RψV(ξ)dλN (ξ)>−∞. In order to prove that, indeed, limN →∞ dV

N =EV , it is sufficient to show that if {dV

N }∞N =1 converges for some subse-quence, the limit is always equal to EV . So, suppose that E♣ := liml→∞ dV

Nl

for some such subsequence {dVNl}∞l=1. Using that λN ⇀μV as N →∞, one

must have that, in particular, λNl⇀μV as l→∞; but, then, as shown above,


EV � lim supl→∞ dVNl

� lim infl→∞ dVNl

�IV[μV]=EV ; thus, by the unicity of the

equilibrium measure, one arrives at E♣=EV , that is, limN →∞ dVN =EV . ��

With the result of Lemma 3.5 at hand, one now establishes, for n∈N and k=1, 2, . . . ,K, the regularity of the family of K equilibrium measures M1(R)�μV .

Lemma 3.6 Let V: R \ {α1, α2, . . . , αK}→R satisfy conditions (6)–(8). Then,for n∈N and k=1, 2, . . . ,K, the equilibrium measure M1(R)�μV is absolutelycontinuous with respect to Lebesgue measure.

Sketch of Proof For n∈N and k=1, 2, . . . ,K, with V: R \ {α1, α2, . . . , αK}→R

satisfying conditions (6)–(8), set

f : N × {1, 2, . . . ,K} × R \ {α1, α2, . . . , αK}→R,

(n,k, x) → f (n,k; x) := f (x)=∏N

j=1(x−x∗j)∏s

q=1(x−αpq)lq,

where N :=(n−1)K+k, {x∗j}Nj=1 (⊂ R \ {α1, α2, . . . , αK}), with x∗i <x∗j fori< j, are the generalised weighted Fekete points described in Lemma 3.5,(cf. Subsection 1.2) lq :=κnkkq

, q=1, 2, . . . , s−1, ls :=κnk, αps:=αk, and the

poles are enumerated as per the ordered partition introduced in Subsection 1.2.One shows, for n∈N and k=1, 2, . . . ,K, proceeding from the definition of fgiven above, and using the identity

⎛⎝ N∑

j=1

1

x−x∗j

⎞⎠

2

−N∑j=1

1

(x−x∗j)2=

N∑j=1

N∑k′=1k′ �= j

(1

x−x∗j− 1

x−x∗k′

)1

x∗j−x∗k′

=N∑j=1

N∑k′=1k′ �= j

1

(x−x∗j)(x−x∗k′)

= 2N∑j=1

N∑k′=1k′ �= j

1

(x−x∗j)(x∗j−x∗k′)

,

that

f ′′(x)f ′(x)

= fN(x)fD(x)

,

where

fN(x) := 2N∑j=1

N∏l=1l �= j

(x−x∗l )N∑

k′=1k′ �= j

1

x∗j−x∗k′−2

N∑j=1

N∏l=1l �= j

(x−x∗l )s∑

q=1

lq

x−αpq


+N∏j=1

(x−x∗j)

⎛⎜⎝

s∑q=1

lq

(x−αpq)2+⎛⎝ s∑

q=1

lq

x−αpq

⎞⎠

2⎞⎟⎠ ,

fD(x) :=N∑j=1

N∏k′=1k′ �= j

(x−x∗k′)−N∏j=1

(x−x∗j)s∑

q=1

lq

x−αpq

,

whence, using the fact that, for m=1, 2, . . . ,N ,

N∑j=1

N∏l=1l �= j

(x−x∗l )N∑

k′=1k′ �= j

1

x∗j−x∗k′

∣∣∣∣∣∣∣x=x∗m

=N∏l=1l �=m

(x∗m−x∗l )N∑

k′=1k′ �=m

1

x∗m−x∗k′,

N∑j=1

N∏l=1l �= j

(x−x∗l )s∑

q=1

lq

x−αpq

∣∣∣∣∣∣∣x=x∗m

=N∏l=1l �=m

(x∗m−x∗l )s∑

q=1

lq

x∗m−αpq

,

N∑j=1

N∏k′=1k′ �= j

(x−x∗k′)−N∏j=1

(x−x∗j)s∑

q=1

lq

x−αpq

∣∣∣∣∣∣∣x=x∗m

=N∏

k′=1k′ �=m

(x∗m−x∗k′),

one arrives at, for n∈N and k=1, 2, . . . ,K,

1

2

f ′′(x∗m)f ′(x∗m)

=N∑

k′=1k′ �=m

1

x∗m−x∗k′−

s∑q=1

lq

x∗m−αpq

, m=1, 2, . . . ,N (43)

(recall that {x∗j}Nj=1 ∩ {αp1 , αp2 , . . . , αps}=∅). Recall from the (sketch of the)

proof of Lemma 3.5 that, for n∈N and k=1, 2, . . . ,K,

δVN = sup

{xl}Nl=1 ⊂R\{α1,α2,...,αK}

⎛⎝∏

i< j

(f(xi, x j))2 e−2(N −1)

∑Nj=1 V(x j)

⎞⎠

1N (N −1)

,

where

f(x, y)= f(y, x)=|x−y| 1n

( |x−y||x−αps

||y−αps|) ls−1

ns−1∏q=1

( |x−y||x−αpq ||y−αpq |

) lqn

:


one shows from this formula for δVN that, for n∈N and k=1, 2, . . . ,K,

− (N −1)V ′(x∗m)+N

n

N∑k′=1k′ �=m

1

x∗m−x∗k′−(N −1)

s−1∑q=1

lq/nx∗m−αpq

− (N −1)(ls−1)/nx∗m−αps

=0, m=1, 2, . . . ,N . (44)

Combining (43) and (44), one shows that, for n∈N and k=1, 2, . . . ,K,

f ′′(x∗m)+2n

N

⎛⎝− (N −1)V ′(x∗m)+

s∑q=1

lq/nx∗m−αpq

+ (N −1)/nx∗m−αps

⎞⎠ f ′(x∗m)=0,

m=1, 2, . . . ,N ,

whence, since {x∈R; f (x)=0}={x∗1, x∗2, . . . , x∗N }, via an ODE argument à laPolya (see Chapter VI of [8]), one arrives at, for n∈N and k=1, 2, . . . ,K,

f ′′(x)+2n

N

⎛⎝−(N −1)V ′(x)+

s∑q=1

lq/nx−αpq

+ (N −1)/nx−αps

⎞⎠ f ′(x)=Q(x) f (x),

(45)

with Q(x∗m) �=0, m=1, 2, . . . ,N , to be determined below. Writing, for n∈N

and k=1, 2, . . . ,K,

Q(x)= f ′′(x)f (x)

+2n

N

⎛⎝−(N −1)V ′(x)+

s∑q=1

lq/nx−αpq

+ (N −1)/nx−αps

⎞⎠ f ′(x)

f (x),

where, for x /∈{x∗1, x∗2, . . . , x∗N },f ′(x)f (x)

=N∑j=1

1

x−x∗j−

s∑q=1

lq

x−αpq

,

f ′′(x)f (x)

=⎛⎝ N∑

j=1

1

x−x∗j−

s∑q=1

lq

x−αpq

⎞⎠

2

−N∑j=1

1

(x−x∗j)2+

s∑q=1

lq

(x−αpq)2,

one arrives at, for n∈N and k=1, 2, . . . ,K, after a straightforward calculation,

Q(x)=2(N −1)

N

N∑j=1

s−1∑q=1

lq

(x∗j−αpq)(x−αpq)

+2(N −1)

N

N∑j=1

(ls−1)

(x∗j−αps)(x−αps

)+

s∑q=1

lq

(x−αpq)2


+ (N −2)

N

⎛⎝ s∑

q=1

lq

x−αpq

⎞⎠

2

− 2(N −1)

N

1

x−αps

s∑q=1

lq

x−αpq

+2(N −1)

NnV ′(x)

s∑q=1

lq

x−αpq

− 2n

N(N −1)

N∑j=1

V ′(x)−V ′(x∗j)x−x∗j

. (46)

Since, from the (sketch of the) proof of Lemma 3.5, for n∈N andk=1, 2, . . . ,K, {x∗j}Nj=1 ⊂ R \ {α1, α2, . . . , αK}, choose 2(N1+1) real points

(N1∈Z+0 ) {B j−1, A j}N1+1

j=1 enumerated so that −∞<B0<A1< · · ·<B j−1<

A j< · · ·<BN1<AN1+1<+∞, with R \ {α1, α2, . . . , αK} ⊃ ∪N1+1j=1 [B j−1, A j] ⊇

{x∗1, x∗2, . . . , x∗N } and [B j−1, A j] ∩ {α1, α2, . . . , αK}=∅, j∈{1, 2, . . . , N1+1}.(∪N1+1

j=1 [B j−1, A j], which is the disjoint union of N1+1 compact real intervals,is the ‘pre-confinement domain’ for the generalised weighted Fekete points{x∗j}Nj=1.) Let Z (x) :=∏N1+1

j=1 (z−B j−1)(z−A j). A calculation shows that Z (x)

can be presented as Z (x)=∑0p′=2(N1+1) c2(N1+1)−p′xp′

, where

cq′ =∑

ki,lm=0,1i,m∈{1,2,...,N1+1}∑N1+1

i=1 ki+∑N1+1

m=1 lm=q′

(−1)q′(

N1+1∏i=1

(1

ki

)(Bi−1)

ki

)(N1+1∏m=1

(1

lm

)(Am)

lm

),

q′ = 0, 1, . . . , 2(N1+1),

with c0=1 and c2(N1+1)=∏N1+1r=1 Br−1 Ar. For n∈N and k=1, 2, . . . ,K, one

proceeds to manipulate the term 2 nN (N −1)

∑Nj=1

V ′(x)−V ′(x∗j )x−x∗j

, which appears

in the expression for Q(x) (cf. (46)):

2n

N(N −1)

N∑j=1

V ′(x)−V ′(x∗j)x−x∗j

= 2n

N

(N −1)

Z (x)

N∑j=1

(V ′(x)Z (x)−V ′(x∗j)Z (x)

x−x∗j

)

=2n

N

(N −1)

Z (x)

N∑j=1

(V ′(x)Z (x)−V ′(x∗j)Z(x

∗j)

x−x∗j

)

− 2n

N

(N −1)

Z (x)

N∑j=1

(V ′(x∗j)(Z (x)−Z (x∗j))

x−x∗j

);

recalling the expression above for Z (x), and using the identity an − b n =(a−b)(an−1+an−2b+· · ·+ab n−2+b n−1), one shows that

Z (x)−Z (x∗j)x−x∗j

=3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝

p′−1∑r=0

xp′−r−1(x∗j)r

⎞⎠

+ c2(N1+1)−2(x+x∗j)+c2(N1+1)−1;


thus, via (44),

2n

N(N −1)

N∑j=1

V′(x)−V′(x∗j )x−x∗j

= 2n

N

(N −1)

Z (x)

N∑j=1

(V′(x)Z (x)−V′(x∗j )Z (x∗j )

x−x∗j

)

− 2n

N

1

Z (x)

N∑j=1

⎛⎜⎜⎜⎝

N

n

N∑k′=1k′ �= j

1

x∗j −x∗k′−(N −1)

s−1∑q=1

lq/n

x∗j −αpq

− (N −1)(ls−1)/nx∗j −αps

)(Z (x)−Z (x∗j )

x−x∗j

)

= 2n

N

(N −1)

Z (x)

N∑j=1

(V′(x)Z (x)−V′(x∗j )Z (x∗j )

x−x∗j

)

− 2

Z (x)

N∑j=1

N∑k′=1k′ �= j

1

x∗j −x∗k′

(Z (x)−Z (x∗j )

x−x∗j

)

+ 2

Z (x)(N −1)

N

N∑j=1

s−1∑q=1

lqx∗j −αpq

(Z (x)−Z (x∗j )

x−x∗j

)

+ 2

Z (x)(N −1)

N

N∑j=1

(ls−1)

x∗j −αps

(Z (x)−Z (x∗j )

x−x∗j

)

= 2n

N

(N −1)

Z (x)

N∑j=1

(V′(x)Z (x)−V′(x∗j )Z (x∗j )

x−x∗j

)

− 2

Z (x)

N∑j=1

N∑k′=1k′ �= j

1

x∗j −x∗k′

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

xp′−r−1(x∗j )r⎞⎠+ c2(N1+1)−2(x+x∗j )

+ c2(N1+1)−1

⎞⎠+ 2

Z (x)(N −1)

N

N∑j=1

s−1∑q=1

lqx∗j −αpq

×⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝

p′−1∑r=0

xp′−r−1(x∗j )r⎞⎠


+ c2(N1+1)−2(x+x∗j )+c2(N1+1)−1

⎞⎠+ 2

Z (x)(N −1)

N

×N∑j=1

(ls−1)

x∗j −αps

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

xp′−r−1(x∗j )r⎞⎠+ c2(N1+1)−2(x+x∗j )

+ c2(N1+1)−1

⎞⎠ .

For n∈N and k=1, 2, . . . ,K, let

A :=N∑j=1

N∑k′=1k′ �= j

1

x∗j−x∗k′

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝

p′−1∑r=0


⎞⎠

+ c2(N1+1)−2(x+x∗j)+c2(N1+1)−1

⎞⎠ .

For n∈N and k=1, 2, . . . ,K, one manipulates the expression for A thus:

A =N∑j=1

N∑k′=1k′ �= j

1

x∗j−x∗k′

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝xp′−1+xp′−2x∗j

+p′−1∑r=2


⎞⎠ +c2(N1+1)−2(x+x∗j)+ c2(N1+1)−1

⎞⎠

=3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎜⎝

N∑j=1

N∑k′=1k′ �= j

xp′−1

x∗j−x∗k′+

N∑j=1

N∑k′=1k′ �= j

xp′−2x∗jx∗j−x∗k′

+p′−1∑r=2

N∑j=1

N∑k′=1k′ �= j


x∗j−x∗k′

⎞⎟⎠+ c2(N1+1)−2

N∑j=1

N∑k′=1k′ �= j

x+x∗jx∗j−x∗k′

+ c2(N1+1)−1

N∑j=1

N∑k′=1k′ �= j

1

x∗j−x∗k′;


recall from the (sketch of the) proof of Lemma 3.5 that, for n∈N and k=1, 2, . . . ,K, given an arbitrary function h, say,

∑j<k′

h(x∗j) :=N −1∑

j=1

N∑k′= j+1

h(x∗j) = (N −1)h(x∗1)+N −1∑m=2

h(x∗m)(N −m),

∑j<k′

h(x∗k′) :=N −1∑

j=1

N∑k′= j+1

h(x∗k′) = (N −1)h(x∗N )+N −1∑m=2

h(x∗m)(m−1),

∑j<k′

h(x∗j)+∑j<k′

h(x∗k′) = (N −1)N∑

m=1

h(x∗m),

whence one derives, via the identity an−b n=(a−b)(an−1+an−2b+· · ·+ab n−2+b n−1), the following relations:

N∑j=1

N∑k′=1k′ �= j

1

x∗j−x∗k′= 0,

N∑j=1

N∑k′=1k′ �= j

x+x∗jx∗j−x∗k′

=N∑j=1

N∑k′=1k′ �= j

x∗jx∗j−x∗k′

=N −1∑

j=1

N∑k′= j+1

1= N (N −1)

2,

N∑j=1

N∑k′=1k′ �= j

(x∗j)r

x∗j−x∗k′=

N −1∑j=1

N∑k′= j+1

(x∗j)r−(x∗k′)

r

x∗j−x∗k′=

N −1∑j=1

N∑k′= j+1

r−1∑m=0

(x∗j)r−m−1(x∗k′)

m

=N −1∑

j=1

N∑k′= j+1

(x∗j)r−1+

N −1∑j=1

N∑k′= j+1

(x∗k′)r−1

+N −1∑

j=1

N∑k′= j+1

r−1∑m=2

(x∗j)r−m(x∗k′)

m−1

= (N −1)N∑j=1

(x∗j)r−1+

N −1∑j=1

N∑k′= j+1

r−1∑m=2


m−1, r�2;


hence, via the above relations, one shows that, for n∈N and k=1, 2, . . . ,K,

A =3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝N (N −1)

2xp′−2+

p′−1∑r=2

⎛⎝(N −1)

N∑j=1

(x∗j)r−1

+N −1∑

j=1

N∑k′= j+1

r−1∑m=2


m−1

⎞⎠xp′−r−1

⎞⎠+ N (N −1)

2c2(N1+1)−2,

whence, for n∈N and k=1, 2, . . . ,K, one arrives at

2n

N(N −1)

N∑j=1

V′(x)−V′(x∗j )x−x∗j

= 2n

N

(N −1)

Z (x)

N∑j=1

(V′(x)Z (x)−V′(x∗j )Z (x∗j )

x−x∗j

)

− 2

Z (x)

⎧⎨⎩

3∑p′=2(N1+1)

c2(N1+1)−p′

⎛⎝N (N −1)

2xp′−2

+p′−1∑r=2

⎛⎝(N −1)

N∑j=1

(x∗j )r−1 +N −1∑

j=1

N∑k′= j+1

·r−1∑m=2

(x∗j )r−m(x∗k′ )m−1

⎞⎠ xp′−r−1

⎞⎠+ N (N −1)

2

× c2(N1+1)−2

⎫⎬⎭+

2

Z (x)(N−1)

N

⎧⎨⎩

3∑p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

N∑j=1

s−1∑q=1

lq(x∗j )r

x∗j −αpqxp′−r−1

⎞⎠+ c2(N1+1)−2

×N∑j=1

s−1∑q=1

lq(x+x∗j )x∗j −αpq

+c2(N1+1)−1

N∑j=1

s−1∑q=1

lqx∗j −αpq

⎫⎬⎭

+ 2(ls−1)

Z (x)(N −1)

N

⎧⎨⎩

3∑p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

N∑j=1

(x∗j )r

x∗j −αps

xp′−r−1

⎞⎠+ c2(N1+1)−2

×N∑j=1

x+x∗jx∗j −αps

+c2(N1+1)−1

N∑j=1

1

x∗j −αps

⎫⎬⎭ . (47)


Substituting (47) into (46), one arrives at, for n∈N and k=1, 2, . . . ,K:

Q(x)= 1

Z (x)

⎧⎨⎩⎛⎝ 0∑

p′=2(N1+1)

c2(N1+1)−p′xp′

⎞⎠⎛⎝2(N −1)

N

N∑j=1

s−1∑q=1

lq

(x∗j−αpq)(x−αpq)

+ 2(N −1)

N

N∑j=1

(ls−1)


)+

s∑q=1

lq

(x−αpq)2+ (N −2)

N

×⎛⎝ s∑

q=1

lq

x−αpq

⎞⎠

2

− 2(N −1)

N

1

x−αps

s∑q=1

lq

x−αpq

+ 2(N −1)

NnV ′(x)

×s∑

q=1

lq

x−αpq

⎞⎠− 2n

(N −1)

N

N∑j=1

(V ′(x)Z (x)−V ′(x∗j)Z (x

∗j)

x−x∗j

)

+ 2

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝N (N −1)

2xp′−2 +

p′−1∑r=2

⎛⎝(N −1)

N∑j=1

(x∗j)r−1

+N −1∑

j=1

N∑k′= j+1

r−1∑m=2


m−1

⎞⎠xp′−r−1

⎞⎠+ N (N −1)

2c2(N1+1)−2

⎞⎠

− 2(N −1)

N

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝

p′−1∑r=0

N∑j=1

s−1∑q=1

lq(x∗j)r

x∗j−αpq

xp′−r−1

⎞⎠

+ c2(N1+1)−2

N∑j=1

s−1∑q=1

lq(x+x∗j)x∗j−αpq

+c2(N1+1)−1

N∑j=1

s−1∑q=1

lq

x∗j−αpq

⎞⎠

− 2(ls−1)(N −1)

N

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝

p′−1∑r=0

N∑j=1

(x∗j)r

x∗j−αps

xp′−r−1

⎞⎠

+ c2(N1+1)−2

N∑j=1

x + x∗jx∗j−αps

+c2(N1+1)−1

N∑j=1

1

x∗j − αps

⎞⎠⎫⎬⎭ . (48)

For n∈N and k=1, 2, . . . ,K, recall the second-order, non-constant coefficientODE for f (x) given in (45), with Q(x) given in (48). As per the proof ofLemma 2.15 in [62], set, for n∈N and k=1, 2, . . . ,K,

f (x)=F(x) exp

(−1

2

∫ x

P(ξ)dξ), (49)


where

P(x) :=2n

N

⎛⎝−(N −1)V ′(x)+

s∑q=1

lq/nx−αpq

+ (N −1)

n1

x−αps

⎞⎠ .

Integrating, and substituting (49) into (45), one arrives at, for n∈N andk=1, 2, . . . ,K, the following second-order, non-constant coefficient ODEfor F(x):

F ′′(x)=(

Q(x)+ 1

2P ′(x)+ 1

4(P(x))2

)F(x), (50)

where, after simplification,

Q(x)+ 1

2P ′(x)+ 1

4(P(x))2 = − (N −1)

NnV ′′(x)− 1

N

(N −1)

N

1

(x−αps)2

+ (N −1)

N

s∑q=1

lq

(x−αpq)2+(

N −1

N

)2

(nV ′(x))2

+ 2nV ′(x)(

N −1

N

)2 s∑q=1

lq

x−αpq

−2

(N −1

N

)2

× 1

x−αps

s∑q=1

lq

x−αpq

−2

(N −1

N

)2 nV ′(x)x−αps

+(

N −1

N

)2⎛⎝ s∑

q=1

lq

x−αpq

⎞⎠

2

+ 2(N −1)

N

×N∑j=1

s−1∑q=1

lq

(x∗j−αpq)(x−αpq)+ 2(N −1)

N

×N∑j=1

(ls−1)


)+ 1

Z (x)

{−2n(N −1)

N

×N∑j=1

(V ′(x)Z (x)−V ′(x∗j)Z (x

∗j)

x−x∗j

)

+ 2

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

⎛⎝N (N −1)

2xp′−2


+p′−1∑r=2

⎛⎝(N −1)

N∑j=1

(x∗j)r−1+

N −1∑j=1

N∑k′= j+1

r−1∑m=2

(x∗j)r−m

× (x∗k′)m−1

⎞⎠ xp′−r−1

⎞⎠+N (N −1)

2c2(N1+1)−2

⎞⎠

− 2(N −1)

N

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

N∑j=1

s−1∑q=1

lq(x∗j)r

x∗j−αpq

xp′−r−1

⎞⎠+ c2(N1+1)−2

×N∑j=1

s−1∑q=1

lq(x+x∗j)x∗j−αpq

+c2(N1+1)−1

N∑j=1

s−1∑q=1

lq

x∗j − αpq

⎞⎠

− 2(ls−1)(N −1)

N

⎛⎝ 3∑

p′=2(N1+1)

c2(N1+1)−p′

×⎛⎝

p′−1∑r=0

N∑j=1

(x∗j)r

x∗j−αps

xp′−r−1

⎞⎠+ c2(N1+1)−2

×N∑j=1

x+x∗jx∗j−αps

+ c2(N1+1)−1

N∑j=1

1

x∗j−αps

⎞⎠⎫⎬⎭ . (51)

Since, for n∈N and k=1, 2, . . . ,K, {x∗1, x∗2, . . . , x∗N } ⊆ ∪N1+1j=1 [B j−1, A j] (⊂

R \ {α1, α2, . . . , αK}) and x∗i <x∗j for i< j, with i, j∈{1, 2, . . . ,N }, it followsfrom the (sketch of the) proof of Lemma 3.5 that, for any two con-secutive generalised weighted Fekete points x∗j and x∗j+1, there are, forr∈{1, 2, . . . , N1+1}, the following cases to consider: (1) Br−1 �x∗j<x∗j+1 �12 (Br−1+Ar); (2) 1

2 (Br−1+Ar)�x∗j<x∗j+1 � Ar; and (3) Br−1 �x∗j<12 (Br−1+

Ar)<x∗j+1 � Ar. This is the (up to a linear scaling) ‘one-interval-case’ resultgiven in Lemma 2.15 of [62], wherein a lower bound for the distance betweentwo consecutive Fekete points is estimated; in order to use the result ofLemma 2.15 of [62], one needs to: (i) map the compact real intervals [Br−1, Ar],r=1, 2, . . . , N1+1, onto the compact real interval [−1, 1]; and (ii) for n∈N andk=1, 2, . . . ,K, estimate an upper bound for Q(x)+ 1

2P ′(x)+ 14 (P(x))

2, whichis given by (51). For the former problem (i), one makes the linear change ofvariables λr : C→C, x →λr(x) :=(2x−(Ar+Br−1))/(Ar−Br−1), r = 1, 2, . . . ,N1 + 1, which maps, one-to-one, the compact real intervals [Br−1, Ar] ontothe compact real interval [−1, 1]; and, for the latter problem (ii), noting that,


for n∈N and k=1, 2, . . . ,K, inf{|αpq−x∗j |; q=1, 2, . . . , s, j=1, 2, . . . ,N }>0 and inf{|αpq−x|; q=1, 2, . . . , s, x∈[Br−1, Ar], r=1, 2, . . . , N1+1}>0, writ-ing Z (x)=(x−Br−1)(x−Ar)

∏N1+1j=1j�=r

(x−B j−1)(x−A j), r∈{1, 2, . . . , N1+1}, and

using the real analyticity of V: R \ {α1, α2, . . . , αK}→R, one shows from (51)that, for n∈N and k=1, 2, . . . ,K,∣∣∣∣Q(x)+ 1

2P ′(x)+ 1

4(P(x))2

∣∣∣∣� C�r (n,k)N 2

(x−Br−1)(Ar−x), r∈{1, 2, . . . , N1+1},

where C�r (n,k)>0. One now uses this inequality in conjunction with the resultof Lemma 2.15 of [62] to show that, for n∈N and k=1, 2, . . . ,K, two con-secutive generalised weighted Fekete points x∗j and x∗j+1 satisfy the following‘nearest-neighbour-distance’ inequality:

x∗j+1 − x∗j �min{(3C�r (n,k))−1/2, 1/4}

((n−1)K+k)

((Ar−x∗j

)(Ar−x∗j+1

))1/2,

r∈{1, 2, . . . , N1+1}, j =1, 2, . . . ,N −1 (52)

(note that ((Ar−x∗j)(Ar−x∗j+1))1/2>0)10. One now uses the estimate (52) and

the fact (cf. Lemma 3.5) that the associated normalised counting measureconverges (as n→∞) weakly (in the weak-∗ topology of measures) to theassociated equilibrium measure μV in order to proceed, mutatis mutandis, as inthe proof of Lemma 2.26 of [62] to conclude that, for n∈N and k=1, 2, . . . ,K,the family of equilibrium measures μV is absolutely continuous with respect toLebesgue measure. ��

Lemma 3.7 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8). Suppose, furthermore, that V is regular. For n∈N and k=1, 2, . . . ,K, set, for the associated equilibrium measure μV ∈M1(R), J :=supp(μV), with J a proper compact subset of R \ {α1, α2, . . . , αK}. Then,for n∈N and k=1, 2, . . . ,K : (1)(1)(1) J=∪N+1

j=1 [b j−1, a j], with N∈Z+0 and fi-

nite, [b j−1, a j] ∩ {α1, α2, . . . , αK}=∅, j=1, 2, . . . , N+1, [bi−1, ai] ∩ [b j−1, a j]=∅, i �= j∈{1, 2, . . . , N+1}, and −∞<b0<a1<b1<a2< · · ·<bN<aN+1<+∞,and {b j−1, a j}N+1

j=1 satisfy the locally solvable system of 2(N+1) real momentequations

∫J

ξ j

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ=0,

j=0, 1, . . . , N,

10If, for n∈N and k=1, 2, . . . ,K, two consecutive generalised weighted Fekete points x∗jand x∗j+1 lie, respectively, in the disjoint compact real intervals [Br1−1, Ar1 ] and [Br2−1, Ar2 ],r1<r2 (∈{1, 2, . . . , N1+1}), then x∗j+1−x∗j �|Br2−1−Ar1 |>0.


∫J

ξN+1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ

= −2

((n−1)K+k

n

),

∫ b j

a j

⎛⎝(R(ς))1/2

∫J(R(ξ))−1/2

+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ V ′(ξ)iπ

⎞⎠ dξξ−ς

⎞⎠dς = 2

(κnk−1

n

)ln

∣∣∣∣b j−αk

a j−αk

∣∣∣∣+2s−1∑q=1

κnkkq

nln

∣∣∣∣b j−αpq

a j−αpq

∣∣∣∣

+(V(b j)−V(a j)), j=1, 2, . . . , N,

where

(R(z))1/2 :=⎛⎝N+1∏

j=1

(z−b j−1)(z−a j)

⎞⎠

1/2

,

with (R(z))1/2± := limε↓0(R(z±iε))1/2, and the branch of the square root is cho-sen so that z−(N+1)(R(z))1/2∼C±�z→∞±1; and (2)(2)(2) the associated equilibriummeasure, which is absolutely continuous with respect to Lebesgue measure, isgiven by

dμV(x) :=ψV(x)dx= 1

2π i(R(x))1/2+ hV(x)1J(x)dx,

where

hV(z) =1

2

((n−1)K+k

n

)−1 ∮CV

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ iV ′(ξ)π

⎞⎠ (R(ξ))−1/2

ξ−zdξ

(real analytic for z∈R \ {α1, α2, . . . , αK}), with CV (⊂ C \ {α1, α2, . . . , αK}) thesimple boundary of any open (s+1)-connected punctured region of the typeCV =∂ U

�R ∪ (∪s

q=1∂ U�δq(αpq)), where U

�R :={z∈C; |z|<R}, with ∂ U

�R ={z∈

C; |z|=R} oriented clockwise, and, for q=1, 2, . . . , s, U�δq(αpq) :={z∈C; |z−

αpq |<δq}, with ∂ U�δq(αpq)={z∈C; |z−αpq |=δq} oriented counter-clockwise,

and where the numbers 0<δq<R<+∞, q=1, 2, . . . , s, are chosen so that, for(any) non-real z in the domain of analyticity of V (that is, C \ {α1, α2, . . . , αK}),• ∂ U

�δi(αpi) ∩ ∂ U

�δ j(αpj)=∅, i �= j∈{1, 2, . . . , s},

• ∂ U�δ j(αpj) ∩ ∂ U

�R =∅, j∈{1, 2, . . . , s},


• (U�δ j(αpj) ∪ ∂ U

�δ j(αpj)) ∩ J=∅, j∈{1, 2, . . . , s},

• ∂ U�R ∩ J=∅,

• int(CV) :=(

ext(∪sq=1(U

�δq(αpq) ∪ ∂ U

�δq(αpq))) ∩ U

�R

)⊃ {z} ∪ J,

1J(x) is the characteristic function of the set J, and ψV(x)�0 (resp., ψV(x)>0)for x∈ J (resp., x∈ int(J)).

Proof One begins by showing that, for n∈N and k=1, 2, . . . ,K, the supportof each member of the family of equilibrium measures, that is, supp(μV) =:J, consists of the union of a finite number of disjoint and bounded (real)intervals. Recalling from Lemma 3.1 that, for n∈N and k=1, 2, . . . ,K, J⊂R \ ({|x|�TM} ∪ (∪s

q=1O 1TM(αpq))), for some TM (= TM(n,k)) >1, and that

V: R \ {α1, α2, . . . , αK}→R is real analytic, thus real analytic on J, and withan analytic extension to, say, the following open neighbourhood of J, U :={z∈C; infq∈J |z−q|<r}, with r∈(0, 1) chosen (small enough) so that U ∩{α1, α2, . . . , αK}=∅, it follows from the result of Lemma 3.5 and a calculationanalogous to that subsumed in the proof of Lemma 2.26 of [62] that, for n∈N

and k=1, 2, . . . ,K, the densities of the elements of the family of equilibriummeasures have the representation dμV(x) :=ψV(x)dx, x∈supp(μV), with ψVdetermined below.

For n∈N and k=1, 2, . . . ,K, set

S : N × {1, 2, . . . ,K} × C \ (J ∪ {αp1 , . . . , αps−1 , αk})→C,

(n,k, z) →S(n,k; z) :=S(z) = −4iπ

((n−1)K+k

n

)⎛⎝ (κnk−1)

n(z−αk)

+s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ψV(z)+ 4i

((n−1)K+k

n

)2

ψV(z)(H ψV)(z), (53)

where H : L 2M2(C)(∗∗∗)→L 2

M2(C)(∗∗∗), f →(H f )(z) :=∫

R

f (ξ)z−ξ

dξπ

denotes the

Hilbert transform, with∫

the principle value integral. For n∈N and k=1, 2, . . . ,K, set

H : N × {1, 2, . . . ,K} × C \ (J ∪ {αp1 , . . . , αps−1 , αk})→C,

(n,k, z) →H(n,k; z) :=H(z) = (F(z))2−∫

J

S(ξ)

ξ−zdξ2π i, (54)


where, from the proof of Lemma 3.4,

F : N × {1, 2, . . . ,K} × C \ (J ∪ {αp1 , . . . , αps−1 , αk})→C,

(n,k, z) →F(n,k; z) := F(z) = − 1

iπ

⎛⎝(κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

+((n−1)K+k

n

)∫J

dμV(ξ)

ξ−z

⎞⎠ , (55)

with∫

JdμV(ξ)

ξ−z the Stieltjes transform of the equilibrium measure. Via the distri-butional identities (x−(x0±i0))−1=(x−x0)

−1±iπδ(x−x0), where δ(·) is theDirac delta function, and

∫ λ2

λ1f (ξ)δ(ξ−λ)dξ= f (λ) if λ∈(λ1, λ2) or 0 if λ∈

R \ (λ1, λ2), it follows that

H±(z)=

⎧⎪⎨⎪⎩(F±(z))2−

∫J

S(ξ)ξ − z

dξ2π i∓

12S(z), z∈ J,

(F(z))2−∫JS(ξ)ξ − z

dξ2π i , z /∈ J,

where !±(z) := limε↓0 !(z±i0), !∈{H,F}. Using the representation dμV(x)=ψV(x)dx, x∈ J, recall the definition of g given in Lemma 3.4:

g : N × {1, 2, . . . ,K} × C \(−∞,max

{max

q=1,2,...,s{αpq},max{J}

})→C,

(n,k, z) →g(n,k; z) :=g(z) =∫

Jln

⎛⎝(z−ξ) 1

n

((z−ξ)

(z−αk)(ξ−αk)

)κnk−1n

×s−1∏q=1

((z−ξ)


)κnkkqn

⎞⎠ψV(ξ)dξ ;

taking note of the above distributional identities and the fact that∫J ψV(ξ)dξ = 1, one shows that (assuming differentiation commutes with

taking boundary values)

(g±(z))′ =⎧⎪⎪⎪⎨⎪⎪⎪⎩

− (κnk−1)n(z−αk)

−s−1∑q=1

κnkkq

n(z−αpq )−((n−1)K+k

n

) ∫J

ψV(ξ)

ξ−z dξ∓iπ((n−1)K+k

n

)ψV(z), z∈ J,

− (κnk−1)n(z−αk)

−s−1∑q=1

κnkkq

n(z−αpq )−((n−1)K+k

n

) ∫J

ψV(ξ)

ξ−z dξ, z /∈ J,


where (g±(z))′ := limε↓0(g)′(z±iε), whence one concludes that

(g++g−)′(z) = −2(κnk−1)

n(z−αk)−2

s−1∑q=1

κnkkq

n(z−αpq )−2

((n−1)K+k

n

)∫J

ψV(ξ)

ξ−zdξ

= −2(κnk−1)

n(z−αk)−2

s−1∑q=1

κnkkq

n(z−αpq )+ 2π

((n−1)K+k

n

)(H ψV)(z), z∈ J,

(g+−g−)′(z) ={−2π i

((n − 1)K + k

n

)ψV(z), z∈ J,

0, z /∈ J.

Demanding that, in a piecewise-continuous sense (see Lemma 3.8 below),(g++g−)′(z)−V ′(z)=0, z∈ J, one shows from the above formula that, for z∈ J,⎛

⎝g′(z)+ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq )

⎞⎠

++⎛⎝g′(z)+ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq )

⎞⎠

−

= 2π

((n−1)K+k

n

)(H ψV)(z) =

2(κnk−1)

n(z−αk)+2

s−1∑q=1

κnkkq

n(z−αpq )+V′(z)⇒

(H ψV)(z)=1

2π

((n−1)K+k

n

)−1⎛⎝2(κnk−1)

n(z−αk)+ 2

s−1∑q=1

κnkkq

n(z−αpq )+V′(z)

⎞⎠, z∈ J. (56)

From (54) and the above distributional identities, one shows that

F±(z)=⎧⎪⎪⎨⎪⎪⎩

− 1iπ

((κnk−1)n(z−αk)

+s−1∑q=1

κnkkqn(z−αpq )

−π((n−1)K+k

n

)(H ψV)(z)

)∓((n−1)K+k

n

)ψV(z), z∈ J,

− 1iπ

((κnk−1)n(z−αk)

+s−1∑q=1

κnkkqn(z−αpq )

+((n−1)K+k

n

) ∫JψV (ξ)

ξ−z dξ

), z /∈ J;

(57)

thus, for z∈R \ (J ∪ {αp1 , . . . , αps−1 , αk}), F+(z)=F−(z)=F(z). Hence, for z /∈J ∪ {αp1 , . . . , αps−1 , αk}, one shows that, upon recalling (53), H+(z)=H−(z). Forz∈ J, one notes that

H+(z)−H−(z)=(F+(z))2−(F−(z))2−S(z),

and

(F±(z))2 = − 1

π2

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠

2

+ 2

π

((n−1)K+k

n

)⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ (H ψV)(z)


∓ 2iπ

((n−1)K+k

n

)⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ψV(z)

± 2i((n−1)K+k

n

)2

(H ψV)(z)ψV(z)

−((n−1)K+k

n

)2

((H ψV)(z))2+((n−1)K+k

n

)2

(ψV(z))2,

whence, for z∈ J,

(F+(z))2−(F−(z))2 =−4iπ

((n−1)K+k

n

)⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ψV(z)

+ 4i((n−1)K+k

n

)2

(H ψV)(z)ψV(z),

which implies that H+(z)−H−(z)=0; thus, for z∈ J, H+(z)=H−(z). The aboveargument shows, therefore, that H(z) is analytic across R \ {αp1 , . . . , αps−1 , αk};in fact, H(z) is entire (resp., meromorphic) for z∈C \ {αp1 , . . . , αps−1 , αk} (resp.,z∈C). Recalling that, for n∈N and k=1, 2, . . . ,K, μV ∈M1(R), in particu-lar,

∫J(ξ−αk)

−m dμV(ξ)=∫

J(ξ−αk)−mψV(ξ)dξ <∞, m∈N, for ξ ∈ J and z /∈

J such that |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|%min{minq=1,2,...,s−1{|αk−αpq |}, infξ∈J{||ξ |−αk|}}), via the expansion 1

(z−αk)−(ξ−αk)=−∑l

j=0(z−αk)

j

(ξ−αk) j+1 +(z−αk)

l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 ,

(F(z))2 =z→αk

− 1

(z−αk)2

(1

π

(κnk−1

n

))2

+ 1

(z−αk)

(2

π2

(κnk−1

n

)

×⎛⎝s−1∑

q=1

κnkkq

n(αpq−αk)−((n−1)K+k

n

)∫J(ξ−αk)

−1ψV(ξ)dξ)⎞⎠

+ O(1),

whence, recalling the definition of H(z) (cf. (54)), in particular, for ξ ∈ J andz /∈ J such that |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|%min{minq=1,2,...,s−1{|αk−αpq |}, infξ∈J{||ξ |−αk|}}), via the expansion 1


j=0(z−αk)

j

(ξ−αk) j+1 +(z−αk)

l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 ,

∫J

S(ξ)

ξ−zdξ2π i

=z→αk

O(1),

it follows that

H(z) =z→αk

− 1

(z−αk)2

(1

π

(κnk−1

n

))2

+ 1

(z−αk)

(2

π2

(κnk−1

n

)


×⎛⎝s−1∑

q=1

κnkkq


n

)∫J(ξ−αk)

−1ψV(ξ)dξ)⎞⎠

+ O(1),

which shows that H(z) has a pole of order 2 at z=αk, with

Res(H(z);αk) = 2

π2

(κnk−1

n

)⎛⎝s−1∑

q=1

κnkkq


n

)

×∫

J

ψV(ξ)

ξ−αkdξ

⎞⎠ , k = 1, 2, . . . ,K.

One learns, then, from the analysis above that, for n∈N and k=1, 2, . . . ,K,∏s−1q=1(z−αpq)

2(z−αk)2H(z) is entire; look, in particular, at the behaviour of∏s−1

q=1(z−αpq)2(z−αk)

2H(z) as z→∞. Towards this end, though, some nota-tion is requisite (cf. Subsection 1.2).

Proceeding as per the detailed discussion of Subsection 1.2, write, for n∈N

and k=1, 2, . . . ,K, the ordered disjoint partition for the repeated real polesequence (with the convention ∪0

m=1{∗∗∗,∗∗∗, . . . ,∗∗∗} :=∅):

1{α1, α2, . . . , αK︸︷︷︸K

} ∪ · · · ∪ n−1{α1, α2, . . . , αK︸︷︷︸K

} ∪ n{α1, α2, . . . , αk︸︷︷︸k

}

:=s−1⋃q=1

{αi(q)kq, αi(q)kq

, . . . , αi(q)kq︸︷︷︸lq=κnkkq


, . . . , αi(s)ks︸︷︷︸ls=κni(s)ks

}

:=s−1⋃q=1

{αpq , αpq , . . . , αpq︸︷︷︸lq=κnkkq

} ∪ {αk, αk, . . . , αk︸︷︷︸ls=κnk

},

where∑s

q=1 lq=∑s−1q=1 lq+ls=∑s−1

q=1 κnkkq+κnk=(n−1)K+k. With this or-

dered disjoint partition, one writes, for n∈N and k=1, 2, . . . ,K, via (53), uponsetting S(z) := iS(z) and noting that deg(

∏s−1q=1(z−αpq)

2(z−αk)2)=2s,

H(z) = − (κnk−1)2

π2n2(z−αk)2− 2(κnk−1)

π2n(z−αk)

s−1∑q=1

κnkkq

n(z−αpq)− 1

π2

⎛⎝s−1∑

q=1

κnkkq

n(z−αpq)

⎞⎠

2

− 2(κnk−1)

π2n(z−αk)

((n−1)K+k

n

)∫J

dμV(ξ)

ξ−z− 2

π2

((n−1)K+k

n

)


×s−1∑q=1

κnkkq

n(z−αpq)

∫J

dμV(ξ)

ξ−z− 1

π2

((n−1)K+k

n

)2 (∫J

dμV(ξ)

ξ−z

)2

−∫

J

S(ξ)

ξ−zdξ2π, z ∈ C \ (J ∪ {αp1 , . . . , αps−1 , αk}). (58)

Recalling that μV ∈ M1(R), in particular,∫

J ξm dμV(ξ) =

∫J ξ

mψV(ξ)dξ <∞,m ∈ Z

+0 , for ξ ∈ J and z /∈ J such that |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1

{|bj−1−a j|},maxq=1,2,...,s{|αpq |}}), via the expansion 1ξ−z =−∑l

j=0ξ j

z j+1 + ξ l+1

zl+1(ξ−z) ,

l∈Z+0 , one shows that, for n∈N and k=1, 2, . . . ,K, H(z)=z→∞

∑2s

m=1 hm(n,k)z−m+O(z−(2s+1)), where, in particular, h1(n,k)= 1

π2 ((n−1)K+k

n )∫

JV′(ξ)ψV

(ξ)dξ ; hence,∏s−1

q=1(z−αpq)2(z−αk)

2H(z)−∑2s−1m=0 ρm(n,k)zm =z→∞ O(z−1),

where, in particular, ρ2s−1(n,k)= 1π2 ((n−1)K+k

n )∫

J V ′(ξ)ψV(ξ)dξ , which impliesthat, for n∈N and k=1, 2, . . . ,K, since


2(z−αk)2H(z) is entire,

via a generalisation of Liouville’s theorem,

H(z)=∑2s−1

m=0 ρm(n,k)zm


2(z−αk)2, z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}). (59)

It remains, therefore, to determine ρm(n,k), m=0, 1, . . . , 2s−1. Recalling(58) for H(z), one shows that, for n∈N and k=1, 2, . . . ,K, upon recall-ing that μV ∈M1(R), in particular,

∫J ξ

m dμV(ξ)=∫

J ξmψV(ξ)dξ <∞, m∈Z

+0 ,

for ξ ∈ J and z /∈ J such that |ξ/z|%1 (e.g., |z|&max{max j=1,2,...,N+1{|bj−1−a j|},maxq=1,2,...,s{|αpq |}}), via the expansion 1

ξ−z =−∑lj=0

ξ j

z j+1 + ξ l+1

zl+1(ξ−z) , l∈Z+0 ,

−s−1∏q=1

(z−αpq)2(z−αk)

2

(∫J

S(ξ)

ξ−zdξ2π

)

=z→∞

2s−1∑m=0

(2s−1∑q=m

ωq−m(n,k)cq+1(n,k)

)zm+O(z−1),

− 1

π2

((n−1)K+k

n

)2 s−1∏q=1


2

(∫J

dμV(ξ)

ξ−z

)2

=z→∞

2(s−1)∑m=0

(2(s−1)∑q=m

μq−m(n,k)c�

q+2(n,k)

)zm+O(z−1),

−s−1∏q=1


2

⎛⎝ 2(κnk−1)

π2n(z−αk)

((n−1)K+k

n

)∫J

dμV(ξ)

ξ−z


+ 2

π2

((n−1)K+k

n

) s−1∑q=1

κnkkq

n(z−αpq)

∫J

dμV(ξ)

ξ−z

⎞⎠

=z→∞

2(s−1)∑m=0

(2(s−1)∑q=m

νq−m(n,k)c�

q+1(n,k)

)zm+O(z−1),

−s−1∏q=1


2

⎛⎝ (κnk−1)2

π2n2(z−αk)2+ 2(κnk−1)

π2n(z−αk)

s−1∑q=1

κnkkq

n(z−αpq)

+ 1

π2

⎛⎝s−1∑

q=1

κnkkq

n(z−αpq)

⎞⎠

2⎞⎟⎠ =

z→∞

2(s−1)∑m=0

c m(n,k)zm+O(z−1),

where

ωr(n,k) := 1

π2

((n−1)K+k

n

)∫Jξ r V ′(ξ)ψV(ξ)dξ, r∈Z

+0 ,

μm(n,k) =m∑

q=0

νq(n,k)νm−q(n,k), m=0, 1, . . . , 2s−2,

νr(n,k) :=∫

Jξ rψV(ξ)dξ, r∈Z

+0 (ν0(n,k)=1),

cl(n,k) :=∑

ir=0,1,2r∈{1,2,...,s}∑sr=1 ir=2s−l

(−1)2s−l(2!)s(

s∏m=1

1

im!(2−im)!

)s−1∏j=1

(αpj)i j(αk)

is,

l = 0, 1, . . . , 2s,

c�l (n,k) := − 1

π2

((n−1)K+k

n

)2 ∑ir=0,1,2r∈{1,2,...,s}∑sr=1 ir=2s−l

(−1)2s−l(2!)ss∏

m=1

1

im!(2−im)!

×s−1∏j=1

(αpj)i j(αk)

is , l = 0, 1, . . . , 2s,

c�l (n,k) :=2

π2

(κnk−1

n

)((n−1)K+k

n

) ∑ir=0,1,2

is=0,1r∈{1,2,...,s−1}∑sr=1 ir=2s−l−1

(−1)2s−l−1(2!)s−1

is!(1−is)!


×s−1∏m=1

1

im!(2−im)!s−1∏j=1

(αpj)i j(αk)

is+ 2

π2

((n−1)K+k

n

)

×s−1∑q=1

κnkkq

n

∑iq=0,1ir=0,1,2

r∈{1,2,...,s}\{q}∑sm=1 im=2s−l−1

(−1)2s−l−1(2!)s−1

iq!(1−iq)!s∏

m=1m �=q

1

im!(2−im)!

×s−1∏j=1

(αpj)i j(αk)

is , l = 0, 1, . . . , 2s−1,

c l (n,k) := − 1

π2

(κnk−1

n

)2 ∑iq=0,1,2

q∈{1,2,...,s−1}∑s−1r=1 ir=2(s−1)−l

(−1)2(s−1)−l(2!)s−1

×s−1∏m=1

1

im!(2−im)!s−1∏j=1

(αpj)i j− 2

π2

(κnk−1

n

) s−1∑q=1

κnkkq

n

×∑

iq,is=0,1ir=0,1,2

r∈{1,2,...,s−1}\{q}∑sm=1 im=2(s−1)−l

(−1)2(s−1)−l(2!)s−2

iq!(1−iq)!is!(1−is)!s−1∏m=1m �=q

1

im!(2−im)!s−1∏j=1

(αpj)i j(αk)

is

− 1

π2

s−1∑q=1

(κnkkq

n

)2 ∑ir,is=0,1,2

r∈{1,2,...,s−1}\{q}∑sm=1m �=q

im=2(s−1)−l

(−1)2(s−1)−l(2!)s−1

×s∏

m=1m �=q

1

im!(2−im)!s−1∏j=1j�=q

(αpj)i j(αk)

is− 2

π2

s−2∑p′=1

s−1∑q=p′+1

κnkkp′

n

κnkkq

n

×∑

ip′ ,iq=0,1ir ,is=0,1,2

r∈{1,2,...,s−1}\{q,p′ }ip′ +iq+is+∑s−1

r=1r �=q,p′

ir=2(s−1)−l

(−1)2(s−1)−l(2!)s−2

ip′ !(1−ip′)!iq!(1−iq)!is!(2−is)!

×s−1∏m=1

m �=q,p′

1

im!(2−im)!(αpp′)ip′ (αpq)

iqs−1∏j=1

j�=q,p′

(αpj)i j(αk)

is, l = 0, 1, . . . , 2s−2.


Hence, via (54) and (59), one arrives at, for n∈N and k=1, 2, . . . ,K,

(F(z))2 =∫

J

S(ξ)

ξ−zdξ2π

+∑2s−1

m=0 ρm(n,k)zm


2(z−αk)2,

z ∈ C\(J∪{αp1 , . . . , αps−1 , αk}),where

ρ2s−1(n,k)=ω0(n,k)= 1

π2

((n−1)K+k

n

)∫J

V ′(ξ)ψV(ξ)dξ,

and

ρm(n,k) = ω2s−m−1(n,k)+ c m(n,k)+2(s−1)∑q=m

(ωq−m(n,k)cq+1(n,k)

+ νq−m(n,k)c�

q+1(n,k)+ μq−m(n,k)c�

q+2(n,k)), m = 0, 1, . . . , 2s − 2,

with ωr(n,k), μm(n,k), νr(n,k), cl(n,k), c�l (n,k), c�l (n,k), and c l (n,k) definedabove. Recalling that S(z) := iS(z), where S(z) is defined in (53), one showsthat, via (56), for n∈N and k=1, 2, . . . ,K, S(z)= 2

π( (n−1)K+k

n )V ′(z)ψV(z),whence

(F(z))2 = 1

π2

((n−1)K+k

n

)∫J

V ′(ξ)ψV(ξ)

ξ−zdξ

+∑2s−1

m=0 ρm(n,k)zm


2(z−αk)2, z ∈ C \ (J ∪ {αp1 , . . . , αps−1 , αk}). (60)

But, via (55),

1

π2

((n−1)K+k

n

)∫J

V ′(ξ)ψV(ξ)

ξ−zdξ

= 1

π2

((n−1)K+k

n

)∫J

(V ′(ξ)−V ′(z))ψV(ξ)

ξ−zdξ

+ V ′(z)π2

(((n−1)K+k

n

)∫J

ψV(ξ)

ξ−zdξ)

︸︷︷︸=−iπF(z)− (κnk−1)

n(z−αk) −s−1∑q=1

κnkkqn(z−αpq )

= 1

π2

((n−1)K+k

n

)∫J

(V ′(ξ)−V ′(z))ψV(ξ)

ξ−zdξ

− iπ

V ′(z)F(z)− V ′(z)π2

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ :


substituting the above into (60), one arrives at, for n∈N and k=1, 2, . . . ,K,upon completing the square and re-arranging terms,

(F(z)+ iV ′(z)

2π

)2

+ qV(z)π2

=0, z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}), (61)

where

qV(z) :=(

V ′(z)2

)2

+V ′(z)

⎛⎝(κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠

−((n−1)K+k

n

)∫J

(V ′(ξ)−V ′(z))ψV(ξ)

ξ−zdξ

−∑2s−1

m=0 ρm(n,k)zm


2(z−αk)2

=(

V ′(z)2

)2

+V ′(z)

⎛⎝(κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠

−((n−1)K+k

n

)∫J

∫ 1

0V ′′(λξ+(1−λ)z)dλdμV(ξ)

−∑2s−1

m=0 ρm(n,k)zm


2(z−αk)2,

where ρm(n,k) :=π2ρm(n,k), m=0, 1, . . . , 2s−1. Perusing the expression forρm(n,k), m=0, 1, . . . , 2s−1, derived above (see, in particular, the formu-lae for ωr(n,k) and νr(n,k), r∈Z

+0 ), one notes that integrals of the type∫

J ξr V ′(ξ)dμV(ξ) and

∫J ξ

r dμV(ξ), r∈Z+0 , are encountered; in fact, one may

use such integrals in order to evaluate the Stieltjes transform of the equilibriummeasure on the real pole set {α1, α2, . . . , αK}, that is,

∫J(ξ−αpq)

−1 dμV(ξ),q=1, 2, . . . , s, as follows. Recalling that, for n∈N and k=1, 2, . . . ,K, μV ∈M1(R), in particular,

∫J(ξ−αk)

−mψV(ξ)dξ <∞ and∫

J ξmψV(ξ)dξ <∞, m∈

N, with∫

J ψV(ξ)dξ=1, for ξ ∈ J and z /∈ J such that |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|%min{min j=1,2,...,N+1{||bj−1−a j|−αk|},minq=1,2,...,s−1{|αpq−αk|}}), via the expansion 1


j=0(z−αk)

j

(ξ−αk) j+1 + (z−αk)l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 ,

one shows, via (58) and (59), that

H(z) =z→αk

− (κnk−1)2

(z−αk)2π2n2+ 2(κnk−1)

(z−αk)π2n

s−1∑q=1

κnkkq

n(αpq−αk)

− 2(κnk−1)

(z−αk)π2n

((n−1)K+k

n

)∫J

dμV(ξ)

ξ−αk+O(1),


H(z) =z→αk

1

(z−αk)2

P(αk)

F(αk)+ 1

(z−αk)

(P′(αk)

F(αk)−(

P(αk)

F(αk)

)F ′(αk)

F(αk)

)

+ O(1),

where P(αk)=∑2s−1m=0 ρm(n,k)(αk)

m, P′(αk)=∑2s−1m=1 mρm(n,k)(αk)

m−1, F(αk)=∏s−1q=1(αk−αpq)

2, and F ′(αk)(F(αk))−1 =2

∑s−1q=1(αk−αpq)

−1, and, for q=1, 2,. . . , s−1,

H(z) =z→αpq

−(κnkkq

)2

π2n2(z−αpq )2+ 2(κnk−1)

π2n(αk−αpq )

κnkkq

n(z−αpq )− 2

π2

κnkkq

n(z−αpq )

×s−1∑r=1r �=q

κnkkr

n(αpq −αpr )− 2

π2

((n−1)K+k

n

)κnkkq

n(z−αpq )

∫J

dμV(ξ)

ξ−αpq

+O(1),

H(z) =z→αpq

1

(z−αpq )2

P(αpq )

G(αpq )+ 1

(z−αpq )

(P ′(αpq )

G(αpq )−(

P(αpq )

G(αpq )

)G′(αpq )

G(αpq )

)

+O(1),

where P(αpq)=∑2s−1

m=0 ρm(n,k)(αpq)m, P ′(αpq)=

∑2s−1m=1 mρm(n,k)(αpq)

m−1,G(αpq)=(αpq−αk)

2 ∏s−1r=1r �=q

(αpq−αpr )2, and G′(αpq)(G(αpq))

−1=2(αpq−αk)−1+

2∑s−1

r=1r �=q

(αpq−αpr )−1; whence, equating coefficients of like powers of (z−αk)

−1

and (z−αpq)−1, q=1, 2, . . . , s−1, respectively, in the above pair of asymptotic

expansions for H(z), one obtains, for n∈N and k=1, 2, . . . ,K, the followingset of 2(s+1) real linear equations:

2s−1∑m=0

ρm(n, k)(αk)m=− 1

π2

(κnk−1

n

)2 s−1∏q=1

(αk−αpq )2,

2s−1∑m=0

mρm(n, k)(αk)m−1= "(n, k),

2s−1∑m=0

ρm(n, k)(αpq )m=−

(κnkkq)2

π2n2(αpq −αk)

2s−1∏r=1r �=q

(αpq−αpr )2,q = 1, 2, . . . , s−1,

2s−1∑m=0

mρm(n, k)(αpq )m−1=Ξ(n, k;q),q = 1, 2, . . . , s−1,

where

"(n, k) := − 2

π2

(κnk−1

n

) s−1∏q=1

(αk−αpq )2

(s−1∑r=1

(κnk+κnkkr−1)

n(αk−αpr )

+((n−1)K+k

n

)∫J

dμV(ξ)

ξ−αk

),


Ξ(n, k;q) := − 2

π2

κnkkq

n(αpq −αk)

2s−1∏r=1r �=q

(αpq −αpr )2

⎛⎜⎜⎝

s−1∑r=1r �=q

(κnkkq+κnkkr

)

n(αpq −αpr )

+(κnk+κnkkq

−1)

n(αpq −αk)+((n−1)K+k

n

)∫J

dμV(ξ)

ξ−αpq

⎞⎟⎟⎠ ,q = 1, 2, . . . , s−1,

which can be presented in the form⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 αk (αk)2 . . . (αk)

2s−1

1 αp1 (αp1)2 . . . (αp1)

2s−1

1 αp2 (αp2)2 . . . (αp2)

2s−1

......

.... . .

...

1 αps−1 (αps−1)2 . . . (αps−1)

2s−1

0 1 2αk . . . (2s−1)(αk)2s−2

0 1 2αp1 . . . (2s−1)(αp1)2s−2

0 1 2αp2 . . . (2s−1)(αp2)2s−2

......

.... . .

...

0 1 2αps−1 . . . (2s−1)(αps−1)2s−2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

︸︷︷︸:= D

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ρ0(n,k)ρ1(n,k)ρ2(n,k)...

ρs−1(n,k)ρs(n,k)ρs+1(n,k)ρs+2(n,k)...

ρ2s−1(n,k)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

− (κnk − 1)2

π2n2

∏s−1q=1(αk−αpq)

2

− (κnkk1)2

π2n2(αp1−αk)

2 ∏s−1q=1q �=1

(αp1−αpq)2

− (κnkk2)2

π2n2(αp2−αk)

2 ∏s−1q=1q �=2

(αp2−αpq)2

...

− (κnkks−1)2

π2n2(αps−1−αk)

2 ∏s−1q=1

q �=s−1

(αps−1−αpq)2

Γ (n,k)Ξ(n,k; 1)Ξ(n,k; 2)...

Ξ (n,k; s−1)

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (62)

For n∈N and k=1, 2, . . . ,K, via the formula for ρm(n,k), m=0, 1, . . . , 2s−1,derived above, the first s equations of system (62) give rise to remarkableidentities, whereas the latter s equations of system (62) allow one to ex-press the Stieltjes transforms,

∫J(ξ−αpq)

−1 dμV(ξ), q=1, 2, . . . , s (with αps:=

αk), in terms of the moment integrals∫

J ξr1 V ′(ξ)dμV(ξ), r1=0, 1, . . . , 2s−1,

and∫

J ξr2 dμV(ξ), r2=0, 1, . . . , 2s−2, which proves, incidentally, and as a


by-product of the above calculations, that the Stieltjes transforms,∫

J(ξ−αpq)

−1 dμV(ξ), q=1, 2, . . . , s, are real valued and bounded. Alternatively,given the Stieltjes transforms,

∫J(ξ−αpq)

−1 dμV(ξ), q=1, 2, . . . , s, one mayregard system (62) as a means by which real linear combinations of themoment integrals

∫J ξ

r1 V ′(ξ)dμV(ξ), r1=0, 1, . . . , 2s−1, and∫

J ξr2 dμV(ξ),

r2=0, 1, . . . , 2s−2, may be expressed in terms of them (also as real linear com-binations); thus, one must verify that the determinant of the 2s × 2s coefficientmatrix, D, is not equal to zero. Towards this end, one uses Theorem 20 of [71]to show that

det(D)=(−1)s(s−1)

2

s−1∏q=1

(αpq−αk)4

s−1∏i, j=1

i< j

(αpj−αpi)4 �= 0.

Note that, since V: R \ {α1, α2, . . . , αK}→R satisfies conditions (6)–(8), itfollows from the identity zl

1−zl2=(z1−z2)(z

l−11 +zl−2

1 z2+· · ·+z1zl−22 +zl−1

2 ),l∈N, that, for n∈N and k=1, 2, . . . ,K, qV(z) is real analytic on R \ {α1,

α2, . . . , αK} ⊃ J. For x∈ J, set z :=x±iε, and consider the ε↓0 limit of(61): limε↓0(F(x±iε)+ iV ′(x±iε)

2π )2=(F±(x)+ iV ′(x)2π )

2 (as V is real analytic on J);recalling that (cf. (57)), for n∈N and k=1, 2, . . . ,K, F±(x)=− 1

iπ ((κnk−1)n(x−αk)

+∑s−1q=1

κnkkq

n(x−αpq )−π( (n−1)K+k

n )(H ψV)(x))∓( (n−1)K+kn )ψV(x), x∈ J, via (56), it

follows that F±(x)=− iV ′(x)2π ∓( (n−1)K+k

n )ψV(x), which implies that (F±(x)+iV ′(x)

2π )2=( (n−1)K+k

n )2(ψV(x))2, whence (ψV(x))

2=−(π( (n−1)K+kn ))−2qV(x), x∈ J,

whereupon, using the fact that ψV(x)�0 ∀ x∈ J, it follows that qV(x)�0,x∈ J; moreover, as a by-product, decomposing qV(x), for x∈ J, into itspositive and negative parts, that is, qV(x)=(qV(x))

+−(qV(x))−, x∈ J, where

(qV(x))± :=max{±qV(x), 0} (�0), one learns from the above analysis that,

for x∈ J, (qV(x))+≡0 and ψV(x)=(π( (n−1)K+k

n ))−1((qV(x))−)1/2; and, since∫

J ψV(ξ)dξ=1, it follows that (π( (n−1)K+kn ))−1

∫J((qV(ξ))

−)1/2 dξ=1, whichgives rise to the interesting fact that the function (qV(x))

− �≡0 on J. Forx /∈ J, set z :=x±iε, and, again, study the ε↓0 limit of (61): in this case,limε↓0(F(x±iε)+ iV ′(x±iε)

2π )2=(F±(x)+ iV ′(x)2π )

2=(F(x)+ iV ′(x)2π )

2; recalling that(cf. (57)), for n∈N and k=1, 2, . . . ,K, F(x)=− 1

iπ ((κnk−1)n(x−αk)

+∑s−1q=1

κnkkq

n(x−αpq )+

( (n−1)K+kn )

∫JψV(ξ)

ξ−x dξ)= iπ( (κnk−1)

n(x−αk)+∑s−1

q=1

κnkkq

n(x−αpq ))−i( (n−1)K+k

n )(H ψV)(x), x /∈ J,

which implies that, via (61), (F(x)+ iV ′(x)2π )

2=( 1π( (κnk−1)

n(x−αk)+∑s−1

q=1

κnkkq

n(x−αpq ))−

( (n−1)K+kn )(H ψV)(x)+ V ′(x)

2π )2=qV(x)/π

2, x /∈ J (since V ′ is real analyticon (R \ {α1, α2, . . . , αK}) \ J, it follows that qV(x), too, is real analytic on(R \ {α1, α2, . . . , αK}) \ J, in which case, this latter relation merely states that,for x=αk, k=1, 2, . . . ,K, +∞=+∞), whence qV(x)>0, x /∈ J. Now, recallingthat, on a compact subset of R, in particular, one whose intersection with{α1, α2, . . . , αK} equals ∅, a rational, in fact, mermorphic, function changes signan at most countable number of times, it follows from the argument above, thefact that V: R \ {α1, α2, . . . , αK}→R satisfies conditions (6)–(8), in particular,


V is real analytic in the open neighbourhood U :={z∈C; infq∈J|z−q|<r},with r∈(0, 1) chosen (small enough) so that U ∩ {α1, α2, . . . , αK}=∅, μVhas compact support (cf. Lemma 3.1), and mimicking a part of the calcu-lations subsumed in the proof of Theorem 1.38 in [62], that, for n∈N andk=1, 2, . . . ,K, J :=supp(μV)={x∈R \ {α1, α2, . . . , αK}; qV(x)�0} consists ofthe disjoint union of a finite number of bounded real (compact) intervals,with representation J=∪N+1

j=1 J j, where J j :=[bj−1, a j], and {b0, a1,b1, a2, . . . ,

bN, aN+1}={x∈R \ {α1, α2, . . . , αK}; qV(x) = 0}, with N(=N(n,k)) ∈ Z+0 and

finite, J j ∩ {α1, α2, . . . , αK}=∅, b0 :=min{J} /∈{−∞, α1, α2, . . . , αK}, aN+1 :=max{J} /∈{α1, α2, . . . , αK,+∞}, and −∞<b0<a1<b1<a2< · · ·<bN<aN+1<

+∞. Furthermore, as a by-product of the above representation for J, it followsthat, since J ∩ {α1, α2, . . . , αK}=∅ and Ji ∩ J j=∅, i �= j∈{1, 2, . . . , N+1},meas(J)=∑N+1

j=1 |bj−1−a j| (<+∞).It remains, still, to determine the 2(N+1) equations satisfied by the end-

points of the support of the equilibrium measure, {b j−1, a j}N+1j=1 . Towards

this end, one proceeds as follows. For n∈N and k=1, 2, . . . ,K, fromthe definition of F(z) given in (55), and recalling that M1(R)�μV , inparticular,

∫J ξ

m dμV(ξ)<∞ and∫

J(ξ−αk)−m dμV(ξ)<∞, m∈N: (i) for

ξ ∈ J and z /∈ J such that |ξ/z|%1 (e.g., |z|&max{max j=1,2,...,N+1{|bj−1−a j|},maxq=1,2,...,s{|αpq |}}), via the expansion 1

ξ−z =−∑lj=0

ξ j

z j+1 + ξ l+1

zl+1(ξ−z) , l∈Z+0 ,

one gets that

F(z) =z→∞

1

iπnz− 1

iπz

∞∑m=1

⎛⎝(

κnk−1

n

)(αk)

m+s−1∑q=1

κnkkq

n(αpq)

m

−((n−1)K+k

n

)∫JξmψV(ξ)dξ

⎞⎠ z−m; (63)

and (ii) for ξ ∈ J and z /∈ J such that |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|% min{minq=1,2,...,s−1{|αk−αpq |},min j=1,2,...,N+1{||bj−1−a j|−αk|}}), via the

expansion 1(z−αk)−(ξ−αk)

=−∑lj=0

(z−αk)j

(ξ−αk) j+1 + (z−αk)l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 , one gets that

F(z)=z→αk − (κnk−1)iπn(z−αk)

+O(1), and F(z)=z→αpq− κnkkq

iπn(z−αpq )+O(1), q=1, 2, . . . ,

s−1. Recalling, too, the formulae for F±(z) given in (57), one deducesthat, via (56), F+(z)+F−(z)= V ′(z)/iπ , z∈ J, and F+(z)−F−(z)=0, z /∈ J.Thus, one learns that, for n∈N and k=1, 2, . . . ,K, F : N × {1, 2, . . . ,K} ×C \ (J ∪ {αp1 , . . . , αps−1 , αk})→C solves the following (scalar and homoge-neous) RHP: (1) F(z) is holomorphic (resp., meromorphic) for z∈C \(J ∪ {αp1 , . . . , αps−1 , αk}) (resp., z∈C \ J); (2) F±(z) := limε↓0 F(z±iε) satisfythe boundary condition F+(z)+F−(z)= V ′(z)/iπ , z∈ J, and F+(z)=F−(z)=F(z), z /∈ J; (3) F(z)=z→∞ 1

iπnz +O(z−2); and (4) Res(F(z);αk)=−(κnk−1)/iπn,and, for q ∈ {1, 2, . . . , s − 1}, Res(F(z);αpq) = −κnkkq

/iπn. For n∈N and


k=1, 2, . . . ,K, an explicit representation for the solution of this RHP can bepresented as (see, for example, Chapter VI of [72])

F(z) = − 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠+ (R(z))1/2

∫J

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)

+s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠ (R(ξ))−1/2

+ξ−z

dξ2π i,

z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}), (64)

where (R(z))1/2 is defined in item (1)(1)(1) of the Lemma, with (R(z))1/2± :=limε↓0(R(z±iε))1/2, and the branch of the square root is chosen so that(for |z|&max{max j=1,2,...,N+1{|bj−1−a j|},maxq=1,2,...,s{|αpq |}}) z−(N+1)(R(z))1/2

∼C±�z→∞±1. It follows from the above integral representation (64) that, forn∈N and k=1, 2, . . . ,K, for ξ ∈ J and z /∈ J such that |ξ/z|%1 (e.g., |z|&max{max j=1,2,...,N+1{|bj−1−a j|},maxq=1,2,...,s{|αpq |}}), via the expansion 1

ξ−z =−∑l

j=0ξ j

z j+1 + ξ l+1

zl+1(ξ−z) , l∈Z+0 ,

F(z) =z→∞− (z

N+1+· · · )2π iz

∫J

1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠

×(

1+ ξz+ ξ

2

z2+· · ·+ ξ

N

zN

)dξ− 1

z1

iπ

(κnk−1

n

)− 1

z

s−1∑q=1

κnkkq

iπn− 1

z(1+· · · )

2π i

×∫

J

ξN+1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ − 1

π iz

×∞∑

m=1

⎛⎝(

κnk−1

n

)(αk)

m+s−1∑q=1

κnkkq

n(αpq)

m

⎞⎠ 1

zm− (1+· · · )

2π iz2

∫J

ξN+2

(R(ξ))1/2+

×⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠(

1+ ξz+· · ·

)dξ :

now, recalling from above that F(z)=z→∞ 1iπnz +O(z−2), it follows, upon

removing secular terms, that

∫J

ξ j

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ=0,

j=0, 1, . . . , N,


which gives N+1 real moment equations, and, upon equating z−1 terms,

∫J

ξN+1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ

=−2

((n−1)K+k

n

);

it remains, therefore, to determine an additional 2(N+1)−(N+1)−1=N realmoment equations. From the integral representation (64), a residue calculationshows that, for n∈N and k=1, 2, . . . ,K, for non-real z in the domain ofanalyticity of V (that is, C \ {α1, α2, . . . , αK}),

F(z) = V ′(z)2π i

+ (R(z))1/2

2

∮CV

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ V ′(ξ)iπ

⎞⎠ (R(ξ))−1/2

ξ−zdξ2π i, (65)

where CV (⊂ C \ {α1, α2, . . . , αK}) is the simple boundary of an open (s+1)-connected punctured region of the type CV =∂ U

�R ∪ (∪s

q=1∂ U�δq(αpq)), with

U�R ={z∈C; |z|<R}, and ∂ U

�R ={z∈C; |z|=R} oriented clockwise, and, for

q=1, 2, . . . , s, U�δq(αpq) :={z∈C; |z−αpq |<δq}, with ∂ U

�δq(αpq)={z∈C; |z−

αpq |=δq} oriented counter-clockwise, and where the numbers 0<δq (suffi-ciently small) <R (sufficiently large) <+∞, q=1, 2, . . . , s, are chosen so that,for (any) non-real z in the domain of analyticity of V,

• ∂ U�δi(αpi) ∩ ∂ U

�δ j(αpj)=∅, i �= j∈{1, 2, . . . , s},

• ∂ U�δ j(αpj) ∩ ∂ U

�R =∅, j∈{1, 2, . . . , s},

• (U�δ j(αpj) ∪ ∂ U

�δ j(αpj)) ∩ J=∅, j∈{1, 2, . . . , s},

• ∂ U�R ∩ J=∅,

• int(CV) :=(ext(∪sq=1(U

�δq(αpq) ∪ ∂ U

�δq(αpq))) ∩ U

�R) ⊃ {z} ∪ J.

Recall from (57) that, for z /∈ J,

F+(z) = F−(z)=F(z) = − 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

+((n−1)K+k

n

)∫J

ψV(ξ)

ξ−zdξ

⎞⎠ ,

whence, via (55),

F(z)+ 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠=−i

((n−1)K+k

n

)(H ψV)(z);


thus, using (64), one arrives at, for z∈R \ J,

(H ψV)(z) = i((n−1)K+k

n

)−1

(R(z))1/2∫

J

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)

+s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠ (R(ξ))−1/2

+ξ−z

dξ2π i.

A contour integration argument shows that, for j=1, 2, . . . , N,

∫ bj

a j

⎛⎝(H ψV)(ξ)−

1

2π

((n−1)K+k

n

)−1⎛⎝2(κnk−1)

n(ξ−αk)

+ 2s−1∑q=1

κnkkq

n(ξ−αpq)+V ′(ξ)

⎞⎠⎞⎠dξ=0, (66)

whence, using the above expression for (H ψV)(z), z∈∪Nj=1(a j,bj), for n∈N

and k=1, 2, . . . ,K,

∫ bj

a j

⎛⎝(R(ς))1/2

∫J

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠

× (R(ξ))−1/2+

ξ−ς dξ

⎞⎠ dς =

∫ bj

a j

⎛⎝2(κnk−1)

n(ξ−αk)+2

s−1∑q=1

κnkkq


⎞⎠ dξ,

j=1, 2, . . . , N.

If no αk, k=1, 2, . . . ,K, belongs to any gap (a j,bj), j=1, 2, . . . , N, then, viathe real analyticity of V : R \ {α1, α2, . . . , αK}→R and an application of thefundamental theorem of calculus, the N integrals on the right-hand side ofthe expression above are easily evaluated; if, however, an αk, k=1, 2, . . . ,K,belongs to any gap (a j,bj), j=1, 2, . . . , N, then the integrals appearing on theright-hand side of the expression above,

∫ b j

a j!(ξ)dξ , j=1, 2, . . . , N, need to be

understood in the Cauchy principal value sense, limε↓0((∫ αk−ε

a j+∫ b j

αk+ε)!(ξ)dξ),j=1, 2, . . . , N; in either case, one arrives at, for n∈N and k=1, 2, . . . ,K,

∫ bj

a j

⎛⎝(R(ς))1/2

∫J

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠

× (R(ξ))−1/2+

ξ−ς dξ

⎞⎠dς = 2

(κnk−1

n

)ln

∣∣∣∣bj−αk

a j−αk

∣∣∣∣+2s−1∑q=1

κnkkq

nln

∣∣∣∣bj−αpq

a j−αpq

∣∣∣∣

+(V(bj)−V(a j)), j = 1, 2, . . . , N,


which yields the remaining N real moment equations determining the2(N+1) end-points of the support of the equilibrium measure. SinceJ ∩ {α1, α2, . . . , αK}=∅ and V: R \ {α1, α2, . . . , αK}→R is real analyticon J, (R(x))1/2=x↓bj−1 O((x−bj−1)

1/2) and (R(x))1/2=x↑a j O((a j−x)1/2), j=1, 2, . . . , N+1, which shows that all the integrals above constituting the systemof 2(N+1) real moment equations for the end-points of the support ofμV haveintegrable singularities at bj−1, a j, j=1, 2, . . . , N+1.

Recall from (57) that, for n∈N and k=1, 2, . . . ,K,

F±(z) = − 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)−π

((n−1)K+k

n

)(H ψV)(z)

⎞⎠

∓((n−1)K+k

n

)ψV(z), z∈ J :

using, for z∈ J, (56), it follows from the real analyticity of V: R \ {α1,

α2, . . . , αK}→R that, for n∈N and k=1, 2, . . . ,K,

F±(z)= V ′(z)2π i

∓((n−1)K+k

n

)ψV(z), z∈ J. (67)

From (64) and the real analyticity of V: R \ {α1, α2, . . . , αK}→R, it followsthat, for n∈N and k=1, 2, . . . ,K, with z∈ J,

F±(z) = V ′(z)2π i

+ (R(z))1/2±

2

∮CV

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠

+ V ′(ξ)iπ

⎞⎠ (R(ξ))−1/2

ξ−zdξ2π i

;

thus, equating the two expressions above for F±(z), z∈ J, one arrives at,for n∈N and k=1, 2, . . . ,K, ψV(x)= 1

2π i (R(x))1/2+ hV(x)1J(x), where hV(z) is

defined in item (2)(2)(2) of the Lemma, and 1J(x) is the characteristic function of theset J, which gives rise to the formula for the equilibrium measure, dμV(x)=ψV(x)dx (the integral representation for hV(z) shows that it is analytic insome open subset of C \ {αp1 , . . . , αps−1 , αk} containing J). Now, recallingthat V: R \ {α1, α2, . . . , αK}→R satisfies conditions (6)–(8), and that, for ξ ∈ J(resp., ξ ∈ int(J)), ψV(ξ)�0 (resp., ψV(ξ)>0) and (R(ξ))1/2+ = i(|R(ξ)|)1/2∈ iR(resp., (R(ξ))1/2+ = i(|R(ξ)|)1/2∈ iR±), it follows from the formula ψV(ξ)=

12π i (R(ξ))

1/2+ hV(ξ)1J(ξ) and the regularity assumption, that is, hV(ξ) �=0 for

ξ ∈ J, that (|R(ξ)|)1/2hV(ξ)�0, ξ ∈ J (resp., (|R(ξ)|)1/2hV(ξ)>0, ξ ∈ int(J)).Finally, it will be shown that, if J=∪N+1

j=1 [bj−1, a j], the end-points of thesupport of the equilibrium measure, which satisfy the system of 2(N+1)real moment equations stated in item (1)(1)(1) of the Lemma, are real-analyticfunctions of zo, thus establishing the (local) solvability of the system of 2(N+1)


real moment equations. Towards this end, one follows closely the idea ofthe proof of Theorem 1.3 (iii) in [63] (see, also, Section 8 of [62]). Recallfrom Subsection 1.3 that V(z) :=zoV(z), where zo : N × N→R+, (N,n) →zo :=N/n, and, in the double-scaling limit N,n→∞, zo=1+o(1). Furthermore,from the analysis above, it was shown that the end-points of the support ofthe equilibrium measure were the simple roots of the (meromorphic) func-tion qV(z), that is, with the enumeration −∞<b 0<a1<b1<a2< · · ·<bN<

aN+1<+∞, {b 0, a1,b1, a2, . . . ,bN, aN+1}={x∈R \ {α1, α2, . . . , αK}; qV(x)=0}(these are the only roots for the regular case studied in this work). The(meromorphic) function qV(x)∈R(x) (the algebra of rational functions inthe indeterminate x with coefficients in R) is real rational (resp., real an-alytic) on R (resp., R \ {α1, α2, . . . , αK}), it has analytic extension (indepen-dent of zo) to the open neighbourhood U

=∪N+1j=1 U

j, where U

j :={z∈C \{α1, α2, . . . , αK}; infp∈(bj−1,a j)|z− p|<r j}, r j∈(0, 1), j=1, 2, . . . , N+1, with U

i ∩U

j=∅, i �= j∈{1, 2, . . . , N+1}, and depends continuously on zo; thus, its simpleroots, that is, bj−1=bj−1(zo) and aj=aj(zo), j=1, 2, . . . , N+1, are continuousfunctions of zo.

Write the large-z (e.g., |z|&max{max j=1,2,...,N+1{|bj−1−a j|},maxq=1,2,...,s{|αpq |}})asymptotic expansion for F(z) (cf. (64)) as follows: for n∈N and k=1, 2, . . . ,K,

F(z) =z→∞ − 1

iπz

⎛⎝(

κnk−1

n

)+

s−1∑q=1

κnkkq

n

⎞⎠− 1

iπz

∞∑m=1

⎛⎝(

κnk−1

n

)(αk)

m

+s−1∑q=1

κnkkq

n(αpq)

m

⎞⎠ z−m − (R(z))

1/2

2π iz

∞∑j=0

T jz− j,

where

T j :=∫

J

ξ j

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ, j∈Z

+0 .

(68)

Set, for n∈N and k=1, 2, . . . ,K,

N j :=∫ b j

a j

⎛⎝(H ψV)(ξ)−

1

2π

((n−1)K+k

n

)−1⎛⎝2(κnk−1)

n(ξ−αk)

+ 2s−1∑q=1

κnkkq


⎞⎠⎞⎠dξ, j = 1, 2, . . . , N.

With the above definitions, the system of 2(N+1) real moment equationsare, thus, equivalent to T j=0, j=0, 1, . . . , N, TN+1=−2((n−1)K+k)/n, andN j=0, j=1, 2, . . . , N. It will first be shown that, for regular V: R \ {α1, α2,


. . . , αK}→R satisfying conditions (6)–(8), the Jacobian of the transfor-mation{b0(zo),b1(zo), . . . ,bN(zo),a1(zo),a2(zo), . . . ,aN+1(zo)} →{T0,T1, . . . ,

TN+1,N1,N2, . . . ,NN} is non-zero whenever bj−1=bj−1(zo) and a j=a j(zo),j=1, 2, . . . , N+1, are chosen so that J=∪N+1

j=1 [bj−1, a j]. Using the equation(cf. (55))

(H ψV)(z)=1

2π

((n−1)K+k

n

)−1⎛⎝2π iF(z)+ 2(κnk−1)

n(z−αk)+2

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ ,

and (64), one follows the analysis on pp. 778–779 of [63] to show that, for n∈N

and k=1, 2, . . . ,K, with i=1, 2, . . . , N+1:

∂T j

∂bi−1=bi−1

∂T j−1

∂bi−1+ 1

2T j−1, j∈N, (69)

∂T j

∂ai=ai∂T j−1

∂ai+ 1

2T j−1, j∈N, (70)

∂F(z)∂bi−1

=− 1

2π i

(∂T0

∂bi−1

)(R(z))1/2

z−bi−1,

z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}), (71)

∂F(z)∂ai

=− 1

2π i

(∂T0

∂ai

)(R(z))1/2

z−ai,

z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}), (72)

∂N j

∂bi−1=− 1

2π

((n−1)K+k

n

)−1 (∂T0

∂bi−1

)∫ bj

a j

(R(ξ))1/2

ξ−bi−1dξ,

j=1, 2, . . . , N, (73)

∂N j

∂ai=− 1

2π

((n−1)K+k

n

)−1 (∂T0

∂ai

)∫ bj

a j

(R(ξ))1/2

ξ−aidξ,

j=1, 2, . . . , N; (74)

furthermore, if one evaluates (69) and (70) on the solution of the system of2(N+1) real moment equations, that is, T j=0, j=0, 1, . . . , N, TN+1= −2((n−1)K+k)/n, and Nl =0, l=1, 2, . . . , N, one arrives at, for i=1, 2, . . . , N+1,

∂T j

∂bi−1=(bi−1)

j ∂T0

∂bi−1and

∂T j

∂ai=(ai)

j ∂T0

∂ai, j=0, 1, . . . , N+1.

(75)


Via (73), (74), and (75), one computes, for n∈N and k=1, 2, . . . ,K, theJacobian of the transformation {b0(zo),b1(zo), . . . ,bN(zo), a1(zo), a2(zo), . . . ,

aN+1(zo)} →{T0,T1, . . . ,TN+1,N1,N2, . . . ,NN} on the solution of the systemof 2(N+1) real moment equations:

Jac(T0,T1, . . . ,TN+1,N1,N2, . . . ,NN)

:= ∂(T0,T1, . . . ,TN+1,N1,N2, . . . ,NN)

∂(b0,b1, . . . ,bN, a1, a2, . . . , aN+1)

=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

∂T0∂b0

∂T0∂b1

· · · ∂T0∂bN

∂T0∂a1

∂T0∂a2

· · · ∂T0∂aN+1

∂T1∂b0

∂T1∂b1

· · · ∂T1∂bN

∂T1∂a1

∂T1∂a2

· · · ∂T1∂aN+1

....... . .

......

.... . .

...

∂TN+1

∂b0

∂TN+1

∂b1· · · ∂TN+1

∂bN

∂TN+1

∂a1

∂TN+1

∂a2· · · ∂TN+1

∂aN+1

∂N1∂b0

∂N1∂b1

· · · ∂N1∂bN

∂N1∂a1

∂N1∂a2

· · · ∂N1∂aN+1

∂N2∂b0

∂N2∂b1

· · · ∂N2∂bN

∂N2∂a1

∂N2∂a2

· · · ∂N2∂aN+1

....... . .

......

.... . .

...∂NN∂b0

∂NN∂b1

· · · ∂NN∂bN

∂NN∂a1

∂NN∂a2

· · · ∂NN∂aN+1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= (−1)N

(2π)N

((n−1)K+k

n

)−N(

N+1∏m=1

∂T0

∂bm−1

∂T0

∂am

)

×⎛⎝ N∏

j=1

∫ b j

a j

(R(ξ j))1/2 dξ j

⎞⎠� d(ξ), (76)

where

�

d(ξ) :=

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

1 1 · · · 1 1 1 · · · 1

b0 b1 · · · bN a1 a2 · · · aN+1

......

. . ....

......

. . ....

(b0)N+1 (b1)

N+1 · · · (bN)N+1 (a1)

N+1 (a2)N+1 · · · (aN+1)

N+1

1ξ1−b0

1ξ1−b1

· · · 1ξ1−bN

1ξ1−a1

1ξ1−a2

· · · 1ξ1−aN+1

1ξ2−b0

1ξ2−b1

· · · 1ξ2−bN

1ξ2−a1

1ξ2−a2

· · · 1ξ2−aN+1

......

. . ....

......

. . ....

1ξN−b0

1ξN−b1

· · · 1ξN−bN

1ξN−a1

1ξN−a2

· · · 1ξN−aN+1

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

. (77)


The above determinant � d(ξ) has been calculated on pg. 780 of [63]:

�

d(ξ)=

(∏N+1j=1

∏N+1i=1 (bi−1−a j)

)(∏N+1i, j=1

j<i

(ai−a j)(bi−1−bj−1)

)(∏Ni, j=1

j<i

(ξi−ξ j)

)

(−1)N∏N

j=1

∏N+1i=1 (ξ j−ai)(ξ j−bi−1)

;(78)

but, for −∞ < b 0 < a1 < ξ1 < b1 < a2 < ξ2 < b2 < · · · < bN−1 < aN < ξN <

bN < aN+1< +∞, � d(ξ) �=0 (which means that it is of fixed sign), and∫ bj

a j(R(ξ j))

1/2 dξ j �=0, j=1, 2, . . . , N; thus, for n∈N and k=1, 2, . . . ,K,

⎛⎝ N∏

j=1

∫ bj

a j

(R(ξ j))1/2 dξ j

⎞⎠� d(ξ) �=0. (79)

It remains to show that ∂T0/∂bj−1 and ∂T0/∂a j, j=1, 2, . . . , N+1, too, arenon-zero: for this purpose, one exploits the fact that, for n∈N and k=1, 2, . . . ,K,

T0=∫

J

1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ (80)

is independent of z. It follows from (64), the integral representation forhV(z) given in item (2)(2)(2) of the Lemma, and (71) and (72) that, for n∈N andk=1, 2, . . . ,K,

(z−bj−1

)√

R(z)

∂F(z)∂bj−1

= − 1

2π i

((n−1)K+k

n

)((z−bj−1

) ∂hV(z)∂bj−1

− 1

2hV(z)

), j = 1, 2, . . . , N+1,

(z−a j)√R(z)

∂F(z)∂a j

= − 1

2π i

((n−1)K+k

n

)((z−a j)

∂hV(z)∂a j

− 1

2hV(z)

), j = 1, 2, . . . , N+1 :

using, now, the z-independence of T0, and the fact that, for the case of regularV: R \ {α1, α2, . . . , αK}→R satisfying conditions (6)–(8), hV(bj−1), hV(aj) �=0,j=1, 2, . . . , N+1, one shows that, for n∈N and k=1, 2, . . . ,K,

(z−bj−1)√R(z)

∂F(z)∂bj−1

∣∣∣∣z=bj−1

= 1

4π i

((n−1)K+k

n

)hV(bj−1) �= 0, j=1, 2, . . . , N+1,


(z−a j)√R(z)

∂F(z)∂a j

∣∣∣∣z=a j

= 1

4π i

((n−1)K+k

n

)hV(a j) �= 0, j=1, 2, . . . , N+1;

thus, via (71) and (72), one arrives at, for n∈N and k=1, 2, . . . ,K,

∂T0

∂bj−1=−1

2

((n−1)K+k

n

)hV(b j−1), j=1, 2, . . . , N+1,

and

∂T0

∂a j=−1

2

((n−1)K+k

n

)hV(a j), j=1, 2, . . . , N+1,

whence, for n∈N and k=1, 2, . . . ,K,

N+1∏m=1

∂T0

∂bm−1

∂T0

∂am=(

1

2

((n−1)K+k

n

))2(N+1) N+1∏j=1

hV(bj−1)hV(a j) �=0.

Hence, Jac(T0,T1, . . . ,TN+1,N1,N2, . . . ,NN) �=0.In order to invoke the implicit function theorem, it remains, still, to show

that T j, j=0, 1, . . . , N+1, and Ni, i=1, 2, . . . , N, are (real) analytic functionsof {bj−1(zo), a j(zo)}N+1

j=1 . From the definition of T j, j∈Z+0 , above, using the

fact that they are independent of z, thus giving rise to zero residue contri-butions, a Residue Calculus calculation shows that, equivalently, for n∈N andk=1, 2, . . . ,K,

T j= 1

2

∮CV

ξ j

(R(ξ))1/2

⎛⎝ 2

iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠dξ, j∈Z

+0 ,

where the (simple closed) contour CV has been defined heretofore: theonly factor depending on {bj−1, aj}N+1

j=1 is (R(z))1/2. As (R(z))1/2 is analyticfor z∈C \ ∪N+1

j=1 [bj−1, aj], and since CV ⊂ C \ ∪N+1j=1 [bj−1, aj], with int(CV) ⊃

{z} ∪ (∪N+1j=1 [bj−1, aj]), it follows that, in particular, (R(z))1/2�CV

is an analyticfunction of {bj−1, aj}N+1

j=1 , which implies, via the above (equivalent) contourintegral representation for T j, j∈Z

+0 , that, for n∈N and k=1, 2, . . . ,K, Ti,

i=0, 1, . . . , N+1, are (real) analytic functions of {bj−1, aj}N+1j=1 . Recalling from

the analysis above that, for n∈N and k=1, 2, . . . ,K, with z∈∪Nj=1(aj,bj),

2π

((n−1)K+k

n

)(H ψV)(z)−V ′(z)− 2(κnk−1)

n(z−αk)−2

s−1∑q=1

κnkkq

n(z−αpq

)

=−((n−1)K+k

n

) (R(z)

)1/2hV(z),


it follows from the definition of N j, j=1, 2, . . . , N, that

N j=− 1

2π

∫ bj

aj

(R(ξ))1/2hV(ξ)dξ, j=1, 2, . . . , N :

making the linear change of variables uj : C→C, ξ →uj(ξ) :=(bj−aj)−1(ξ−aj),

j=1, 2, . . . , N, which take each of the compact intervals [aj,bj], j=1, 2, . . . , N,onto [0, 1], and setting

(R(z))1/2 :=⎛⎝

j∏i1=1

(z−bi1−1)

j−1∏i2=1

(z−ai2)

N+1∏i3= j+1

(ai3−z)N+1∏

i4= j+2

(bi4−1−z)

⎞⎠

1/2

,

one arrives at, for n∈N and k=1, 2, . . . ,K,

N j = − 1

2π(bj−aj)

2∫ 1

0

(uj(1−uj)

)1/2(

R((bj−aj)uj+aj

))1/2

× hV

((bj−aj)uj+aj

)duj, j=1, 2, . . . , N.

Recalling that hV(z) is real analytic on R \ {α1, α2, . . . , αK}, in particular,hV(bj−1), hV(aj) �=0, j=1, 2, . . . , N+1, and that it is an analytic function of{bi−1(zo), ai(zo)

}N+1i=1 , and noting from the definition of (R(z))1/2 above that,

it, too, is an analytic function of (bj−aj)uj+aj, and thus an analytic functionof{bi−1(zo), ai(zo)

}N+1i=1 , it follows that N j, j=1, 2, . . . , N, are (real) analytic

functions of{bi−1(zo), ai(zo)

}N+1i=1 .

Thus, as the Jacobian of the transformation{b0(zo),b1(zo), . . . ,bN(zo),

a1(zo), a2(zo), . . . , aN+1(zo)} →{

T0,T1, . . . ,TN+1,N1,N2, . . . ,NN}

is non-

zero whenever{bj−1(zo), aj(zo)

}N+1j=1 are chosen so that, for regular V: R \

{α1, α2, . . . , αK}→R satisfying conditions (6)–(8), J= ∪N+1j=1

[bj−1(zo), aj(zo)

],

and T j, j=0, 1, . . . , N+1, and Ni, i=1, 2, . . . , N, are (real) analytic functions

of{b m−1(zo), am(zo)

}N+1m=1 , it follows, via the implicit function theorem, that, for

n∈N and k=1, 2, . . . ,K, bj−1(zo), aj(zo), j=1, 2, . . . , N+1, are real analyticfunctions of zo. ��

Remark 3.2 It turns out that, for a V: R \ {αp1 , . . . , αps−1 , αps} → R (with

αps:= αk) satisfying conditions (6)–(8) of the form

V(z) =s∑

q=1

−1∑j=−2mq

ρ j,q(z − αpq

) j +2m∞∑j=0

ρ j,∞z j,


where, for q=1, 2, . . . , s, mq∈N, m∞∈N, and ρ−2mq,q, ρ2m∞,∞>0, a closed-form expression for

hV(z) =1

2

((n−1)K+k

n

)−1 ∮CV

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq

)⎞⎠

+ iV′(ξ)π

)(R(ξ))−1/2

ξ−zdξ

can be derived, where the definition of the contour CV (⊂ C \ {α1, α2, . . . , αK})is given in item (2)(2)(2) of Lemma 3.7, (R(z))1/2 :=(∏N+1

j=1 (z−bj−1)(z−aj))1/2,

and {bj−1, aj}N+1j=1 satisfy the locally solvable system of 2(N+1) real moment

equations stated in item (1)(1)(1) of Lemma 3.7. A Residue Calculus calculationshows that

hV(z) =z2m∞−N−2

ℵ 2m∞−N−2∑

j=0

(2m∞− j )ρ2m∞− j,∞∑′

k0,k1,...,kN

∑′

l0,l1,...,lN

0�|k|+|l|�2m∞− j−N−2ki�0, li�0, i∈{0,1,...,N}

·⎛⎝ N∏

p′=0

kp′−1∏jp′=0

(1

2+ jp′

)⎞⎠⎛⎝ N∏

q′=0

lq′−1∏mq′=0

(1

2+mq′

)⎞⎠

×(∏N

p′′=0(bp′′)kp′′) (∏N

q′′=0(aq′′+1)lq′′)

(∏Nl′=0 kl′ !

) (∏Nm′=0 lm′ !

) z− j−|k|−|l|

− 1

ℵ s∑

q=1

(−1)N (αpq )(∏N+1

i=1 |bi−1−αpq ||ai−αpq |)−1/2

(z−αpq)2mq+1

·········0∑

j=−2mq+1

(2mq+ j)$−2mq− j,q

∑′′

k0,k1,...,kN

∑′′

l0,l1,...,lN

0�|k|+|l|�2mq+ jki�0, li�0, i∈{0,1,...,N}

·(∏N

p′=0

∏kp′−1jp′=0

(12 + jp′

)) (∏Nq′=0

∏lq′−1mq′=0

(12 +mq′

))(∏N

p′′=0(bp′′ −αpq)kp′′) (∏N

q′′=0(aq′′+1−αpq)lq′′) (∏N

l′=0 kl′ !) (∏N

m′=0 lm′ !)


× (z−αpq)|k|+|l|− j+ 2

ℵ s−1∑q=1

κnkkq

n

(−1)N (αpq )(∏N+1

i=1 |bi−1−αpq ||ai−αpq |)−1/2

(z−αpq)

+ 2

ℵ (

κnk−1

n

) (−1)N (αk)(∏N+1

i=1 |bi−1−αk||ai−αk|)−1/2

(z−αk),

z∈C \ {αp1 , . . . , αps−1 , αk},

where ℵ :=((n−1)K+k)/n,

|k| :=k0+k1+· · ·+kN, |l| := l0+l1+· · ·+lN,

N (αpq )=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0, αpq ∈(aN+1,+∞),N+1, αpq ∈(−∞, b0),

N− j+1, αpq ∈(aj, bj),

q=1, 2, . . . , s, j=1, 2, . . . , N,

and the primes (resp., double primes) on the summations mean that allpossible sums over {kl}N

l=0 and {lk}Nk=0 must be taken for which 0�k0+k1+· · ·+

kN+l0+l1+· · ·+lN �2m∞− j−N−2, j=0, 1, . . . , 2m∞−N−2, ki �0, li �0, i∈{0, 1, . . . , N} (resp., 0�k0+k1+· · ·+kN+l0+l1+· · ·+lN �2mq+ j, j=−2mq+1,−2mq+2, . . . , 0, ki �0, li �0, i∈{0, 1, . . . , N}). It is important to note that allof the above sums are finite sums: any sums for which the upper limit is lessthan the lower limit are defined to be equal to zero, and any products in whichthe upper limit is less than the lower limit are defined to be equal to one; forexample,

∑−1j=0(∗∗∗) :=0 and

∏−1j=0(∗∗∗) :=1. �

Remark 3.3 It is interesting to note that, for n∈N and k=1, 2, . . . ,K, one mayderive explicit formulae for the various moments of the associated equilibriummeasure, that is,

∫Rξm dμV(ξ)=

∫J ξ

mψV(ξ)dξ and∫

R(ξ−αpq)

−m dμV(ξ)=∫J(ξ−αpq)

−mψV(ξ)dξ , m∈N, q=1, 2, . . . , s, in terms of the external field

V: R \ {α1, α2, . . . , αK}→R satisfying conditions (6)–(8), and the multi-valuedfunction (R(z))1/2 :=(∏N+1

i=1 (z−bi−1)(z−ai))1/2, where {bj−1, aj}N+1

j=1 satisfy thelocally solvable system of 2(N+1) moment equations stated in item (1)(1)(1)of Lemma 3.7. Without loss of generality, and for demonstrative purposesonly, consider, say, the following non-trivial moments:

∫J(ξ−αpq)

−r dμV(ξ)

and∫

J ξr dμV(ξ), q=1, 2, . . . , s, r=1, 2, 3 (the scheme of the calculations

below generalises to the moments∫

J(ξ−αpq)−(r+3) dμV(ξ) and

∫J ξ

r+3 dμV(ξ),q=1, 2, . . . , s, r∈N). Recall the representations for F(z) given in (the proof of)Lemma 3.7 (cf. (55) and (64)):

F(z) = − 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)+((n−1)K+k

n

)∫J

ψV(ξ)

ξ−zdξ

⎞⎠ ,


F(z) = − 1

iπ

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠+ (R(z))1/2

∫J

(2

iπ

((κnk−1)

n(ξ−αk)

+s−1∑q=1

κnkkq

n(ξ−αpq)

⎞⎠+ V ′(ξ)

iπ

⎞⎠ (R(ξ))−1/2

+ξ−z

dξ2π i.

Recalling that, for n ∈ N and k=1, 2, . . . ,K, μV ∈ M1(R), in particular,∫J(ξ−αpq)

−mdμV(ξ)<∞ and∫

J ξmdμV(ξ)<∞, m∈N, q=1, 2, . . . , s, with∫

J dμV(ξ)=1, one derives, for n∈N and k=1, 2, . . . ,K, the following as-ymptotic expansions: (1) for ξ ∈ J and z /∈ J such that |(z−αk)/(ξ−αk)|% 1(e.g., 0 < |z − αk| % min{min j=1,2,...,N+1{|bj−1 − αk|},min j= 1,2,...,N+1{|aj − αk|},min j=1,2,...,N+1{||bj−1−aj|−αk|},minq=1,2,...,s−1{|αpq−αk|}}), via the expansions

1(z−αk)−(ξ−αk)

=−∑lj=0

(z−αk)j

(ξ−αk) j+1 + (z−αk)l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 , and ln(1−∗∗∗)=−∑∞

m=1∗∗∗m

m ,

|∗∗∗|%1,

F(z) =z→αk

− 1

iπ

⎛⎝(κnk−1)

n(z−αk)−

s−1∑q=1

κnkkq

n(αpq −αk)+((n−1)K+k

n

)∫J(ξ−αk)

−1ψV(ξ) dξ

−⎛⎝s−1∑

q=1

κnkkq

n(αpq −αk)2−((n−1)K+k

n

)∫J(ξ−αk)

−2ψV(ξ) dξ

⎞⎠(z−αk)

−⎛⎝s−1∑

q=1

κnkkq


n

)∫J(ξ−αk)

−3ψV(ξ) dξ

⎞⎠(z−αk)

2

−⎛⎝s−1∑

q=1

κnkkq


n

)∫J(ξ−αk)

−4ψV(ξ) dξ

⎞⎠ (z−αk)

3

+O((z−αk)4)

⎞⎠ ,

and

F(z) =z→αk

− 1

iπ(κnk − 1)

n(z − αk)+ 1

iπ

s−1∑q=1

κnkkq

n(αpq − αk)+(−1)N (αk)Q�0

×⎛⎝N+1∏

j=1

|bj−1 − αk||aj − αk|⎞⎠

1/2

+⎛⎝ 1

iπ

s−1∑q=1

κnkkq

n(αpq −αk)2

+ (−1)N (αk)(

Q�1+α�Q�0) ⎛⎝N+1∏

j=1

|bj−1−αk||aj−αk|⎞⎠

1/2⎞⎟⎠ (z−αk)


+⎛⎜⎝ 1

iπ

s−1∑q=1

κnkkq

n(αpq −αk)3+(−1)N (αk)

(Q�2+α�Q�1+

(β�+ 1

2(α�)2

)Q�0

)

×⎛⎝N+1∏

j=1

|bj−1−αk||aj−αk|⎞⎠

1/2⎞⎟⎠ (z−αk)

2+O((z−αk)3),

where (recall that αps:=αk) N (αk) is defined in Remark 3.2,

α� := −1

2

N∑j=1

(1

bj−1−αk+ 1

aj−αk

),

β� := −1

4

N∑j=1

(1(

bj−1−αk)2 +

1

(aj−αk)2

),

Q�j :=∫

J

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ V′(ξ)

iπ

⎞⎠

× (R(ξ))−1/2+

(ξ−αk)j+1

dξ2π i, j = 0, 1, 2;

and (2) for ξ ∈ J and z /∈ J such that |ξ/z|%1 (e.g., |z|&max{maxq=1,2,...,s{|αpq |},max j=1,2,...,N+1{|bj−1 − aj|}}), via the expansions 1

ξ−z = −∑lj=0

ξ j

z j+1 + ξ l+1

zl+1(ξ−z) ,

l∈Z+0 , and ln(1−∗∗∗)=−∑∞

m=1∗∗∗m

m , |∗∗∗|%1,

F(z) =z→∞

1

iπnz− 1

iπ

⎛⎝(

κnk − 1

n

)αk +

s−1∑q=1

κnkkq

nαpq −

((n − 1)K + k

n

)

×∫

JξψV(ξ) dξ

⎞⎠ 1

z2− 1

iπ

⎛⎝(

κnk − 1

n

)(αk)

2 +s−1∑q=1

κnkkq

n(αpq )

2

−((n − 1)K + k

n

)∫Jξ2ψV(ξ) dξ

⎞⎠ 1

z3− 1

iπ

⎛⎝(

κnk − 1

n

)(αk)

3

+s−1∑q=1

κnkkq

n(αpq )

3 −((n − 1)K + k

n

)∫Jξ3ψV(ξ) dξ

⎞⎠ 1

z4+ O(z−5),


and, using the following N+2 moment equations (cf. (the proof of) Lemma 3.7)

∫J

ξ j

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ V′(ξ)

iπ

⎞⎠ dξ=0,

j=0, 1, . . . , N,

∫J

ξN+1

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ V′(ξ)

iπ

⎞⎠dξ

= −2

((n−1)K+k

n

),

one arrives at

F(z) =z→∞

1

iπnz−⎛⎝Q�0 −

((n − 1)K + k

n

)α�

iπ+ 1

iπ

⎛⎝(

κnk − 1

n

)αk

+s−1∑q=1

κnkkq

nαpq

⎞⎠⎞⎠ 1

z2−⎛⎝Q�1 + α�Q�0 −

((n − 1)K + k

n

) (β� + 12 (α�)2)

iπ

+ 1

iπ

⎛⎝(

κnk − 1

n

)(αk)

2 +s−1∑q=1

κnkkq

n(αpq )

2

⎞⎠⎞⎠ 1

z3−(

Q�2 + α�Q�1

+(β� + 1

2 (α�)2)

Q�0 −((n − 1)K + k

n

) (γ � + α�β� + 13! (α

�)3)

iπ

+ 1

iπ

⎛⎝(

κnk − 1

n

)(αk)

3 +s−1∑q=1

κnkkq

n(αpq )

3

⎞⎠⎞⎠ 1

z4+ O(z−5),

where

α� :=−1

2

N∑j=1

(bj−1+aj

), β� := 1

4

N∑j=1

(b 2

j−1+a2j

), γ � :=− 1

3!N∑

j=1

(b 3

j−1+a3j

),

Q�j :=∫

J

ξN+2+ j

(R(ξ))1/2+

⎛⎝ 2

iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ V′(ξ)

iπ

⎞⎠ dξ

2π i, j=0, 1, 2.

Equating coefficients of like powers of (z − αk)i, i = −1, 0, 1, 2, and z− j,

j = 1, 2, 3, 4, in the above asymptotic expressions for F(z) as z → αk,k = 1, 2, . . . ,K, and as z → ∞, respectively, one arrives at, for n ∈ N and


k = 1, 2, . . . ,K, the following expressions for the ‘first three’ non-trivialmoments of the equilibrium measure:

∫J

ψV(ξ)

ξ−αkdξ =

(−1)N (αk)(∏N+1

j=1 |bj−1−αk||aj−αk|)1/2

2(ℵ /n)∫

J

(R(ξ))−1/2+

(ξ−αk)

×⎛⎝2iπ

⎛⎝(κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ,

∫J

ψV(ξ)

(ξ−αk)2

dξ =(−1)N (αk)

(∏N+1j=1 |bj−1−αk||aj−αk|

)1/2

2(ℵ /n)

⎛⎝∫

J

(R(ξ))−1/2+

(ξ−αk)2

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

− 1

2

∫J

(R(ξ))−1/2+

(ξ−αk)

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠

+ iV′(ξ)π

⎞⎠dξ

N+1∑m=1

(1

bm−1−αk+ 1

am−αk

)⎞⎠ ,

∫J

ψV(ξ)

(ξ−αk)3

dξ =(−1)N (αk)

(∏N+1j=1 |bj−1−αk||aj−αk|

)1/2

2(ℵ /n)

⎛⎝∫

J

(R(ξ))−1/2+

(ξ−αk)3

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

− 1

2

∫J

(R(ξ))−1/2+

(ξ−αk)2

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠

+iV′(ξ)π

⎞⎠dξ

N+1∑m=1

(1

b m−1−αk+ 1

am−αk

)+∫

J

(R(ξ))−1/2+

(ξ−αk)

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

×⎛⎝1

8

(N+1∑m=1

(1

b m−1−αk+ 1

am−αk

))2

− 1

4

(N+1∑m=1

(1

(bm−1−αk)2+ 1

(am−αk)2

))2⎞⎠⎞⎠ ,


∫JξψV(ξ) dξ = 1

2(ℵ /n)∫

J

ξN+2

(R(ξ))1/2+

⎛⎝2iπ

⎛⎝ (κnk − 1)

n(ξ − αk)+

s−1∑q=1

κnkkq

n(ξ − αpq )

⎞⎠

+ iV′(ξ)π

⎞⎠dξ − 1

2

N+1∑j=1

(bj−1 + aj),

∫Jξ2ψV(ξ) dξ = 1

2(ℵ /n)

⎛⎝∫

J

ξN+3

(R(ξ))1/2+

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠

+ iV′(ξ)π

⎞⎠dξ + 1

2

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠∫

J

ξN+2

(R(ξ))1/2+

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

⎞⎠

− 1

4

N+1∑j=1

(b2

j−1+a2j

)+ 1

8

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠

2

,

∫Jξ3ψV(ξ) dξ = 1

2(ℵ /n)

⎛⎝∫

J

ξN+4

(R(ξ))1/2+

⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠

+ iV′(ξ)π

⎞⎠dξ − 1

2

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠∫

J

ξN+3

(R(ξ))1/2+

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

−⎛⎜⎝1

4

N+1∑j=1

(b 2j−1+a2

j )−1

8

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠

2⎞⎟⎠∫

J

ξN+2

(R(ξ))1/2+

×⎛⎝2iπ

⎛⎝ (κnk−1)

n(ξ−αk)+

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠+ iV′(ξ)

π

⎞⎠dξ

⎞⎠

− 1

8 · 3!

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠

3

− 1

3!N+1∑j=1

(b 3

j−1+a3j

)

+ 1

8

⎛⎝N+1∑

j=1

(bj−1+aj)

⎞⎠(

N+1∑m=1

(b2m−1+a2

m)

),

where ℵ :=((n−1)K+k)/n. Note that, for n∈N and k=1, 2, . . . ,K, all of the‘moment’ integrals above are real valued (since, for ξ ∈ J, with J ∩ {α1, α2, . . . , αK}=∅,(R(ξ))1/2+ = i(|R(ξ)|)1/2∈ iR) and bounded (since, for j=1, 2, . . . , N+1,(R(ξ))1/2=ξ↓bj−1 O((ξ−bj−1)

1/2) and (R(ξ))1/2=ξ↑aj O((a j−ξ)1/2), that is, there


are integrable singularities at the end-points of the support of the equilibriummeasure). �

Lemma 3.8 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy con-ditions (6)–(8). Suppose, furthermore, that V is regular. For n∈N andk=1, 2, . . . ,K, let the associated equilibrium measure, μV , and its sup-port, supp(μV)=: J=∪N+1

j=1 [bj−1, aj] (⊂ R \ {α1, α2, . . . , αK}), be as described inLemma 3.7, and let there exist � : N × {1, 2, . . . ,K}→R, the associated varia-tional constant, such that

2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)

dμV(ξ)−2

(κnk−1

n

)ln|z−αk|

−2s−1∑q=1

κnkkq

nln|z−αpq | + 2

s−1∑q=1

κnkkq

nln|αpq −αk|−V(z)−�=0, z∈ J,

(81)

2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)

dμV(ξ)−2

(κnk−1

n

)ln|z−αk|

−2s−1∑q=1

κnkkq

nln|z−αpq | + 2

s−1∑q=1

κnkkq

nln|αpq −αk|−V(z)−��0, z∈R \ J,

where, for V regular, the inequality in the second of (81) is strict. Then, forn∈N and k=1, 2, . . . ,K : (i) g+(z)+g−(z)−P+

0 −P−0 −V(z)−�=0, z∈ J, where

P±0 are defined in Lemma 3.4, and g±(z) := limε↓0 g(z±iε); (ii) g+(z) + g−(z)−

P+0 −P−

0 −V(z)−��0, z∈R \ J, where equality holds for at most a finite numberof points, and, for V regular, the inequality is strict; (iii)

g+(z)−g−(z)+P−0 −P+

0 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

2π iG�(z), z∈(bj−1, aj), j=1, 2, . . . , N+1,2π iG�(z), z∈(ai,bi), i=1, 2, . . . , N,2π iG�(z), z∈(aN+1,+∞),2π iG�(z), z∈(−∞,b 0),

where

G�(z) :=((n−1)K+k

n

)∫ aN+1

zψV(ξ) dξ−

((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z),

G�(z) :=((n−1)K+k

n

)∫ aN+1

bi

ψV(ξ) dξ−((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z),


G�(z) := −((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n

−(

κnk−1

n

)1R<αk(z),

G�(z) :=((n−1)K+k

n

)−((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ −∑

q∈Δ(k;z)

κnkkq

n

+∑

q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z),

with Δ2(k)={ j∈{1, 2, . . . , s−1}; αpj>αk}, and Δ(k; z)={ j∈{1, 2, . . . , s−1};z<αpj}; and (iv)

i(

g+(z)−g−(z)+P−0 −P+

0 +2π i((n−1)K+k

n

)1R<αk(z)+2π i

∑q∈Δ(k;z)

κnkkq

n

)′

=

⎧⎪⎨⎪⎩

2π

((n−1)K+k

n

)ψV(z)�0, z∈ J,

0, z∈R \ J,

where ′′ denotes differentiation with respect to z, and, for V regular, equalityholds for at most a finite number of points.

Proof For n∈N and k=1, 2, . . . ,K, set supp(μV)=: J=∪N+1j=1 J j, where Jj=

[bj−1, aj] (the jth band), with (cf. Lemma 3.7) N∈Z+0 and finite, Jj ∩ {α1,

α2, . . . , αK}=∅, j=1, 2, . . . , N+1, Ji ∩ J j=∅, i �= j∈{1, 2, . . . , N+1},and −∞<b0<a1<b1<a2< · · ·<bN<aN+1<+∞, and {bj−1, aj}N+1

j=1 satisfythe locally solvable system of 2(N+1) real moment equations given initem (1)(1)(1) of Lemma 3.7. Present R as follows: R=(−∞,b0) ∪ (aN+1,+∞)∪(∪N+1

j=1 (aj,bj)) ∪ J. Consider the following cases: (1)(1)(1) z∈ J j, j=1, 2, . . . , N+1;

(2)(2)(2) z∈(ai,bi), i=1, 2, . . . , N; (3)(3)(3) z∈(aN+1,+∞); and (4)(4)(4) z∈(−∞,b0).

(1)(1)(1) Recall the definition of g(z) given in Lemma 3.4:

g(z) :=∫

Jln

⎛⎝(z−ξ) 1

n

((z−ξ)

(z−αk)(ξ−αk)

)κnk−1n

×s−1∏q=1

((z−ξ)


)κnkkqn

⎞⎠ψV(ξ)dξ,


where the representation (cf. Lemma 3.7) dμV(x)=ψV(x)dx, x∈ J, was used.For z∈ J j=[bj−1, aj], j=1, 2, . . . , N+1, since J j ∩ {α1, α2, . . . , αK}=∅, oneshows from the above expression for g(z) that

g±(z) =((n−1)K+k

n

)∫J

ln(|z−ξ |)ψV(ξ)dξ

−(

κnk−1

n

)∫J

ln(|ξ−αk|)ψV(ξ)dξ − iπ(

κnk−1

n

)∫J∩R<αk

ψV(ξ)dξ

−s−1∑q=1

κnkkq

n

∫J

ln(|ξ−αpq |)ψV(ξ)dξ − iπs−1∑q=1

κnkkq

n

∫J∩R<αpq

ψV(ξ)dξ

−s−1∑q=1

κnkkq

nln|z−αpq | ± iπ

((n−1)K+k

n

)∫ aN+1

zψV(ξ)dξ

∓ iπ∑

q∈Δj(k;z)

κnkkq

n−(

κnk−1

n

)(ln|z−αk|±iπ1R<αk

(z)),

where g±(z) := limε↓0 g(z±iε), Δj(k; z) :={i∈{1, 2, . . . , s−1}; αpi>z∈ J j}, and,since card(Δj(k; z)) is finite, the sum

∑q∈Δj(k;z) κnkkq

/n is finite for everyordered three-tuple (n,k,q)∈N × {1, 2, . . . ,K} ×Δj(k; z). (Note: the setΔ j(k; z) accounts for the fact that some of the real poles may lieto the right of J j�z, j=1, 2, . . . , N+1, in which case Δj(k; z) �=∅ andcard(Δj(k; z)) �=0; if, however, there are no real poles to the right ofJ j�z, j=1, 2, . . . , N+1, then Δj(k; z)=∅ and card(Δj(k; z))=0, in whichcase

∑q∈Δj(k;z) κnkkq

/n :=0.) Via the definition of P±0 given in Lemma 3.4

and the above expression for g±(z), one arrives at, for n∈N andk=1, 2, . . . ,K,

1

2π i

(g+(z)− g−(z)+ P−

0 − P+0

) =((n−1)K+k

n

)∫ aN+1

zψV(ξ)dξ

−((n−1)K+k

n

)∫J∩R>αk

ψV(ξ)dξ

−∑

q∈Δj(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n

−(

κnk − 1

n

)1R<αk(z),

where Δ2(k) :={i∈{1, 2, . . . , s−1}; αpi>αk}, which shows that g+(z)−g−(z)+P−

0 −P+0 ∈ iR and Re(g+(z)−g−(z)+P−

0 −P+0 )=0; moreover, via the


Fundamental Theorem of Calculus and the fact that ψV(x)�0, x∈ J, oneshows that, for z∈ J j, j=1, 2, . . . , N+1,

i(

g+(z)−g−(z)+P−0 −P+

0 +2π i(

κnk−1

n

)1R<αk(z)+2π i

∑q∈Δj(k;z)

κnkkq

n

)′′

= 2π

((n−1)K+k

n

)ψV(z)�0.

Furthermore, using the first of (the variational) conditions (81), one showsthat, for n∈N and k=1, 2, . . . ,K,

g+(z)+g−(z)−P−0 −P+

0 −V(z)−� = 2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)ψV(ξ) dξ

+ 2s−1∑q=1

κnkkq

nln|αpq −αk| − 2

s−1∑q=1

κnkkq

nln|z − αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�=0, (82)

which gives the formula for the variational constant � : N × {1, 2, . . . ,K}→R, which is the same [63] (see, also, Section 7 of [62]) for each compactinterval J j, j=1, 2, . . . , N+1; in particular,

� = 2

((n − 1)K + k

n

)∫J

ln

⎛⎜⎝∣∣∣∣∣∣∣

1

2(bN + aN+1)− ξξ − αk

∣∣∣∣∣∣∣

⎞⎟⎠ψV(ξ)dξ

+ 2s−1∑q=1

κnkkq

nln|αpq − αk| − 2

s−1∑q=1

κnkkq

nln

∣∣∣∣12 (bN + aN+1)− αpq

∣∣∣∣

− 2

(κnk − 1

n

)ln

∣∣∣∣12 (bN + aN+1)− αk

∣∣∣∣− V(

1

2(bN + aN+1)

).

(2)(2)(2) For n∈N and k=1, 2, . . . ,K, one shows that, for z∈(aj,bj), j =1, 2, . . . , N,

g±(z) =((n−1)K+k

n

)∫J

ln(|z−ξ |)ψV(ξ) dξ

−(

κnk−1

n

)∫J

ln(|ξ−αk|)ψV(ξ) dξ − iπ(

κnk−1

n

)∫J∩R<αk

ψV(ξ) dξ

−s−1∑q=1

κnkkq

n

∫J

ln(|ξ−αpq |)ψV(ξ) dξ − iπs−1∑q=1

κnkkq

n

∫J∩R<αpq

ψV(ξ) dξ


−s−1∑q=1

κnkkq

nln|z−αpq | ± iπ

((n−1)K+k

n

)∫ aN+1

bj

ψV(ξ)dξ

∓ iπ∑

q∈Δj(k;z)

κnkkq

n−(

κnk−1

n


(z)),

whence, via the definition of P±0 given in Lemma 3.4, one arrives at

1

2π i

(g+(z)−g−(z)+P−

0 −P+0

) =((n−1)K+k

n

)∫ aN+1

bj

ψV(ξ)dξ

−((n−1)K+k

n

)∫J∩R>αk

ψV(ξ)dξ

−∑

q∈Δj(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n

−(

κnk−1

n

)1R<αk(z),

which shows that g+(z)−g−(z)+P−0 −P+

0 = i const ., where const .∈R, andRe(g+(z)−g−(z)+P−

0 −P+0 )=0; moreover, for z∈(aj,bj), j=1, 2, . . . , N,

i

⎛⎝g+(z)−g−(z)+P−

0 −P+0 +2π i

(κnk−1

n

)1R<αk(z)+2π i

∑q∈�j(k;z)

κnkkq

n

⎞⎠

′=0.

Furthermore, for n∈N and k=1, 2, . . . ,K, one shows that, for z∈(aj,bj), j=1, 2, . . . , N,

g+(z)+g−(z)−P−0 −P+

0 −V(z)−� = 2

((n−1)K+k

n

)∫J


− 2

((n−1)K+k

n

)∫J

ln(|ξ−αk|)ψV(ξ) dξ

+ 2s−1∑q=1

κnkkq


s−1∑q=1

κnkkq

nln|z−αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�. (83)

Recalling that (cf. the proof of Lemma 3.7) H : L 2M2(C)(∗)→L 2

M2(C)(∗), f →

(H f )(z) := limε↓0

∫|z−ξ |>ε

f (ξ)z−ξ

dξπ

(the Hilbert transform of f ), one shows that, for

z∈(aj,bj), j=1, 2, . . . , N,∫J

ln(|z−ξ |)ψV(ξ)dξ=π∫ z

aj

(H ψV)(ξ)dξ+∫

Jln(|aj−ξ |)ψV(ξ)dξ.


The problem, however, is that there may be real poles between aj andz∈(aj,bj), in which case, the integral

∫ zaj(H ψV)(ξ)dξ has to handled care-

fully. Towards this end, and for z∈(aj,bj), j=1, 2, . . . , N, let Δ�

j :={αpq , q∈{1, 2, . . . , s}; aj<αpq<z}, j=1, 2, . . . , N, with card(Δ�

j )=:K�

j , and where, forsimplicity, the αpq ’s are enumerated so that aj<αp1<αp2< · · ·<αp

K�

j

<z: if

Δ�

j =∅, then K�

j =0; otherwise, if Δ�

j �=∅, then K�

j �=0 (and finite). ForΔ

�

j �=∅, j=1, 2, . . . , N, set, for αpq ∈Δ�

j , q∈{1, 2, . . . ,K�

j }, U�

δ�

q(αpq) :={x∈

R; |x−αpq |<δ�q }, where δ�q is some arbitrarily fixed, sufficiently small positivereal number chosen so that, for i �=q∈{1, 2, . . . ,K�

j }, U�

δ�

i(αpi) ∩ U

�

δ�

q(αpq)=∅.

With the understanding that, if Δ�

j =∅ (thus K�

j =0), then ∪0q=1U

�

δ�

q(αpq) :=∅,

one writes, for z∈(aj,bj), j=1, 2, . . . , N,

∫ z

aj

(H ψV)(ξ)dξ =⎛⎝∫(aj,z)\∪

K�

jq=1 U

�

δ�

q(αpq )

+∫∪K�

jq=1 U

�

δ�

q(αpq )

⎞⎠ (H ψV)(ξ)dξ

=⎛⎜⎝

K�

j∑q=1

∫U

�

δ�

q(αpq )

+K�

j −1∑i=1

∫ αpi+1−δ�i+1

αpi+δ�i+∫ αp1−δ�1

aj

+∫ z

αpK�

j

+δ�K�

j

⎞⎟⎠

× (H ψV)(ξ)dξ

=⎛⎜⎝

K�

j∑q=1

∫U

�

δ�

q(αpq )

+K�

j −1∑i=1

∫ αpi+1−δ�i+1

αpi+δ�i+∫ αp1−δ�1

aj

+∫ z

αpK�

j

+δ�K�

j

⎞⎟⎠

× limε↓0

∫{ω∈J; |ξ−ω|>ε}

ψV(ω)

ξ−ωdωπ

:

now, setting ε j :=min{ε,maxq=1,2,...,K�

j{δ�q }}, j=1, 2, . . . , N, and considering

the ε j→0 limit, one evaluates, via Fubini’s theorem, the above integrals (givingrise to integrable logarithmic singularities) to arrive at

⎛⎜⎝∫ αp1

aj

+K�

j −1∑i=1

∫ αpi+1

αpi

+∫ z

αpK�

j

⎞⎟⎠ (H ψV)(ξ)dξ=

∫ z

aj

(H ψV)(ξ)dξ ;

whence, for z∈(aj,bj), j=1, 2, . . . , N, it follows that the relation∫

J ln(|z−ξ |)ψV(ξ)dξ=π

∫ zaj(H ψV)(ξ)dξ+

∫J ln(|aj−ξ |)ψV(ξ)dξ holds true. Substitut-


ing the latter relation into (83), one shows that, for n∈N and k=1, 2, . . . ,K,with z∈(aj,bj), j=1, 2, . . . , N,

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� = 2π

((n−1)K+k

n

)∫ z

aj

(H ψV)(ξ) dξ

+2

((n−1)K+k

n

)∫J

ln(|aj−ξ |)ψV(ξ) dξ

− 2

((n−1)K+k

n

)∫J


+ 2s−1∑q=1

κnkkq


s−1∑q=1

κnkkq

nln|z−αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�

= 2π

((n−1)K+k

n

)∫ z

aj

(H ψV)(ξ) dξ

+ 2

((n−1)K+k

n

)∫J

ln(|aj−ξ |)ψV(ξ) dξ

− 2

((n−1)K+k

n

)∫J


+ 2s−1∑q=1

κnkkq

nln|αpq −αk|−V(aj)− �−

∫ z

aj

V′(ξ) dξ

− 2

(κnk−1

n

)(ln|aj−αk|+

∫ z

aj

1

ξ−αkdξ

)

− 2s−1∑q=1

κnkkq

n

(ln|aj−αpq |+

∫ z

aj

1

ξ−αpq

dξ

)

= 2π

((n−1)K+k

n

)∫ z

aj

(H ψV)(ξ) dξ

−∫ z

aj

V′(ξ) dξ − 2

(κnk−1

n

)∫ z

aj

1

ξ−αkdξ

− 2s−1∑q=1

κnkkq

n

∫ z

aj

1

ξ−αpq

dξ +(

2

((n−1)K+k

n

)

×∫

Jln(|aj−ξ |)ψV(ξ) dξ−2

((n−1)K+k

n

)

×∫

Jln(|ξ−αk|)ψV(ξ) dξ + 2

s−1∑q=1

κnkkq

nln|αpq −αk|


− 2

(κnk−1

n

)ln|aj−αk|−2

s−1∑q=1

κnkkq

nln|aj−αpq |

− V(aj)−�),

which, via (82), simplifies to

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =∫ z

aj

⎛⎝2π

((n−1)K+k

n

)(H ψV)(ξ)−V′(ξ)

− 2(κnk−1)

n(ξ−αk)−2

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠dξ. (84)

From (the proof of) Lemma 3.7, (55), one gets that, for z∈C \ (J ∪{αp1 , . . . , αps−1 , αk}),

((n−1)K+k

n

)(H ψV)(z)= iF(z)+ 1

π

⎛⎝ (κnk−1)

n(z−αk)+

s−1∑q=1

κnkkq

n(z−αpq)

⎞⎠ ,

and, from (the proof of) Lemma 3.7, (65), and the integral representation forhV(z) given in item (2)(2)(2) of Lemma 3.7, one gets that

F(z)= V ′′(z)2π i

− 1

2π i

((n−1)K+k

n

)(R(z))1/2hV(z),

whence, one shows that, for z∈(aj,bj), j=1, 2, . . . , N,

2π

((n−1)K+k

n

)(H ψV)(z) = V ′(z)+ 2(κnk−1)

n(z−αk)+2

s−1∑q=1

κnkkq

n(z−αpq)

−((n−1)K+k

n

)(R(z))1/2hV(z). (85)

(In fact, the latter (85) is true for all z∈C \ (J ∪ {αp1 , . . . , αps−1 , αk}), and notjust for z∈(aj,bj), j=1, 2, . . . , N; see, also, below.) Substituting (85) into (84),one arrives at, for n∈N and k=1, 2, . . . ,K, with z∈(aj,bj), j=1, 2, . . . , N,

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� = −((n−1)K+k

n

)∫ z

aj

(R(ξ))1/2hV(ξ) dξ (<0).

Since (in the regular case) hV : R \ {α1, α2, . . . , αK}→R is real analytic (andclearly not identically zero), it follows that one has equality only at pointsz∈∪N

j=1(aj,bj) for which hV(z)=0, which are finitely denumerable, that is,

card({z∈∪Nj=1(aj,bj); hV(z)=0})<∞. (Note that, for z∈∪N

j=1(aj,bj), (R(z))1/2+ =

(R(z))1/2− =(R(z))1/2.) One also shows, using (83), that, for n∈N and k=1,2, . . . ,K, ξ ∈ J, z∈(aj,bj), j=1, 2, . . . , N, and |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|%min{min j=1,2,...,N+1{||bj−1−aj|−αk|},minq=1,2,...,s−1{|αpq−αk|}}), via the


expansions 1(z−αk)−(ξ−αk)

=−∑lj=0

(z−αk)j

(ξ−αk) j+1 + (z−αk)l+1

(ξ−αk)l+1(z−ξ) , l∈Z+0 , and ln(1−∗∗∗)=

−∑∞m=1

∗∗∗m

m , |∗∗∗|%1, since J ∩ {α1, α2, . . . , αK}=∅,

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =z→αk

−(

V(z)−(

κnk−1

n

)ln(|z − αk|−2+1)

)

+O(1),

which, via condition (8) and the fact that V: R \ {α1, α2, . . . , αK}→R isreal analytic, shows that g+(z)+g−(z)−P+

0 −P−0 −V(z)−�<0, z∈(aj,bj), j=

1, 2, . . . , N.

(3)(3)(3) For n∈N and k=1, 2, . . . ,K, one shows that, for z∈(aN+1,+∞),g±(z) =

((n−1)K+k

n

)∫J


−(

κnk−1

n

)∫J


κnk−1

n

)∫J∩R<αk

ψV(ξ) dξ

−s−1∑q=1

κnkkq

n

∫J


κnkkq

n

∫J∩R<αpq

ψV(ξ) dξ

−s−1∑q=1

κnkkq

nln|z−αpq |∓ iπ

∑q∈Δ(k;z)

κnkkq

n−(

κnk−1

n

)(ln|z−αk|

± iπ1R<αk(z)),

where Δ(k; z) :={ j∈{1, 2, . . . , s−1}; αpj>z} (if Δ(k; z)=∅, that is, card(Δ(k;z))=0, then

∑q∈Δ(k;z) κnkkq

/n :=0), whence, via the definition of P±0 given in

Lemma 3.4, one arrives at1

2π i(g+(z)−g−(z)+P−

0 −P+0 ) = −

((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z),


0 = i const ., where const .∈R, and Re(g+(z)−g−(z)+P−

0 −P+0 )=0; moreover,

i

⎛⎝g+(z)−g−(z)+P−

0 −P+0 +2π i

(κnk−1

n

)1R<αk(z)+2π i

∑q∈Δ(k;z)

κnkkq

n

⎞⎠

′=0,

z∈(aN+1,+∞).Furthermore, for n∈N and k=1, 2, . . . ,K, one shows that, forz∈(aN+1,+∞),

g+(z)+g−(z)−P−0 −P+

0 −V(z)−� = 2

((n−1)K+k

n

)∫J



− 2

((n−1)K+k

n

)∫J


+ 2s−1∑q=1

κnkkq


s−1∑q=1

κnkkq

nln|z−αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�. (86)

Following through with the analogue of the (removable logarithmic singular-ity) analysis in case (2)(2)(2) above, one shows that, for z∈(aN+1,+∞),

∫J

ln(|z−ξ |)ψV(ξ)dξ=π∫ z

aN+1

(H ψV)(ξ)dξ+∫

Jln(|aN+1−ξ |)ψV(ξ)dξ,

whence, via (86) and the latter relation, one shows that, for n∈N and k=1, 2, . . . ,K, with z∈(aN+1,+∞),

g+(z)+ g−(z)− P+0 −P−

0 −V(z)−� = 2π

((n−1)K+k

n

)∫ z

aN+1

(H ψV)(ξ) dξ

+ 2

((n−1)K+k

n

)∫J

ln(|aN+1−ξ |)ψV(ξ) dξ

− 2

((n−1)K+k

n

)∫J


+ 2s−1∑q=1

κnkkq


s−1∑q=1

κnkkq

nln|z−αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�

= 2π

((n−1)K+k

n

)∫ z

aN+1

(H ψV)(ξ) dξ

−∫ z

aN+1

V′(ξ) dξ − 2

(κnk−1

n

)∫ z

aN+1

1

ξ−αkdξ

− 2s−1∑q=1

κnkkq

n

∫ z

aN+1

1

ξ−αpq

dξ

+(

2

((n−1)K+k

n

)∫J

ln(|aN+1−ξ |)ψV(ξ) dξ

− 2

((n−1)K+k

n

)∫J



+ 2s−1∑q=1

κnkkq


(κnk−1

n

)

× ln|aN+1−αk| − 2s−1∑q=1

κnkkq

nln|aN+1−αpq |

− V(aN+1)−�),

which, via (82), simplifies to

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =∫ z

aN+1

(2π

((n−1)K+k

n

)(H ψV)(ξ)−V′(ξ)

− 2(κnk−1)

n(ξ−αk)− 2

s−1∑q=1

κnkkq

n(ξ−αpq )

⎞⎠ dξ. (87)

Via (85), the latter equation simplifies to, for z∈(aN+1,+∞),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� = −((n−1)K+k

n

)∫ z

aN+1

(R(ξ))1/2hV(ξ) dξ (<0).

Since (in the regular case) hV : R \ {α1, α2, . . . , αK}→R is real analytic (and clearlynot identically zero), it follows that one has equality only at points z∈(aN+1,+∞)for which hV(z)=0, which are finitely denumerable, that is, card({z∈(aN+1,+∞); hV(z)=0})<∞. Furthermore, recalling that, for n∈N and k=1, 2, . . . ,K,μV ∈M1(R), in particular,

∫J(ξ−αk)

−mψV(ξ) dξ <∞ and∫

J ξmψV(ξ) dξ <∞, m∈N,

with∫

J ψV(ξ) dξ=1, one derives the following asymptotic expansions:

(i) for ξ ∈ J, z∈(aN+1,+∞), and |(z−αk)/(ξ−αk)|%1 (e.g., 0< |z−αk|%min{min j=1,2,...,N+1{||bj−1−aj|−αk|},minq=1,2,...,s−1{|αpq −αk|}}), via the expan-

sions 1(z−αk)−(ξ−αk)

= −∑lj=0

(z−αk)j

(ξ−αk)j+1 + (z−αk)

l+1

(ξ−αk)l+1(z−ξ) , l ∈ Z

+0 , and ln(1−∗∗∗)=

−∑∞m=1

∗∗∗m

m , |∗∗∗|%1, and (86),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =z→αk

−(

V(z)−(

κnk−1

n

)ln(|z−αk|−2+1)

)

+O(1),

which, via condition (8), shows that, for n∈N and k=1, 2, . . . ,K, g+(z)+g−(z)−P+

0 −P−0 −V(z)−�<0, z∈(aN+1,+∞);

(ii) for ξ ∈ J, z ∈ (aN+1,+∞), and |ξ/z| % 1 (e.g., |z| & max{max j=1,2,...,N+1{|bj−1−aj|},maxq=1,2,...,s{|αpq |}}), via the expansions 1

ξ−z =−∑lj=0

ξ j

z j+1 + ξ l+1

zl+1(ξ−z), l∈Z

+0 ,

and ln(1−∗∗∗)=−∑∞m=1

∗∗∗m

m , |∗∗∗|%1, and (86),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =z→+∞ −

(V(z)− 1

n ln(z2+1))

+O(1),


0 −P−0 −V(z)−�<0, z∈(aN+1,+∞).


(4)(4)(4) For n∈N and k=1, 2, . . . ,K, one shows that, for z∈(−∞, b0),

g±(z) =((n−1)K+k

n

)∫J


−(

κnk−1

n

)∫J


κnk−1

n

)∫J∩R<αk

ψV(ξ) dξ

−s−1∑q=1

κnkkq

n

∫J


κnkkq

n

∫J∩R<αpq

ψV(ξ) dξ

−s−1∑q=1

κnkkq

nln|z−αpq |±iπ

((n−1)K+k

n

)∓ iπ

∑q∈Δ(k;z)

κnkkq

n

−(

κnk−1

n


(z))

(if Δ(k; z)=∅, that is, card(Δ(k; z))=0, then∑

q∈Δ(k;z) κnkkq/n :=0), whence, via the

definition of P±0 given in Lemma 3.4, one arrives at

1

2π i

(g+(z)−g−(z)+P−

0 −P+0

) =((n−1)K+k

n

)−((n−1)K+k

n

)∫J∩R>αk

ψV(ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z),


0 = i const ., where const .∈R, and Re(g+(z)−g−(z)+P−

0 −P+0 )=0; moreover,

i(

g+(z)−g−(z)+P−0 −P+

0 +2π i(

κnk−1

n

)1R<αk(z)+2π i

∑q∈Δ(k;z)

κnkkq

n

)′′=0,

z∈(−∞, b0).

Furthermore, for n∈N and k=1, 2, . . . ,K, one shows that, for z∈(−∞, b0),

g+(z)+g−(z)−P−0 −P+

0 −V(z)−� = 2

((n−1)K+k

n

)∫J

ln

(∣∣∣∣ z−ξξ−αk

∣∣∣∣)ψV(ξ) dξ

+ 2s−1∑q=1

κnkkq


s−1∑q=1

κnkkq

nln|z−αpq |

− 2

(κnk−1

n

)ln|z−αk|−V(z)−�. (88)

Following through with the analogue of the (removable logarithmic singularity) analysisin case (2)(2)(2) above, one shows that, for z∈(−∞, b0),∫

Jln(|z−ξ |)ψV(ξ) dξ=−π

∫ b0

z(H ψV)(ξ) dξ+

∫J

ln(|b0−ξ |)ψV(ξ) dξ,


whence, via (88), (82), and (85), and this latter relation, one shows that, for n∈N andk=1, 2, . . . ,K, with z∈(−∞, b0),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =((n−1)K+k

n

)∫ b0

z(R(ξ))1/2hV(ξ) dξ (<0) :

equality holds only on the set {z∈(−∞, b0); hV(z)=0}, for which card({z∈(−∞, b0);hV(z)=0})<∞. Furthermore, as in case (3)(3)(3) above, one derives the following asymptoticexpansions:

(i) for ξ ∈ J, z ∈ (−∞, b 0), and |(z − αk)/(ξ − αk)| % 1 (e.g., 0 < |z − αk| % min{min j=1,2,...,N+1{||bj−1−aj|−αk|},minq=1,2,...,s−1{|αpq −αk|}}), via the expansions

1(z−αk)−(ξ−αk)

=−∑lj=0

(z−αk)j

(ξ−αk)j+1 + (z−αk)

l+1

(ξ−αk)l+1(z−ξ) , l∈Z

+0 , and ln(1−∗∗∗)=−∑∞

m=1∗∗∗m

m ,

|∗∗∗|%1, and (88),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =z→αk

−(

V(z)−(

κnk−1

n

)ln(|z−αk|−2+1)

)

+O(1),


0 −P−0 −V(z)−�<0, z∈(−∞, b0);

(ii) for ξ ∈ J, z∈(−∞, b0), and |ξ/z|%1 (e.g., |z|&max{max j=1,2,...,N+1{|bj−1−aj|},maxq=1,2,...,s{|αpq |}}), via the expansions 1

ξ−z =−∑lj=0

ξ j

z j+1 + ξ l+1

zl+1(ξ−z), l∈Z

+0 , and

ln(1−∗∗∗)=−∑∞m=1

∗∗∗m

m , |∗∗∗|%1, and (88),

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =z→−∞ −

(V(z)− 1

nln(z2+1)

)+O(1),


0 −P−0 −V(z)−�<0, z∈(−∞, b 0).

This concludes the proof. ��

4 The Monic FPC ORF Family of Model RHPs

In this section, the monic FPC ORF family of K auxiliary matrix RHPsformulated in Lemma 3.4 is augmented, by means of a sequence of contourdeformations and transformations à la Deift–Venakides–Zhou [57–59], intosimpler, ‘model’ matrix RHPs which, as n→∞, can be solved explicitly interms of Riemann theta functions (associated with the underlying family ofK genus-N hyperelliptic Riemann surfaces) and Airy functions.

Lemma 4.1 Let the external field V: R \ {α1, α2, . . . , αK}→R satisfy condi-tions (6)–(8). Suppose, furthermore, that V is regular. For n∈N and k=1, 2, . . . ,K, let the associated equilibrium measure, μV , and its compact sup-port, supp(μV)=: J=∪N+1

j=1 [bj−1, a j] (⊂ R \ {α1, α2, . . . , αK}), be as describedin Lemma 3.7, and, along with the corresponding variational constant, �,satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let the


corresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. For n∈N

and k=1, 2, . . . ,K, let Y : N × {1, 2, . . . ,K} × C \ R→SL2(C) solve the RHP(Y(n,k; z) :=Y(z),V(z),R) formulated in Lemma 3.4, and set

M(z)={

Y(z)E−σ3 , z∈C+,Y(z)Eσ3 , z∈C−,

where

E :=exp

(iπ((n−1)K+k)

∫J∩R>αk

ψV(ξ)dξ

).

Then, for n∈N and k=1, 2, . . . ,K, M : N × {1, 2, . . . ,K} × C \ R→SL2(C)

solves the following RHP: (i) M(n,k; z) :=M(z) is holomorphic for z∈C \ R;(ii) the boundary values M±(z) := limε↓0 M(z±iε) satisfy the jump condition

M+(z)=M−(z)VM(z), z∈R,

where, for j=1, 2, . . . , N+1 and i=1, 2, . . . , N,

VM : N × {1, 2, . . . ,K} × R → GL2(C), (n,k, z) →VM(n,k; z) := VM(z) =⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

(e−2π i((n−1)K+k)

∫ aN+1z ψV (ξ) dξ 1

0 e2π i((n−1)K+k)∫ aN+1

z ψV (ξ) dξ

), z∈(bj−1, a j),

⎛⎝e

−2π i((n−1)K+k)∫ aN+1

biψV (ξ) dξ en(g+(z)+g−(z)−P+

0 −P−0 −V(z)−�)

0 e2π i((n−1)K+k)

∫ aN+1bi

ψV (ξ) dξ

⎞⎠ , z∈(ai,bi),

I+en(g+(z)+g−(z)−P+0 −P−

0 −V(z)−�)σ+, z∈(−∞,b 0) ∪ (aN+1,+∞),

with g(z), g±(z), and P±0 defined in Lemma 3.4,

±Re

(i∫ aN+1

zψV(ξ)dξ

)>0, z∈C± ∩ (∪N+1

j=1 U j),

where, for j=1, 2, . . . , N+1, U j :={z∈C \ {α1, α2, . . . , αK}; infq∈(b j−1,a j)|z−q|<r j}, r j∈(0, 1), with the r j’s chosen (small enough) so that Ui ∩U j=∅, i �= j∈{1, 2, . . . , N+1}, and g+(z)+g−(z)−P+

0 −P−0 −V(z)−�<0, z∈

(−∞,b0) ∪ (aN+1,+∞) ∪ (∪Ni=1(ai,bi)); (iii) M(z)=C+�z→αk (I+O(z−αk))E

−σ3

and M(z)=C−�z→αk (I+O(z−αk))Eσ3; (iv) M(z)=C±�z→∞O(1); and (v) for q∈

{1, 2, . . . , s−1}, M(z)=C±�z→αpqO(1).

Proof For n∈N and k=1, 2, . . . ,K, via the definition of M(z) in termsof Y(z) stated in the Lemma, item (i) of the Lemma is simply a re-statement of item (i) of Lemma 3.4. Write R=(−∞,b0) ∪ (aN+1,+∞) ∪(∪N

i=1(ai,bi)) ∪ (∪N+1j=1 [bj−1, a j]). Recall from the proof of Lemma 3.8 that,


for V: R\ {α1, α2, . . . , αK}→R, μV , and � as described therein: (1) for j=1, 2, . . . , N+1 and i=1, 2, . . . , N,

g+(z)−g−(z)+P−0 −P+

0

2π i=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

((n−1)K+k

n

)∫ aN+1

zψV (ξ) dξ −

((n−1)K+k

n

)∫J∩R

>αk

ψV (ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z), z∈[bj−1, a j],

((n−1)K+k

n

)∫ aN+1

bi

ψV (ξ) dξ −((n−1)K+k

n

)∫J∩R

>αk

ψV (ξ) dξ

−∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z), z∈(ai,bi),

−((n−1)K+k

n

)∫J∩R

>αk

ψV (ξ) dξ −∑

q∈Δ(k;z)

κnkkq

n+

∑q∈Δ2(k)

κnkkq

n

−(

κnk−1

n

)1R<αk(z), z∈(aN+1,+∞),

((n−1)K+k

n

)−((n−1)K+k

n

)∫J∩R

>αk

ψV (ξ) dξ −∑

q∈Δ(k;z)

κnkkq

n

+∑

q∈Δ2(k)

κnkkq

n−(

κnk−1

n

)1R<αk(z), z∈(−∞,b0),

whereΔ(k; z) and Δ2(k) are defined in Lemma 3.8; and (2) for j=1, 2, . . . , N+1 and i=1, 2, . . . , N,

g+(z)+g−(z)−P+0 −P−

0 −V(z)−� =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

0, z∈[bj−1, a j],−((n − 1)K + k

n

)∫ za j(R(ξ))1/2hV(ξ) dξ (<0), z∈(ai,bi),

−((n − 1)K + k

n

)∫ zaN+1(R(ξ))1/2hV(ξ) dξ (<0), z∈(aN+1,+∞),

((n − 1)K + k

n

)∫ b0z (R(ξ))

1/2hV(ξ) dξ (<0), z∈(−∞,b0).

Recall, also, the formula for the ‘jump matrix’ given in Lemma 3.4:

V(z)=(

e−n(g+(z)−g−(z)+P−0 −P+

0 ) en(g+(z)+g−(z)−P+0 −P−

0 −V(z)−�)

0 en(g+(z)−g−(z)+P−0 −P+

0 )

).

Presenting R as above, and recalling that, for n∈N and k=1, 2, . . . ,K, κnk∈N and κnkkq

∈Z+0 , q∈{1, 2, . . . , s−1}, via the definition of M(z) in terms

of Y(z) stated in the Lemma, one obtains the formula for VM(z) statedin the Lemma, thus item (ii); furthermore, via the definition of M(z) interms of Y(z) stated in the Lemma, items (iii)–(v) are the corresponding re-statements of the respective items (iii)–(v) of Lemma 3.4. It remains, therefore,


to establish that ±Re(i∫ aN+1

z ψV(ξ)dξ)>0 for z∈C± ∩ (∪N+1j=1 U j), where, for

j=1, 2, . . . , N+1, U j :={z∈C \ {α1, α2, . . . , αK}; infq∈(bj−1,a j)|z−q|<r j}, with(0, 1)�r j, j=1, 2, . . . , N+1, some arbitrarily fixed, sufficiently small positivereal numbers chosen so that, for i �= j∈{1, 2, . . . , N+1}, Ui ∩ U j=∅. Recallfrom the proof of Lemma 3.4 that g(z) is uniformly Lipschitz continuous inC±; furthermore, via the Cauchy–Riemann conditions, item (iv) of Lemma 3.4,that is, i(g+(z)−g−(z)+P−

0 −P+0 +2π i

(κnk−1

n

)1R<αk(z)+2π i

∑q∈Δ(k;z)

κnkkq

n )′�0,

z∈∪N+1j=1 [bj−1, a j], implies that the quantity g+(z)−g−(z)+P−

0 −P+0 (since,

by piecewise continuity, exp(2π i(κnk−1)1R<αk(z))=exp(2π i

∑q∈Δ(k;z) κnkkq

)=1) has an analytic continuation, G(z), to an open neighbourhood, UV , ofJ, where UV :=∪N+1

j=1 U j, with U j, j=1, 2, . . . , N+1, defined above, with theproperty that ±Re(G(z))>0, z∈C± ∩ UV . ��

Remark 4.1 Recalling that V: R \ {α1, α2, . . . , αK}→R satisfying conditions(6)–(8) is regular, that is, for n∈N and k=1, 2, . . . ,K, hV(z) �=0 for

z∈∪N+1j=1 [bj−1, a j] and the second inequality of (81) is strict, and g+(z)+

g−(z)− P+0 − P−

0 − V(z)− �< 0, z∈ (−∞,b0) ∪ (aN+1,+∞) ∪ (∪Ni=1(ai,bi)), it

follows thatVM(z) =

n→∞{

e−(2π i((n−1)K+k)

∫ aN+1bi

ψV (ξ)dξ)σ3(I+o(1)σ+), z∈(ai, bi), i=1, 2, . . . , N,

I+o(1)σ+, z∈(−∞, b0) ∪ (aN+1,+∞),where, here and below, o(1) denotes terms that are exponentially small (asn→∞). �


j=1 [bj−1, a j] (⊂ R \ {α1, α2, . . . , αK}), be as describedin Lemma 3.7, and, along with the corresponding variational constant, �,satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let thecorresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. For n∈N

and k=1, 2, . . . ,K, let M : N × {1, 2, . . . ,K} × C \ R→SL2(C) solve the RHP(M(n,k; z) :=M(z),VM(z),R) formulated in Lemma 4.1, and let the deformed(and oriented) contour �� :=R ∪ (∪N+1

j=1 (J�j ∪ J�

j )) be as in Fig. 1 below, with

Fig. 1 Oriented and deformed contour �� :=R ∪ (∪N+1j=1 (J

�j ∪ J�

j ))


∪N+1j=1 (Ω

�j ∪Ω�

j ∪ J�j ∪ J�

j ) ⊂ ∪N+1j=1 U j, where U j, j=1, 2, . . . , N+1, are de-

fined in Lemma 4.1. For n∈N and k=1, 2, . . . ,K, set

m�(z)=⎧⎪⎪⎪⎨⎪⎪⎪⎩

M(z), z∈C \(�� ∪ (∪N+1

j=1 ('�j ∪'�

j )))

,

M(z)(

I−e−2π i((n−1)K+k)∫ aN+1

z ψV (ξ) dξ σ−), z∈C+ ∩

(∪N+1

j=1 '�j

),

M(z)(

I+e2π i((n−1)K+k)∫ aN+1

z ψV (ξ) dξ σ−), z∈C− ∩

(∪N+1

j=1 '�j

).

Then, for n∈N and k=1, 2, . . . ,K, m� : N × {1, 2, . . . ,K} × C \��→SL2(C)

solves the following equivalent RHP: (i) m�(n,k; z) :=m�(z) is holomorphicfor z∈C \��; (ii) the boundary values m

�±(z) := limz′ → z∈��

z′ ∈± side of��m�(z′) satisfy the

jump condition

m�+(z)=m

�−(z)v�(z), z∈��,

where, for j=1, 2, . . . , N+1 and i=1, 2, . . . , N,

v� : N × {1, 2, . . . ,K} ×�� → GL2(C), (n,k, z) →v�(n,k; z) :=v�(z) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

iσ2, z∈(bj−1, a j),

I+e−2π i((n−1)K+k)∫ aN+1

z ψV (ξ) dξ σ−, z∈ J�j ,

I+e2π i((n−1)K+k)∫ aN+1

z ψV (ξ) dξ σ−, z∈ J�j ,⎛

⎝e−2π i((n−1)K+k)

∫ aN+1bi

ψV (ξ) dξ en(g+(z)+g−(z)−P+0 −P−

0 −V(z)−�)

0 e2π i((n−1)K+k)

∫ aN+1bi

ψV (ξ) dξ

⎞⎠ , z∈(ai,bi),

I+en(g+(z)+g−(z)−P+0 −P−

0 −V(z)−�)σ+, z∈(−∞,b 0) ∪ (aN+1,+∞),

with Re(i∫ aN+1

z ψV(ξ)dξ)>0 (resp., Re(i∫ aN+1

z ψV(ξ)dξ)<0), z∈Ω�j (resp., z∈

Ω�j ); (iii) for l∈{+,−}, with t(+) :=�, t(−) :=�, r(+) :=−1, and r(−) :=+1,

m�(z) =z→αk

z∈Cl\∪N+1j=1 (J

t(l)j ∪Ωt(l)

j )

(I+O(z−αk))Er(l)σ3;

(iv)

m�(z) =z→∞

z∈Cl\∪N+1j=1 (J

t(l)j ∪Ωt(l)

j )

O(1);

(v) for q∈{1, 2, . . . , s−1},m�(z) =

z→αpq

z∈Cl\∪N+1j=1 (J

t(l)j ∪Ωt(l)

j )

O(1).

Proof For n∈N and k=1, 2, . . . ,K, items (i), (iii), (iv), and (v) in the for-mulation of the RHP for m�(z) follow from the definition of m�(z) in termsof M(z) given in the Lemma and the respective items (i), (iii), (iv), and (v)for the RHP for M(z) stated in Lemma 4.1; it remains, therefore, to verify


item (ii), namely, the formula for v�(z). Recall from item (ii) of Lemma 4.1that, for n∈N, k=1, 2, . . . ,K, and z∈(bj−1, a j), j=1, 2, . . . , N+1, M+(z)=M−(z)VM(z), where VM(z)=exp(−(2π i((n−1)K+k)

∫ aN+1

z ψV(ξ)dξ)σ3)+σ+:noting the matrix factorisation(

e−2π i((n−1)K+k)∫ aN+1

z ψV(ξ) dξ 10 e2π i((n−1)K+k)

∫ aN+1z ψV(ξ) dξ

)=

(1 0

e2π i((n−1)K+k)∫ aN+1

z ψV(ξ) dξ 1

)(0 1−1 0

)(1 0

e−2π i((n−1)K+k)∫ aN+1

z ψV(ξ) dξ 1

),

it follows that, for z∈(bj−1, a j), j=1, 2, . . . , N+1,

m�+(z)

(1 0

−e−2π i((n−1)K+k)∫ aN+1

z ψV(ξ) dξ 1

)

=m�−(z)

(1 0

e2π i((n−1)K+k)∫ aN+1

z ψV(ξ) dξ 1

)iσ2.

It was shown in Lemma 4.1 that, for n∈N and k=1, 2, . . . ,K, ±Re(i∫ aN+1

zψV(ξ)dξ)>0, z∈C± ∩ U j, j=1, 2, . . . , N+1. The terms ±2π i((n−1)K+k)· ∫ aN+1

z ψV(ξ)dξ , which are pure imaginary for z∈R, and corresponding towhich exp(±2π i((n−1)K+k)

∫ aN+1

z ψV(ξ)dξ) are undulatory, are continuedanalytically to C± ∩ (∪N+1

j=1 U j), respectively, corresponding to which exp(∓2π i((n−1)K+k)

∫ aN+1

z ψV(ξ)dξ) are exponentially decreasing as n→∞. Asper the DZ non-linear steepest-descent method [57, 58] (see, also, the ex-tension [59]), one now ‘deforms’ the original (and oriented) contour R

to the deformed, or extended, (and oriented) contour/skeleton �� :=R∪(∪N+1

j=1 (J�j ∪ J�

j )) (Fig. 1) in such a way that the upper (resp., lower) ‘lips’

of the ‘lenses’ J�j (resp., J�

j ), j=1, 2, . . . , N+1, which are the boundaries of

'�j (resp., '�

j ), j=1, 2, . . . , N+1, respectively, lie within the domain of ana-lytic continuation of g+(z)−g−(z)+P−

0 −P+0 (cf. proof of Lemma 4.1), that

is, ∪N+1j=1 ('

�j ∪'�

j ∪ J�j ∪ J�

j ) ⊂ ∪N+1j=1 U j; in particular, each (oriented and

bounded) open interval (bj−1, a j), j=1, 2, . . . , N+1, in the original (and ori-ented) contour R is ‘split’, or branched, into three, and the new (and oriented)contour �� is the old contour (R) together with the (oriented) boundaryof N+1 lens-shaped regions, one region surrounding each (bounded andoriented) open interval (bj−1, a j). Now, recalling the definition of m�(z) interms of M(z) given in the Lemma, and the expression for VM(z) given inLemma 4.1, one arrives at, for n∈N and k=1, 2, . . . ,K, the formula for v�(z)given in item (ii) of the Lemma. ��

Recalling from Lemma 4.1 that, for n∈N and k=1, 2, . . . ,K, g+(z)+g−(z)−P+

0 −P−0 −V(z)−�<0, z∈(−∞,b0) ∪ (aN+1,+∞) ∪ (∪N

i=1(ai,bi)), andRe(i

∫ aN+1

z ψV(ξ)dξ)>0 (resp., Re(i∫ aN+1

z ψV(ξ)dξ)<0), z∈ J�j (resp., z∈ J�

j ),


j=1, 2, . . . , N+1, one arrives at the following large-n behaviour for the jumpmatrix v�(z); for j=1, 2, . . . , N+1 and i=1, 2, . . . , N:

v�(z) =n→∞

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

iσ2, z∈(bj−1, a j),I+O(e−nc|z|)σ−, z∈ J�

j ∪ J�j ,

e−(2π i((n−1)K+k)∫ aN+1

biψV(ξ) dξ)σ3

(I+O(e−nc|z−ai|)σ+

), z∈(ai,bi) \ Uδk(αk),

e−(2π i((n−1)K+k)∫ aN+1

biψV(ξ) dξ)σ3

(I+O(e−nc|z−αk|−1

)σ+), z∈(ai,bi) ∩ Uδk(αk),

I+O(e−nc|z|)σ+, z∈J \ Uδk(αk),I+O(e−nc|z−αk|−1

)σ+, z∈J ∩ Uδk(αk),

where c is some positive constant, J :=(−∞,b 0) ∪ (aN+1,+∞), and, for k=1, 2, . . . ,K, Uδk(αk) :={z∈C; |z−αk|<δk}, where (0, 1)�δk are some arbitrar-ily fixed, sufficiently small positive real numbers chosen so that, ∀ r1 �=r2∈{1, 2, . . . ,K}, Uδr1

(αr1) ∩ Uδr2(αr2)=∅. The above convergences are normal,

that is, uniform in the respective compact subsets.Recall from Lemma 2.56 of [57] that, for an oriented skeleton in C on

which the jump matrix of an RHP is defined, one may always choose toadd or delete a portion of the skeleton on which the jump matrix equalsI without altering the RHP in the operator sense; hence, neglecting thosejumps on �� tending exponentially quickly (as n→∞) to I, and removing thecorresponding oriented skeletons from��, it becomes more or less transparenthow to construct a parametrix (an approximate solution) for the family of KRHPs (m�(n,k; z) :=m�(z), v�(z),��) stated in Lemma 4.2, that is, the large-nsolution of the family of K RHPs for m� : N × {1, 2, . . . ,K} × C \��→SL2(C)

formulated in Lemma 4.2 should be ‘close to’, in some appropriately definedoperator-theoretic sense, the solution of the following family of K ‘limiting’,or ‘model’, RHPs.


j=1 [bj−1, a j] (⊂ R \ {α1, α2, . . . , αK}), be as describedin Lemma 3.7, and, along with the corresponding variational constant, �,satisfy the variational conditions stated in Lemma 3.8, (81); moreover, let thecorresponding conditions (i)–(iv) stated in Lemma 3.8 be valid. Then, forn∈N and k=1, 2, . . . ,K, � : N × {1, 2, . . . ,K} × C \ J→SL2(C), where J :=J ∪ (∪N

i=1(ai,bi)), solves the following RHP: (i) �(n,k; z) :=�(z) is holomorphic


for z∈C \ J; (ii) the boundary values �±(z) := lim z′→z∈Jz′∈± side of J

�(z′) satisfy the jump

condition

�+(z)=�−(z)υ�(z), z∈J,

where

υ� : N × {1, 2, . . . ,K} × J→GL2(C), (n,k, z) →υ�(n,k; z) :=υ�(z)

={

iσ2, z∈(bj−1, a j), j=1, 2, . . . , N+1,

e−(2π i((n−1)K+k)∫ aN+1

biψV(ξ) dξ)σ3 , z∈(ai,bi), i=1, 2, . . . , N;

(iii) �(z)=C+�z→αk (I+O(z−αk))E−σ3 and �(z)=C−�z→αk (I+O(z−αk))E

σ3;(iv) �(z)=C±�z→∞O(1); and (v) for q∈{1, 2, . . . , s−1}, �(z)=C±�z→αpq

O(1).

It is well known that the family of K model RHPs (�(n,k; z) :=�(z),υ�(z), J) formulated in Lemma 4.3 is explicitly solvable in terms of Riemanntheta functions associated with the family of K two-sheeted genus-N hyper-elliptic Riemann surfaces {(y, z); y2=R(z)=∏N+1

j=1 (z−bj−1)(z−a j)}; see, forexample, [73] (see, also, [66]): this will be considered elsewhere.

Acknowledgements K. T.-R. McLaughlin was supported, in part, by National Science Foun-dation Grant Nos. DMS-0200749 and DMS-0451495, as well as a NATO Collaborative LinkageGrant ‘Orthogonal Polynomials: Theory, Applications, and Generalizations’, Ref. No.PST.CLG.979738. A. H. Vartanian was supported, in part, by a College of Charleston (CofC)Summer Research Stipend. X. Zhou was supported, in part, by National Science Foundation GrantNo. DMS-0300844.

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Spectrum of the Lichnerowicz Laplacianon Asymptotically Hyperbolic Surfaces

Erwann Delay

Received: 8 October 2007 / Accepted: 31 July 2008 / Published online: 12 September 2008© Springer Science + Business Media B.V. 2008

Abstract We show that, on any asymptotically hyperbolic surface, the essentialspectrum of the Lichnerowicz Laplacian �L contains the ray

[14 , +∞[

. If more-over the scalar curvature is constant then −2 and 0 are infinite dimensionaleigenvalues. If, in addition, the inequality 〈�u, u〉L2 � 1

4 ||u||2L2 holds for allsmooth compactly supported function u, then there is no other value in thespectrum.

Keywords Asymptotically hyperbolic surfaces · Lichnerowicz Laplacian ·Symmetric 2-tensor · Essential spectrum · Asymptotic behavior

Mathematics Subject Classifications (2000) 35P15 · 58J50 · 47A53

1 Introduction

This article is a complement of the papers [7, 8] where the study of theLichnerowicz Laplacian �L is given in dimension n greater than 2. We referthe reader to those papers for all the motivations. In the preceding papers,the spectrum was only given for n � 3 because of the natural relation to theprescribed Ricci curvature problem. In dimension 2 this study does not appearbecause the corresponding problem is conform. The present paper, firstly givenfor completeness, appears to be particulary interesting because of the quite bigdifferences with the other dimensions.

E. Delay (B)Laboratoire d’analyse non linéaire et géométrie, Faculté des Sciences,33 rue Louis Pasteur, 84000 Avignon, Francee-mail: [email protected]: http://www.math.univ-avignon.fr/Delay

366 E. Delay

For instance on the hyperbolic space, when n � 3 the spectrum of �L ontrace free symmetric two tensors is the ray

[(n − 1)(n − 9)

4, +∞

[.

This spectrum is essentially characterized by non trivial trace free tensors onthe boundary at infinity. In dimension 2 (so 1 at infinity) those tensors do notexist, and the situation is very different.

Also, in dimension two, the cohomology of the manifold appears naturallyin the spectrum. This situation was already noticed by Avez [2, 3] and Buzanca[5, 6].

The principal result is the following

Theorem 1.1 Let (M, g) be an asymptotically hyperbolic surface. The essentialspectrum of �L on trace free symmetric two tensors contains the ray [1/4, +∞[.If moreover g has constant scalar curvature R = −2 then −2 and 0 are also in theessential spectrum. Moreover their eigenspaces are in one to one correspondencewith the space of harmonic one forms respectively in L4 and in L2 (in particularthey are infinite dimensional). Finally, if in addition, as for the hyperbolic plane,for all smooth compactly supported function u, 〈�u, u〉L2 � 1

4 ||u||2L2 , then thespectrum of �L is

{−2} ∪ {0} ∪ [1

4, +∞[.

Along the paper we also obtain some relative results on more generalsurfaces, with or without constant scalar curvature.

2 Definitions, Notations and Conventions

Let M be a smooth, compact surface with boundary ∂ M. Let M := M\∂ M bea non-compact surface without boundary. In our context the boundary ∂ M willplay the role of a conformal boundary at infinity of M. Let g be a Riemannianmetric on M. The manifold (M, g) is conformally compact if there exists on Ma smooth defining function ρ for ∂ M (that is ρ ∈ C∞(M), ρ > 0 on M, ρ = 0on ∂ M and dρ is nowhere vanishing on ∂ M) such that g := ρ2g is a C2,α(M) ∩C∞(M) Riemannian metric on M . We will denote by g the metric inducedon ∂ M. Now if |dρ|g = 1 on ∂ M, it is well known (see [12] for instance) thatg has asymptotically sectional curvature −1 near its boundary at infinity. Inthis case we say that (M, g) is asymptotically hyperbolic. Along the paper, itwill be assumed sometimes than (M, g) has constant scalar curvature : then theasymptotic hyperbolicity enforces the normalisation

R(g) = −2 , (2.1)

where R(g) is the scalar curvature of g.

Spectrum of �L on A.H. Surfaces 367

The basic asymptotically hyperbolic surface is the real hyperbolic Poincarédisc. In this case M is the unit disc of R

2, with the hyperbolic metric

g0 = ω−2δ , (2.2)

δ is the Euclidean metric, ω(x) = 12

(1 − |x|2δ

).

We denote by T qp the set of rank p covariant and rank q contravariant

tensors. When p = 2 and q = 0, we denote by S2 the subset of symmetric

tensors, and by◦S2 the subset of S2 of trace free symmetric tensors. We use

the summation convention, indices are lowered with gij and raised with itsinverse gij.

The Laplacian is defined as

= −tr∇2 = ∇∗∇,

where ∇∗ is the L2 formal adjoint of ∇. In dimension 2, the LichnerowiczLaplacian acting on trace free symmetric covariant 2-tensors is

L = + 2R,

where R is the scalar curvature of g.For u a covariant 2-tensorfield on M we define the divergence of u by

(div u)i = −∇ juji.

If u is a symmetric covariant 2-tensorfield on M, it can be seen as a one formwith values in the cotangent bundle. Thus we can define its exterior differentialwith

(d∇u

)ijk := ∇iujk − ∇juik,

which is a two form with values the cotangent bundle.For ω, a one form on M, we define its divergence

d∗ω = −∇ iωi,

the symmetric part of its covariant derivative :

(Lω)ij = 1

2(∇iω j + ∇jωi),

(note that L∗ = div) and the trace free part of that last tensor :( ◦Lω

)

ij= 1

2(∇iω j + ∇jωi) + 1

2d∗ωgij.

The well known [4] Weitzenböck formula for the Hodge-De Rham Laplacianon 1-forms, in dimension 2, reads

�Hωi = ∇∗∇ωi + Ric(g)ikωk = ∇∗∇ωi + R

2ωi.

We recall also the Weitzenböck formula

�K := (d∇)∗

d∇ + div∗ div = � + R = �L − R.

368 E. Delay

For a one form ω, we will consider the trace free symmetric covariant twotensor defined by

( ◦S ω

)

ij= ωiωj − |ω|2

2gij.

A TT-tensor (Transverse Traceless tensor) is by definition a symmetricdivergence free and trace free covariant 2-tensor.

L2 denotes the usual Hilbert space of functions or tensors with the product(resp. norm)

〈u, v〉L2 =∫

M〈u, v〉dμg

(

resp.|u|L2 =(∫

M|u|2dμg

) 12

)

,

where 〈u, v〉 (resp. |u|) is the usual product (resp. norm) of functions or tensorsrelative to g, and the measure dμg is the usual measure relative to g (we willomit the term dμg). For k ∈ N, Hk will denote the Hilbert space of functions ortensors with k-covariant derivative in L2, endowed with its standard productand norm.

We will first work near the infinity of M, so it is convenient to define forsmall ε > 0, the manifold

Mε = {x ∈ M, ρ(x) < ε}.It is well know that near infinity, we can choose the defining function ρ to bethe g-distance to the boundary. Thus, if ε is small enough, Mε can be identifiedwith (0, ε) × ∂ M equipped with the metric

g = ρ−2(dρ2 + g(ρ)dθ2

),

where {g(ρ)}ρ∈(0,ε) is a family of smooth, positive functions on ∂ M, withg(0) = g.

Let P be an uniformly degenerate elliptic operator of order 2 on some tensorbundle over M (see [9] for more details). We recall here a criterion for P to besemi-Fredholm. We first need the

Definition 2.1 We say that P satisfies the asymptotic estimate

〈Pu, u〉L2�∞C||u||2L2

(resp. ||Pu||L2�∞C||u||L2

)

if for all ε > 0, there exists δ > 0 such that, for all smooth u with compactsupport in Mδ , we have

〈Pu, u〉L2�(C − ε)||u||2L2

(resp. ||Pu||L2�(C − ε)||u||L2

).

Proposition 2.2 below is standard in the context of non-compact manifolds(see [7] for instance). It shows that the essential spectrum is characterized nearinfinity.


Proposition 2.2 Let P : H2 −→ L2. Then P is semi-Fredholm (ie. has finitedimensional kernel and closed range) if and only if P satisfies an asymptoticestimate

||Pu||L2�∞c||u||L2

for some c > 0.

This proposition will be used to compute the essential spectrum of �L whichis, by definition, the closed set

σe(�L) = {λ ∈ R, �L − λId is not semi-Fredholm}.

3 Commutators of Some Natural Operators

Lemma 3.1 On one forms, we have

div ◦ ◦L = 1

2

(� − R

2

)= 1

2(�H − R).

Proof In local coordinates, 2 div ◦ ◦L(ω) is equal to :

− ∇ i (∇iωj + ∇jωi − ∇kωkgij) = �ω j − ∇k∇jωk + ∇j∇kωk

= (� − Ric)ω j

=(

� − R2

)ω j. �

Recall that in dimension 2 (see Corollary 3.2 of [8] for instance) we have:

Lemma 3.2 Let (M, g) be a Riemannian surface with Levi-Civita connexion∇. Then the following equality holds for trace free symmetric covariants twotensors:

div ◦�L = �H ◦ div .

So we obtain

Corollary 3.3 If h is a trace free symmetric covariant two tensor with �Lh = λhthen �H div h = λ div h.

Lemma 3.4 On a Riemannian surface with Levi-Civita connexion ∇, we have

�L ◦ L = L ◦ �H − ◦S(dR, .),

where◦S(dR, ξ)ij = 1

2 (∇jRξi + ∇i Rξ j − ∇p Rξ pgij). In particular

�L ◦ ◦L = ◦

L ◦�H − ◦S(dR, .).

370 E. Delay

Moreover R is constant iff

�L ◦ L = L ◦ �H so �L ◦ ◦L = ◦

L ◦�H.

Proof The first part comes from [8] lemma 3.3 where here Ric(g) = R(g)

2 g.

Now, if �L ◦ L = L ◦ �H then for any one form ξ ,◦S(dR, ξ) = 0. At any point

x ∈ M, we take an orthonormal basis (e1, e2) on T∗x M, and choose ξ = e1. We

then see that the matrix of◦S(dR, ξ) has the form

(a bb −a

), where (a, b) are

the coordinates of dR. We finally deduce that dR = 0. �

4 Some Decompositions of Trace Free Symmetric Two Tensors

In this section, we recall two well known natural decompositions. We give theirsimple proofs for completeness.

Lemma 4.1 For all k ∈ N,

Hk+1(M,◦S2) = ker div ⊕ Im

◦L,

where the decomposition is orthogonal in L2.

Proof For ω ∈ C∞c (M), we have

∫

M<

◦L(ω), h >=

∫

M< ω, div h > .

Thus◦L∗ = div, which gives

(Im

◦L

)⊥ = Ker div. �

Lemma 4.2 For all k ∈ N,

Hk+1(M, T1) = ker �H ⊕ Im d ⊕ Im(∗d),

where the decomposition is orthogonal in L2.

Remark 4.3 Recall that, from the definition of �H , we have: ker �H = ker d ∩ker d∗.

Proof First, from the definition of d∗, it is clear that (Im d)⊥ = ker d∗, and soHk+2(M, T1) = ker d∗ ⊕ Im d. For all H1 function u and all H1 one forms ω,we have

∫

M〈∗du, ω〉 =

∫

M〈d∗ ∗ u, ω〉 =

∫

M〈∗u, dω〉 =

∫

M〈u, ∗dω〉.


As a consequence, if 〈∗du, ω〉L2 = 0 for all u ∈ C∞c (M), then dω = 0, and if in

addition d∗ω = 0 then �Hω = 0. This shows that ker d∗ = Im(∗d) ⊕ ker �H . �

From Lemma 4.2, any one form ω in H1 can be decomposed in a uniqueway with

ω = η + du + ∗dv, (4.1)

where �Hη = 0.

5 The Spectrum on TT-tensors

Lemma 5.1 Let M be any Riemannian surface. If h ∈ C2(M,

◦S2

), then the

following properties are equivalent:

(i) div h = 0,(ii) d∇h = 0,

(iii) h = ◦S(ω), where ω is a harmonic one form.

They imply

(iv) �Lh = Rh.

Moreover, if h ∈ L2, then (iv) implies (i ), (ii) and (iii).

Proof The first part is due to Avez ([3] Lemma A and Lemma C). The secondpart is simply due to the following Weitzenböck formula [10]:

(d∇)∗

d∇ + div∗ div = �K = �L − R,

and the fact that if h ∈ L2 solves (iv) weakly, then elliptic regularity gives h ∈H∞ ⊂ C∞. �

Corollary 5.2 There exists a non trivial eigen-TT-tensor of �L iff R is constant.In this case any TT-tensor is an eigentensor with eingenvalue R.

Proof The “if” part is clear. For the “only if” direction, assume that h is anon trivial eigen-TT-tensor of �L, so that div h = 0 and �Lh = λh hold forsome λ ∈ R. From Lemma 5.1, (�L − R)h = 0 and then (R − λ)h = 0. If R �= λ

near a point, then h has to be trivial near this point, so from the uniquecontinuation property, h is trivial. This contradicts the assumption on h andproves the result. �

372 E. Delay

6 Spectrum on Im◦L

If R is constant then from Lemma 3.4 and the fact that �H preserves the

decomposition (4.1), it suffices to study the spectrum of �L on Im◦L, restricted

successively to Ker �H , Im d and Im(∗d).

Lemma 6.1 When R is constant then◦L(Ker �H) is in the kernel of �L . If R is

moreover negative,◦L(Ker �H) is in one to one correspondence with Ker �H.

Proof If h = ◦Lη, with η ∈ Ker �H , then from Lemma 3.4 �Lh = 0. Now from

Lemma 3.1 we have

2 div ◦ ◦L(η) =

(� − R

2

)η.

Thus if R < cte < 0, then◦L is injective on H2. �

We are now interested in the spectrum on Im◦L ◦ d. We begin with a lemma.

Lemma 6.2 If h = ◦Lω, with ω ∈ H1 then:

||h||2L2 = 1

2

(||ω||2H1 −

∫

M

(R2

+ 1

)|ω|2

).

In particular, if R = −2 we obtain 2||h||2 = ||ω||2H1 .

Proof Using Lemma 3.4 we compute:∫

M|h|2 =

⟨ ◦Lω,

◦Lω

⟩

L2

=⟨

div◦Lω, ω

⟩

L2

= 1

2

∫

M

(|∇ω|2 − R

2|ω|2

)

= 1

2

(||ω||2H1 −

∫ (R2

+ 1

)|ω|2

). �

Corollary 6.3 On an A.H. surface, for all ε > 0, there exists δ0 > 0 small suchthat, for all δ ∈ (0, δ0) and all one forms ω with compact support in Mδ , if

h = ◦Lω then

||ω||2H1 � 2(1 − ε)||h||2L2 .


Proof

||ω||2H1 −∫

M

(R2

+ 1

)|ω|2 = ||ω||2H1 −

∫

Mδ

O(ρ)|ω|2

� ||ω||2H1 + Cδ||ω||2L2

� (1 + Cδ) ||ω||2H1,

where C is a positive constant. Lemma 6.2 concludes the proof. �

Let us recall a well known lemma.

Lemma 6.4 Let u be a smooth compactly supported function. If

〈�u, u〉L2 � c||u||2L2 ,

then

〈�Hdu, du〉L2 � c||du||2L2

and

〈�H(∗du), (∗du)〉L2 � c|| ∗ du||2L2 .

Proof

〈�Hdu, du〉 = 〈dd∗du, du〉 = 〈d∗du, d∗du〉= ||�u||2 � c||u||||�u|| � c〈u, �u〉 = c||du||2.

〈�H(∗du), (∗du)〉 = 〈d∗d(∗du), ∗du〉 = 〈d ∗ du, d ∗ du〉= ||�u||2 � c||u||||�u|| � c〈u, �u〉 = c||du||2 = c|| ∗ du||2.

�

We would like an equivalent to this lemma when substituting one forms tofunctions. This is achieved by the following lemma and its corollary.

Lemma 6.5 Let ω be a smooth compactly supported one form. If

〈�Hω, ω〉L2 � c||ω||2L2

then⟨�L

◦Lω,

◦Lω

⟩� c

2||ω||2H1 + 1

2c∫

M

(R2

+ 1

)|ω|2

−∫

M

(R2

+ 1

)〈�Hω, ω〉 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2.

374 E. Delay

Proof⟨�L

◦Lω,

◦Lω

⟩

L2=

⟨ ◦L�Hω,

◦Lω

⟩

L2−

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

=⟨�Hω, div

◦Lω

⟩

L2−

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

= 1

2〈�Hω, (�H − R)ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

= 1

2||�Hω||2L2 − 1

2〈R�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

� 1

2c||�Hω||L2 ||ω||L2 − 1

2〈R�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

� 1

2c〈�Hω, ω〉L2 − 1

2〈R�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

� 1

2c||∇ω||2L2 + 1

2c∫

M

R2

|ω|2− 1

2〈R�Hω, ω〉L2

−⟨ ◦

S(dR, ω),◦Lω

⟩

L2

� 1

2c||∇ω||2L2 + 1

2c∫

M

R2

|ω|2 + 〈�Hω, ω〉L2

−1

2〈(R + 2)�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

� 1

2c||∇ω||2L2 + 1

2c∫

M

R2

|ω|2 + c||ω||2L2

−1

2〈(R + 2)�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2

� 1

2c||ω||2H1 + 1

2c∫

M

(R2

+ 1

)|ω|2

−1

2〈(R + 2)�Hω, ω〉L2 −

⟨ ◦S(dR, ω),

◦Lω

⟩

L2.

�

Remark 6.6 Under the assumptions of Lemma 6.4, the assumptions of Lemma6.5 are satisfied by ω = du or ω = ∗du.

Proposition 6.5 together with Lemma 6.2 give:

Corollary 6.7 If R = −2 and 〈�Hω, ω〉L2 � c||ω||2L2 then⟨�L

◦Lω,

◦Lω

⟩

L2� c

∣∣∣∣∣∣

◦Lω

∣∣∣∣∣∣2

L2.


In the A.H. setting we have

Corollary 6.8 On an A.H. surface, for ω ∈ Im d or ω ∈ Im ∗d,

⟨�L

◦Lω,

◦Lω

⟩

L2�∞

1

4

∣∣∣∣∣∣

◦Lω

∣∣∣∣∣∣2

L2.

Proof It is well know that on A.H. surfaces, 〈�u, u〉 �∞ 14 ||u||2L2 holds. Then

(see Lemma 6.4 for instance) 〈�Hω, ω〉L2 �∞ 14 |ω|2L2 . We will show that the

three terms in the right-hand side of Lemma 6.5 do not contribute at infinity.We work with a one form ω compactly supported in Mδ with small δ. We recallthat R + 2 = O(ρ) and ||d(R + 2)|| = O(ρ).

Let us begin with

∫

M

(R2

+ 1

)〈�Hω, ω〉 =

∫

M

(R2

+ 1

)〈�ω, ω〉 +

∫

M

(R2

+ 1

)R2

||ω||2

=∫

Mω j∇ i

(R2

+ 1

)∇iω j +

∫

M

(R2

+ 1

)||∇ω||2

+∫

M

(R2

+ 1

)R2

||ω||2.

The two last terms are clearly bounded in absolute value by C1δ||ω||2H1 . LetA(ω) = ∫

M ω j∇ i(R + 2)∇iω j. Then:

|A(ω)| � ||∇ω||L2 ||d(R + 2)ω||L2

� ||∇ω||L2

(∫

M||d(R + 2)||2||ω||2

)1/2

� C2δ||∇ω||L2

(∫

M||ω||2

)1/2

� C2

2δ||ω||2H1 .

We so get:

∣∣∣∣

∫

M

(R2

+ 1

)〈�Hω, ω〉

∣∣∣∣ � C3δ||ω||2H1 .

The term⟨ ◦S(dR, ω),

◦Lω

⟩L2 proceed in a manner similar to A(ω) to obtain the

same estimate, perhaps with a different constant. Finally, the term∫

M

( R2 +

1)|ω|2 is clearly bounded in absolute value by C4δ||ω||2H1

. The conclusionfollows from Lemma 6.5, the triangular inequality and Corollary 6.3. �

376 E. Delay

Proposition 6.9 Let λ � 14 and C > 0. Let P be the operator �L − λId : H2 −→

L2. There is no asymptotic estimate

|Pu|L2�∞C|u|L2 ,

for P on Im( ◦L ◦ d

).

Proof Let λ � 14 , and μ :=

√λ − 1

4 . The idea of the proof is to construct a

family of tensors {hR} = { ◦L(dfR)

} = { ◦Hess fR

}with compact support in Me−R/2

such that |PhR|L2(M) goes to zero when R goes to infinity but |hR|L2(M) goes toinfinity when R goes to infinity.

It is well known (see [11, lemma 5.1] for example) that we can change thedefining function ρ into a defining function r such that the metric takes theform

g = r−2g = r−2(dr2 + g(r)

),

on Mδ =]0, δ[×∂∞M (reducing δ if necessary), where g(r) is a metric on {r} ×∂∞M.

The non trivial Christoffel symbols of g = r−2[dr2 + g(r)dθ2] are

�rrr = −r−1,

�rθθ = −r2

2

(−2r−3g + r−2g ′) = r−1g − 1

2g ′,

�θθr = 1

2

(−2r−1 + g−1g ′) = −r−1 + 1

2g−1g ′,

where the primes denote r-derivatives. If f is a “radial” function, ie f = f (r),we compute:

Hess f = (f ′′ + r−1 f ′) dr2 +

(−r−1g + 1

2g ′

)f ′dθ2.

We deduce

� f = −r2

(f ′′ + 1

2g−1g ′ f ′

).

We also have◦

Hess f =(

1

2f ′′ + r−1 f ′ − 1

4g−1g ′ f ′

)(dr2 − gdθ2

) =: F f (r)(dr2 − gdθ2

).

This tensor is in the set V2 of [9] page 201: substitute f there by F f here, qthere by dr2 − gdθ2 here, ρ there by r here and the dimension n + 1 there by2 here. Recall that the Lichnerowicz Laplacian in our context is �L = � + K,where K = −4 + O(r). Thus, from [9] Lemma 2.9 page 202, we obtain

(�L − λ)(F(r)q) = I2(F(r))q + rX(F),


where

I2(F) = −r2 F ′′ − 4rF ′ − 2 f F

has for characteristic exponents

s1, s2 = 1

2

(−3 ± √

1 − 4λ)

,

and X = ar2 d2

dr2 + br ddr + c is a second order operator polynomial in r d

dr withg-bounded coefficients depending on g and q.

In particular, if λ � 14 and f (r) = √

r[a cos(μ ln(r)) + b sin(μ ln(r))], where

μ=√

λ− 14 , then F f (r) = r−3/2[A cos(μ ln(r))+B sin(μ ln(r))]+O

(r−1/2

)and

(A, B) �= 0 if (a, b) �= 0. Thus we obtain

I2(F f (r)) = O(r−1/2

).

Let us now define the function

fR(r) = f (r)�R(r),

where �R is as in Lemma 8.1. A simple calculation shows that

F fR(r) = �R(r)F f (r) + O(R−1

)O

(r−3/2

).

Therefore:

I2(F fR(r)) = O(r−1/2

) + O(R−1

)O

(r−3/2

),

and

rX(F fR(r)) = O(r−1/2

).

Then:

(�L − λ)(F fR(r)q) = O(r−1/2

) + O(R−1)O(r−3/2).

We deduce that

||(�L − λ)(F fR(r)q)||2L2 = O(R−1

).

On the other hand, we have

||(F fR(r)q)||2L2 � cR,

where c is a positive constant. Letting R going to infinity, this concludes theproof of the proposition. �

7 Conclusion

From Proposition 6.9 and Corollary 6.8, the essential spectrum of �L restricted

to Im( ◦L ◦ d

)is

[1

4, +∞[.

378 E. Delay

In particular this ray is in the essential spectrum of �L.If R is constant then the A. H. condition forces R = −2. Lemma 6.1 shows

that any tensor in◦L(ker �H) is in the kernel of �L. The eigenspace for 0 is then

infinite dimensional as ker �H (recall that◦L is injective if R < 0).

From Lemma 5.1, any TT-tensor h is an eigentensor for the eigenvalue −2

and there is a harmonic one form omega such that h = ◦S(ω). Moreover ω is in

L4 iff h is in L2.Assume now that

〈�u, u〉L2 � 1

4||u||L2

holds for all smooth compactly supported functions u. Then Lemma 6.4 andCorollary 6.7 give, when ω = df or ω = ∗df ,

⟨�L

◦Lω,

◦Lω

⟩

L2� 1

4|| ◦Lω||L2 .

This proves that there are no eigentensors with eigenvalue less than 14

in Im( ◦L ◦ d

)nor in Im

( ◦L ◦ (∗d)

). Recall that, when R is a constant, the

Lichnerowicz Laplacian commute with the Hodge Laplacian, and also that the

Hodge Laplacian preserves the decomposition (4.1). We so get that on Im◦L,

the essential spectrum of �L is

{0} ∪ [1

4, +∞[.

This concludes the proof of the main Theorem 1.1.

Acknowledgements I am grateful to N. Yeganefar for discussions on forms and to F. Gauterofor his comments on the original manuscript.

Appendix : A Family of Cutoff Functions

In this appendix, we give a family of cutoff functions. Standard in the A.H.context, they can be found in [1], Definition 2.1 p.1362 for instance.

Lemma 8.1 Let (M, g, ρ) be an asymptotically hyperbolic manifold. For R ∈ R

large enough, there exits a cutoff function �R : M → [0, 1] depending only onρ, supported in the annulus {e−8R < ρ < e−R}, equal to 1 in {e−4R < ρ < e−2R}and which satisfies for R large :

∣∣∣∣dk�R

dρk(ρ)

∣∣∣∣ � Ck

Rρk,

for all k ∈ N\{0}, where Ck is independent of R.


Proof Let χ : R −→ [0, 1] be a smooth function equal to 1 on ] − ∞, 1] and 0on [2, +∞[. We define

χR(x) := χ

(ln(ρ(x))

−R

),

we then have χR : M −→ [0, 1] is equal to 1 on ρ � e−R and 0 on ρ � e−2R.Now we define

�R := χ4R(1 − χR)

which satisfies the announced properties. �

References

1. Andersson, L.: Elliptic systems on manifolds with asymptotically negative curvature. IndianaUniv. Math. J. 42(4), 1359–1388 (1993)

2. Avez, A.: Le laplacien de Lichnerowicz. Rend. Sem. Mat. Univ. Politec. Torino (35), 123–127(1976–1977)

3. Avez, A.: Le laplacien de Lichnerowicz sur les tenseurs. C. R. Acad. Sci. Paris Sér. A (284),1219–1220 (1977)

4. Besse, A.L.: Einstein manifolds. Ergebnisse d. Math. 3. folge, vol. 10. Springer, Berlin (1987)5. Buzzanca, C.: Le laplacien de Lichnerowicz sur les surfaces à coubure négative constante.

C. R. Acad. Sci. Paris Sér. A (285), 391–393 (1977)6. Buzzanca, C.: Il laplaciano di Lichnerowicz sui tensori. Boll. Un. Mat. Ital. 6(3-B), 531–541

(1984)7. Delay, E.: Essential spectrum of the Lichnerowicz laplacian on two tensor on asymptotically

hyperbolic manifolds. J. Geom. Phys. 43, 33–44 (2002)8. Delay, E.: TT-eigentensors for the Lichnerowicz laplacian on some asymptotically hyperbolic

manifolds with warped products metrics. Manuscripta Math. 123(2), 147–165 (2007)9. Graham, C.R., Lee, J.M.: Einstein metrics with prescribed conformal infinity on the ball. Adv.

Math. 87, 186–225 (1991)10. Koiso, N.: On the second derivative of the total scalar curvature. Osaka J. Math. 16(2), 413–421

(1979)11. Lee, J.M.: The spectrum of an asymptotically hyperbolic Einstein manifold. Comm. Anal.

Geom. 3, 253–271 (1995)12. Mazzeo, R.: The Hodge cohomology of a conformally compact metric. J. Differ. Geom. 28,

309–339 (1988)


Some Examples of Graded C∗-Algebras

Athina Mageira

Received: 15 July 2008 / Accepted: 15 September 2008 / Published online: 26 October 2008© Springer Science + Business Media B.V. 2008

Abstract We apply the theory of C∗-algebras graded by a semilattice tocrossed products of C∗-algebras. We establish a correspondence betweenthe spectrum of commutative graded C∗-algebras and the spectrum of theircomponents. This will allow us to compute the spectrum of some commutativeexamples of graded C∗-algebras.

Keywords C∗-algebras · Semilattices · Crossed products ·Spectrum of C∗-algebras

Mathematics Subject Classifications (2000) 46L05 · 47A10 · 47L65

1 Introduction

This paper is a continuation of a previous work ([12]), where we studied gradedC∗-algebras by a semilattice and some of their properties.

In the present paper, we study crossed products of graded C∗-algebrasand we construct C∗-algebras graded by semilattices of closed subgroups. Weinvestigate the relation of the spectrum of commutative graded C∗-algebrasand those of its components. Then follows in particular the study of twocommutative examples.

C∗-algebras graded by semilattices appear in the work of A. Boutet deMonvel and V. Georgescu in [2, 3] and [5], M. Damak and V. Georgescu

A. Mageira (B)Institut de Mathématiques de Jussieu–Université Paris-Diderot (Paris 7),175, rue du Chevaleret, 75013 Paris, Francee-mail: [email protected], [email protected]

382 A. Mageira

(cf. [6, 7]) and also V. Georgescu and A. Iftimovici (cf. [8, 9] and [10]) inconnection with the quantum N body problem.

The Hamiltonian of a N body system is of the form H = −� + V where � isthe Laplace-Beltrami operator on X = R

3N and V is the interaction potential.This V has the form V = ∑

1≤ j<k≤NVjk where Vjk ∈ C0(X/Xjk) with X jk = {x ∈

X; x j = xk} which describes the interaction of the particles j and k.The C∗-algebra A generated by these potentials is a commutative C∗-

algebra graded by the subsemilattice of vector subspaces of X generated bythe X jk. The resolvent of H lives then in A � X which was also shown to begraded. This was used to describe some spectral properties of hamiltonians ofthis form and to prove the Mourre estimate for N body systems (see also [4]).We refer to [12] for more references on the use of graded C∗-algebras in the Nbody problem.

The main goal of this paper is

a) To show that the full and reduced crossed product of a graded C∗-algebra(by a group action preserving the grading) is also graded. This is due to thefact that crossed products are compatible with inductive limits and splitexact sequences.

b) To study the spectrum of some commutatives graded C∗-algebras.

i) If A is a commutative C∗-algebra graded by a good semilattice (inthe sense of Definition 3.1 below), we show that its spectrum is thedisjoint union of the spectra of its components.

ii) When the semilattice G is formed by closed subgroups of a locallycompact group G, we give the necessary and sufficient conditionsto have structure morphisms and therefore to construct a G-gradedcommutative C∗-algebra A with components the C∗-algebrasC0(G/H), H ∈ G. Moreover, we study the injectivity of the morphismϕ : A → Cb (G).

iii) We explicitly compute the spectra of two commutativegraded C∗-algebras.

• For the first one we consider a good semilattice. We take a subsemilatticeL of the lattice (for the inclusion as order relation) formed by vectorsubspaces of a finite dimensional vector space X and whose componentsare AY = C0(X/Y) (with Y ∈ L). In particular, we study the case where Xis a plane P and L = {{0}, δ1, ..., δn,P} where δ1, . . . , δn are n vector linesin P . We use the theory of normal forms of a Riemann surface (see [11]and [13]) to show that the spectrum of the associated graded C∗-algebra ishomeomorphic to the torus Tg with g holes where g = n

2 if n is even and ifn is odd this spectrum is homeomorphic to a pinched torus with g = E

( n2

)

holes (i.e. we identify two of its points).• For the second example we take an arbitrary semilattice L and all the com-

ponents Ai equal to C. We identify then the spectrum of the correspondingL-graded C∗-algebra A with the set of non empty final subsemilattices of L.

Some Examples of Graded C∗-Algebras 383

In particular, if L = Q we show that the spectrum of A is in bijection withthe set R

∐Q

∐{−∞}.

Here is a summary of the paper.

• We show in Section 2 that crossed products of C∗-algebras are compatiblewith graded C∗-algebras.

• Section 3 is devoted to a study of the spectrum of commutative gradedC∗-algebras.

• In Section 4 we consider a particular case of semilattices. Here L will bea subsemilattice of the lattice G which is formed by closed subgroups ofa locally compact group G. We examine which conditions give rise to aL-graded C∗-algebra.

• Section 5 consists of applications of the previous sections to vector spaces.We study in particular the spectrum of a commutative C∗-algebra with agrading by vector lines of a plane.

• Finally, we give a quite different commutative example of gradedC∗-algebras by a semilattice (which is not a good semilattice) which isgenerated by copies of C and we study its spectrum. The case where L = Q

is investigated.

Let (L, �) be a partially ordered set. Recall that (L, �) is a semilattice if forall k, � ∈ L the set {m ∈ L; m � k and m � �} has a greatest element notedk ∧ �. An initial segment of L is a subset M of L such that for all a ∈ M wehave {b ∈ L; b � a} ⊂ M. Every initial segment of L is a subsemilattice of L.A final subsemilattice of L is a subsemilattice M which is a final segment i.e.such that, for all a ∈ M, we have {b ∈ L; a � b} ⊂ M.

A L-graded C∗-algebra (or a C∗-algebra graded by L) is a C∗-algebraA equipped with a linearly independent and total family (Ai)i∈L of C∗-subalgebras of A such that Ai A j ⊂ Ai∧ j for all i, j ∈ L. We will denote by(A, (Ai)i∈L) a L-graded C∗-algebra A and we will call the Ai’s the componentsof A. For an arbitrary subset M of L denote by AM the sum of the Ai’s fori ∈ M. We also denote by FL the set of finite subsemilattices of L.

2 Crossed Products of Graded C∗-Algebras

Theorem 2.1 Let L be a semilattice and (A, (Ai)i∈L) a L-graded C∗-algebra,with a continuous action α of a locally compact group G. Assume that for alli ∈ L the C∗-algebra Ai is G-invariant.

Then the full crossed product (A�αG, (Ai�αG)i∈L) and the reduced crossedproduct (A�r,αG, (Ai�r,αG)i∈L) are also L-graded C∗-algebras.

Proof We will only show this for the full or reduced crossed product. The sameproof works equally well for both.

384 A. Mageira

Let k ∈ L, put Lk = { j ∈ L|k � j } and L′k = { j ∈ L|k � j}. Then we have a

split exact sequence (cf. [12] proposition 1.12)

0 → AL′k→ A � ALk → 0

and since crossed products are compatible with split exact sequences we alsohave

0 → AL′k�αG −→ A�αG � ALk�αG → 0.

We will show that :

i) Ak�αG → A�αG is an injective morphism for all k ∈ L.ii) The family of C∗-algebras (Ai�αG)i∈L is linearly independent,

iii) (Ai�αG)(A j�αG) ⊂ Ai∧ j�αG for all i, j ∈ L andiv)

⋃

F∈FL

AF�αG = ∑

i∈LAi�αG is dense in A�αG.

i) Since Ak is an ideal of ALk , Ak�αG is also an ideal of ALk�αG and wehave

Ak�αG � � ��

��ALk�αG

��A�αG.

ii) Let f ∈ A�αG such that f = ∑

i∈Lfi = 0 with fi = 0 only for a finite num-

ber of i ∈ L. Denote by � = { j ∈ L : f j = 0} the support of f . Assumethat � = ∅ and take m ∈ � a maximal element then fm = 0 ⇒ || fm|| = 0.Since Lm ∩ � = {m} and A�αG → ALm�αG is a morphism of C∗-algebras || fm|| � || ∑

i∈Lfi||. In other words fm = 0, which contradicts the

assumption.iii) Let i, j ∈ L and let fi ∈ Cc(G, Ai) , f j ∈ Cc(G, A j). By the definition

of the product of the ∗-algebra Cc(G, A), for all s ∈ G we have( fi ∗ f j)(s) ∈ Ai∧ j. Then Cc(G, Ai)Cc(G, A j) ⊂ Cc(G, Ai∧ j) and since theproduct is continuous we can extend this inclusion to crossed products(Ai�αG)(A j�αG) ⊂ (Ai∧ j�αG).

iv) It is clear since crossed products preserve the inductive limits. ��

Remark 2.2 Let i, j ∈ L with i � j. Denote by ϕi, j : A j → M(Ai) the structuremorphisms of A. We deduce morphisms

ϕαi, j : A j�αG → M(Ai�αG) and ϕ

r,αi, j : A j�r,αG → M(Ai�r,αG).

It is clear that these are the structure morphisms of (A�αG, (Ai�αG)i∈L) and(A�r,αG, (Ai�r,αG)i∈L) respectively.


Moreover, if L has a least element �0 and the structure morphisms ϕi, j ofA satisfy ϕ−1

i, j (Ai) = {0} then by [12], proposition 1.18 the morphism ϕ�0 : A →M(A�0) is injective. Therefore, the morphism ϕ

r,α�0

is injective.

3 Spectrum of Commutative Graded C∗-Algebras

Let L be a semilattice and (A, (Ai)i∈L) a C∗-algebra graded by L. We saw in[12] that A is commutative if and only if for any i ∈ L the component Ai iscommutative.

Definition 3.1 We say that L is a good semilattice if any totally orderednonempty part of L is well-ordered (i.e. any nonempty subset of this part has aleast element).

Proposition 3.2 Let L be a semilattice. The following are equivalent:

a) The set L is a good semilattice.b) Every totally ordered nonempty part of L has a least element.c) Every nonempty subsemilattice of L has a least element.d) Every final nonempty subsemilattice of L has a least element.

Proof It’s easy to see that a) ⇔ b).We show now that b) ⇒ c). For any nonempty subsemilattice F of L, by

Zorn’s lemma, there exists a maximal totally ordered subset A of F . We denoteby a its least element and we take x ∈ F . Since F is a subsemilattice we havea ∧ x ∈ F . The set {a ∧ x} ∪ A is totally ordered and A is maximal, so a ∧ x ∈A and therefore a ≤ a ∧ x, so a ≤ x. This means that a is the least elementof F .

Clearly c) ⇒ b) and c) ⇒ d).To see that d) ⇒ c) we consider a subsemilattice I of L and we set J = { j ∈

L; ∃i ∈ I; i ≤ j }. Note that J is a final subsemilattice of L so by assumption ithas a least element that we denote by j0. Since j0 ∈ J there is i0 ∈ I such thati0 ≤ j0. But we also have I ⊂ J, so i0 ∈ J and j0 ≤ i0. In other words i0 = j0 isthe least element of I. ��

We will show now that the spectrum of commutative graded C∗-algebras isrelated with the spectrum of their components.

Remark 3.3 Let (A, (A�)�∈L) be a commutative graded C∗-algebra. Take i ∈ Land χi a character of Ai. Denote by χi its extension to the multiplier algebraM(Ai). Then, we can define in a unique way a character of A by χ = χi ◦ πi

where πi = ϕi ◦ pi is the morphism A → M(Ai). We get then a continuousinjective map ψi : Sp(Ai) → Sp(A).

Let χ be a character of A and assume that there exists i, j ∈ L such that χ =χi ◦ πi = χ j ◦ π j. Assume also that i ≤ j. Then πi|A j

= 0 and we have χi ◦ πi|A j= 0

386 A. Mageira

which is absurd since χ j ◦ π j|A j= χ j = 0. This means actually that the ψi’s have

disjoint images.

Proposition 3.4 Let (A, (A�)�∈L) be a commutative graded C∗-algebra. If L is agood semilattice then

Sp(A) = ⋃

i∈Lψi(Sp(Ai)).

Proof Let χ ∈ Sp(A). We consider the set I = {i ∈ L; χ|Ai= 0}. Note that I is

a subsemilattice of L since for i, j ∈ I and ai ∈ Ai, a j ∈ A j such that χ(ai) = 0and χ(a j) = 0, we have χ(aia j) = χ(ai)χ(a j) = 0 i.e. χ|Ai∧ j

= 0. It follows thatthe set I has a least element denoted by i0 (Proposition 3.2). We set now χi0 =χ|Ai0

, then χi0 = 0. We will show that χ = ψi0(χi0). By linearity and continuity

we just need to show that these characters coincide on A j for any j ∈ L.So, let j ∈ L and a j ∈ A j. If i0 � j, since i0 is the least element of I, then

j /∈ I and therefore χ|A j= 0. We also have ψi0(χi0)(a j) = χi0 ◦ πi0(a j) = 0.

If i0 ≤ j, since χi0 = 0 there exists ai0 ∈ Ai0 such that χi0(ai0) = χ(ai0) = 1and since πi0(ai0) = ai0 then ψi0(χi0)(ai0) = 1. Then, for a j ∈ A j we have

ψi0(χi0)(a j) = ψi0(χi0)(a j)ψi0(χi0)(ai0) = ψi0(χi0)(a jai0) = χi0(a jai0)

= χ(a jai0) = χ(a j)χ(ai0) = χ(a j)

��

Remark 3.5 Let L be a good semilattice and (A, (A�)�∈L) a commutativegraded C∗-algebra. Assume that the structure morphisms ϕi, j of A are non-degenerate for all i, j ∈ L such that i ≤ j.

Let M be a subsemilattice of L. Assume that AMA = A, then the inclusionι : AM → A implies a continuous map � : Sp(A) → Sp(AM) which is given byχ �→ χ ◦ ι. We will study this map with the help of the decompositions Sp(A) =⋃

i∈Lψi(Sp(Ai)) and Sp(AM) = ⋃

m∈Mρm(Sp(Am)) (the second one is due to the

fact that a subsemilattice of a good semilattice is itself a good semilattice).Take χ ∈ Sp(A). There exists a unique i ∈ L such that χ = ψi(χi) ∈

ψi(Sp(Ai)). Set Li = { j ∈ L; i ≤ j}. Since χ|Ai= 0 and for all j ∈ L such that

i ≤ j the morphisms ϕi, j are non-degenerate, χ is not zero over A j Ai henceχ|A j

= 0. Therefore one has Li = { j ∈ L; i ≤ j} = { j ∈ L; χ|A j= 0}. Set Mi =

M ∩ Li = {m ∈ M; i ≤ m} = { j ∈ M; χ|A j= 0}. Then Mi is a final subsemilat-

tice of M and since L is a good semilattice M has a least element (Proposition3.2). Denote by m(i) = infMi, then χ|Am(i)

= χm(i) = 0.Note that the morphism ϕi,m(i) : Am(i) → M(Ai) gives rise to a continu-

ous map �i : Sp(Ai) → Sp(Am(i)) which is given by �i(χi) = χi ◦ ϕi,m(i). If we


denote by ϕi,m(i) the extension of ϕi,m(i) to the multiplier algebra M(Am(i)), onesees (we only need to check it on the components of AM) that the diagram

AM��

��

A

��M(Am(i)) �� M(Ai)

commutes. In consequence we have commuting diagrams

Sp(Ai)�i

��

ψi

��

Sp(Am(i))

ρm(i)

��

Sp(A)�

�� Sp(AM)

which show that the map � is entirely described by the map ρm(i) and themaps �i.

4 Semilattices of Closed Subgroups

Let G be a locally compact group. Note that the set G of closed subgroupsof G is a complete lattice for the inclusion: if (Hi)i∈I is a family of closedsubgroups of G, it is clear that the greatest element of {H ∈ G; ∀i ∈ I, H ⊂ Hi}is the intersection

⋂i∈I Hi; the least element of {H ∈ G; ∀i ∈ I, Hi ⊂ H} is the

closure of the subgroup generated by the Hi’s.For H ∈ G, set AH = C0(G/H). If H, K ∈ G are such that K ⊂ H, every

left equivalent class x ∈ G/K is contained in a left equivalent class pK,H(x) ∈G/H. This will induce a continuous map pK,H : G/K → G/H and therefore ahomomorphism ϕK,H : AH = C0(G/H) → Cb (G/K) = M(AK).

Lemma 4.1 Let H, K ∈ G. The following conditions are equivalent:

(i) We have ϕH∩K,H(AH)ϕH∩K,K(AK) ⊂ AH∩K.(ii) The map q : x �→ (pH∩K,H(x), pH∩K,K(x)) from G/(H ∩ K) to G/H ×

G/K is closed.(iii) The subset HK = {xy; x ∈ H, y ∈ K} is closed in G and the product map

H × K → HK, (x, y) �→ xy is open.(iv) pH(K) is closed (in G/H) and the restriction map K → pH(K) of pH is

open.(v) pK(H) is closed (in G/K) and the restriction map H → pK(H) of pK is

open.

388 A. Mageira

Proof To prove this lemma we will use some general topological results thatcan be found in [1].

Let us show that we have (ii) ⇒ (i). Since q is continuous, injective andclosed, it is proper. Thus we have a map q∗ : C0(G/H × G/K) → C0(G/(H ∩K)) and for f ∈ C0(G/H), g ∈ C0(G/K) we have q∗( f ⊗ g) = ( f ⊗ g) ◦ q =f (pH∩K,H)g(pH∩K,K) = ϕH∩K,H( f )ϕH∩K,K(g) = fg. The assertion now fol-lows since q∗( f ⊗ g) ∈ C0(G/(H ∩ K)).

For the converse (i) ⇒ (ii) if C is a compact subset of G/H × G/K, thereexists compact subsets C1 of G/H and C2 of G/K such that C ⊂ C1 × C2.Since q is continuous, q−1(C) is a closed set of q−1(C1 × C2). Take nowf ∈ C0(G/H), g ∈ C0(G/K) such that f = 1 on C1 and g = 1 on C2. ThenS = {x ∈ G/(H ∩ K) : ( fg)(x) = 1} is a compact subset of G/(H ∩ K) andsince q−1(C1 × C2) ⊂ S then q−1(C1 × C2) is compact and q is proper.

We show now that (ii) ⇔ (iii). Define a map r : G → G/H × G/K byx �→ (pH(x), pK(x)) and take the quotient map p : G → G/(H ∩ K). Sinceq is injective, p is open and r = q ◦ p, we can replace condition (ii) by theequivalent (ii)′: r(G) is a closed set and r : G → r(G) is open.

Take the open map t : G × G → G/H × G/K defined by (x, y) �→(pH(x), pK(y)) and consider the following diagram

G ��

p

�� r �� G × G

t��

G/(H ∩ K)q

�� G/H × G/K.

Since r(G) = q(G/(H ∩ K))

= {(x, y) ∈ G/H × G/K; ∃g ∈ G : x = gH, y = gK},we have

t−1(r(G)) = {(g1, g2) ∈ G × G; ∃g ∈ G, ∃h ∈ H,

k ∈ K : g1 = gh, g2 = gk}= {

(g1, g2) ∈ G × G; ∃h ∈ H, k ∈ K : g−11 g2 = h−1k

}

= {(g1, g2) ∈ G × G : g−1

1 g2 ∈ HK} = HK.

Thus HK is closed if and only if t−1(r(G)) is closed in G × G, therefore ifand only if r(G) is closed.

The fiber product of G by t−1(r(G)) over r(G) is given by

G×r(G)t−1(r(G)) = G×G/H×G/K(G × G)

= {(x, y, z) ∈ G × G × G; r(x) = t(y, z)}= {(x, y, z) ∈ G × G × G; x−1 y ∈ H, x−1z ∈ K}.


Take (g, h, k)∈G×H×K and set s(g, h, k)=g and ϕ(g, h, k)=(gh, gk)∈t−1(r(G)). Then the map ψ : G × H × K → G×r(G)t−1(r(G)) defined by(g, h, k) �→ (g, gh, gk) is a homeomorphism and gives rise to a cartesiansquare

G × H × Kφ

��

s

�� t−1(r(G))

tr(G)

��

Gr

�� r(G).

Since the maps of this diagram are surjective, tr(G) is open (since t isopen) and G × H × K is the fiber product G×r(G)t−1(r(G)) then r is openif and only if φ is open. If we denote by T : t−1(r(G)) → G × HK the map(g1, g2) �→ (g1, g−1

1 g2) which is a homeomorphism then φ is open if and onlyif T ◦ φ : G × H × K → G × HK where T ◦ φ(g, h, k) = (g, hk) is open. Inother words if and only if H × K → HK is open.

The final equivalence (iii) ⇔ (iv) follows by the fact that p−1H pH(K) = KH

therefore if we pass to the inverse elements of KH we have that pH(K) isclosed if and only if HK is closed and also by the cartesian square:

K × H ��

��

KH

��

K �� pH(K)

where the homeomorphism K × H → K×pH(K)KH is defined by (k, h) �→(k, kh). ��

Take now a subsemilattice L of G and assume that every H, K ∈ L satisfy theequivalent conditions of Lemma 4.1. We have then the following proposition(cf. [12] Theorem 2.2).

Proposition 4.2 There exists a L-graded C∗-algebra (A, (AH)H∈L) whose struc-ture morphisms are the ϕK,H’s defined as above.

The multiplier algebra of C0(G) is the algebra of continuous boundedfunctions on G, Cb (G).

Denote by pH : G → G/H the quotient map for H ∈ L. We can then iden-tify C0(G/H) as a C∗-subalgebra of Cb (G) via the application f �→ f ◦ pH .

We have then morphisms ϕH which are nondegenerate for all H ∈ L andwe can then extend them to morphisms ϕH defined on the multiplier algebraCb (G/H). It is easy to see that ϕH = ϕK ◦ ϕK,H for all H, K ∈ L such thatK ⊂ H. This implies that ϕH( f )ϕK(g) = ϕH∩K( fg) for all K, H ∈ L and f ∈

390 A. Mageira

C0(G/H), g ∈ C0(G/K) and gives rise to a homomorphism ϕ : A → Cb (G) (cf.[12], proposition 1.10).

Proposition 4.3 The homomorphism ϕ : A → Cb (G) is injective if and only iffor any H, K ∈ L satisfying K ⊂ H and H = K the quotient space H/K is notcompact.

We will need the following lemma:

Lemma 4.4 Let K, H be two subgroups of G with K ⊂ H.

a) If H/K is compact we have ϕK,H(C0(G/H)) ⊂ C0(G/K).b) If H/K is not compact we have ϕK,H(C0(G/H)) ∩ C0(G/K) = {0}.

For this we will show the following lemma:

Lemma 4.5 Let G be a locally compact group and let H and K be closedsubgroups of G with K ⊂ H.

a) If H/K is compact, pK,H is proper.b) If H/K is not compact, for all a ∈ G/H the subset p−1

K,H({a}) of G/K is notrelatively compact.

Proof

a) Let C be a compact subset of G/H, since G is locally compact and pH

is open and surjective, there exists a compact subset A of G such thatpH(A) = C. We have p−1

K,H(C) = {ax; a ∈ A, x ∈ H/K}, which is theimage of the compact set A × H/K by the continuous map G × G/K →G/K. Thus p−1

K,H(C) is a compact subset of G/K.b) Take g ∈ G whose class is a, the application x �→ gx is a homeomorphism

of G/K that maps H/K into p−1K,H({a}). But since H/K is closed in G/K

and it is not compact then it is not relatively compact. ��

Proof of Lemma 4.4

a) Let f ∈ C0(G/H). For every ε > 0, {x ∈ G/K; | f ◦ pK,H(x)| � ε} =p−1

K,H({x ∈ G/H; | f (x)| � ε}) is a compact set, therefore f ◦ pK,H ∈C0(G/K).

b) Let f ∈ C0(G/K) ∩ ϕK,H(C0(G/H)). For all a ∈ G/H, the map f which isconstant on p−1

K,H({a}) is then zero. ��

Proof of Proposition 4.3 If there exists H, K with K ⊂ H, H = K and H/Kcompact, then for f ∈ C0(G/H) we have

ϕ( f − ϕK,H( f )) = ϕH( f ) − ϕK(ϕK,H( f )) = 0,

therefore ϕ is not injective.


Conversely assume that for all H, K ∈ L such that K ⊂ H and H = K thequotient space H/K is not compact. We have three cases:

a) If {e} ∈ L. This follows immediately by applying Lemma 4.4 and byproposition 1.18 in [12].

b) Assume that neither of H ∈ L is compact, if we set L′ = L ∪ {{e}} we getto the first case.

c) If there exists H ∈ L with H compact, then for all K ∈ L, the spaceH/(H ∩ K) is compact. But this is possible only if H ∩ K = H. ThereforeH is the least element of L. Then πH : A → Cb (G/H) ⊂ Cb (G) is injective(proposition 1.18 in [12]). ��

Remark 4.6 Assume that we are given a semilattice of closed subgroups thatsatisfy the conditions of Lemma 4.1 and Proposition 4.3. Thanks to Theorem2.1 the family of C∗-subalgebras C0(G/H)�redG ⊂ B(L2(G)) form a gradedC∗-subalgebra of B(L2(G)).

5 Examples

If H is a Hilbert space then B(H) and K(H) will denote the C∗-algebras of allbounded and compact operators on H respectively.

5.1

Now let us reconsider the crossed product studied by M. Damak and V.Georgescu (cf. [6]) and by V. Georgescu and A. Iftimovici (cf. [8], theorem3.12).

Recall that X will be a finite dimensional vector space and G the latticeof all vector subspaces of X for the inclusion. Since in finite dimension, everyvector subspace is closed and every linear surjective map is open, condition (iii)of Lemma 4.1 is satisfied and for all Y, Z ∈ G we have C0(X/Y)C0(X/Z ) ⊂C0(X/(Y ∩ Z )). Thanks to Proposition 4.2, for every subsemilattice L of G wehave a L-graded C∗-algebra

A = ⊕

Y∈LC0(X/Y).

Moreover, since every vector space is not compact, condition b) of Lemma4.5 is also satisfied and we can then realize A as a C∗-subalgebra of Cb (X) =M(C0(X)) ⊂ B(L2(X)).

We take now the crossed product A � X where X acts continuously bytranslation. Then Proposition 2.1 implies that this crossed product is also aL-graded C∗-algebra i.e.

A � X = ⊕

Y∈L(C0(X/Y) � X).

392 A. Mageira

By Remark 2.2,

A � X ⊂ M(C0(X) � X) = M(K(L2(X))) = B(L2(X)).

In other words A � X is a graded C∗-algebra of operators defined on theHilbert space L2(X).

5.2

We take a plane P and n vector lines δi, i = 1, ..., n of P . Then we can form alattice of vector spaces F = {{0}, δ1, ..., δn,P}.

The previous example shows that we have a F-graded C∗-algebra:

A = ⊕

Y∈FC0(P/Y) ⊂ Cb (P)

which is commutative and has a unit. Our goal is to compute the spectrum ofthis algebra.

We will start by studying the case n = 1. Then F = {{0}, δ1,P} and A =C0(P)⊕C0(P/δ1)⊕C.

Consider the sublattice of F , F1 = {{0}, δ1}. Then A1 = C0(P)⊕C0(P/δ1) isa F1-graded C∗-subalgebra of Cb (P) and moreover it is an ideal for A. TheGelfand–Naimark theorem now implies that A1 � C0(Sp B1).

Identify P with R2 and δ1 with the “axis of x” (i.e. {(x, y); y = 0}). Then we

can consider an element of C0(P/δ1) as the function (x, y) �→ g(y) where g ∈C0(R). There exists a unique injective homomorphism ψ : A1 → C0(T × R)

whose restriction to C0(P) and C0(P/δ1) is the natural inclusion ψ0 : C0(P) ↪→C0(T × R) and ψδ1 : C0(P/δ1) ↪→ C0(T × R) respectively ([12], proposition1.10). Note that the product fg is given by the formula ( fg)(x, y) = f (x, y)g(y)

for all (x, y) ∈ T × R.We get the algebra A by adjoining a unit to A1, i.e. A = A1 thus the mor-

phism ψ has a unique extension to a isomorphism ψ : A → C((T × R) ∪ {∞})where (T × R) ∪ {∞} is the one point compactification which is homeomorphictopologically to a pinched sphere.

For n � 2, consider in the first place the vector space E = Rn and the

canonical basis (e1, ..., en). Let I be a subset of {1, ..., n}. Denote by PI thevector subspace generated by (ei)i∈I . Let M be the set {PI; I ⊆ {1, ..., n}}, thenM is a lattice whose least element is {0}. Note that the map I �→ PI is anisomorphism of lattices.

We get then a M-graded C∗-algebra (B, (C0(E/H))H∈M) whose structuremorphisms are the ϕK,H’s defined as above and we have

B = ⊕

H∈MC0(E/H) � C(Sp(B)) ⊂ Cb (E).

Thus Sp(B) is a compactification of E which is homeomorphic to the n dimen-sional torus, T

n. Indeed, take H ∈ F then we can write H = PI where I ⊆ J.The quotient maps E → E/H and the natural inclusions of the quotients


E/H into their compactifications T{1,...,n}\I deduce the following commuting

diagrams

C0(E/H)ψH

��

��

C(Tn)

��

Cb (E).

Note that the morphism C(Tn) → Cb (E) is injective since Tn is a compactifi-

cation of E which implies that the ψH’s are injective. Therefore there existsa unique and injective homomorphism ψ : B → C(Tn) whose restriction onC0(E/H) is ψH ([12], proposition 1.10). Moreover by the Stone-Weiestrasstheorem ψ is surjective.

We go back now to the case of A. Define n linear forms fi, i = 1, ..., n of Pwith kernels equal to the lines δi, i = 1, ..., n respectively. To the proper inclu-sion ( f1, ..., fn) : P → E corresponds a surjective homomorphism Cb (E) →Cb (P) which induces together with the inclusion B ⊂ Cb (E) a morphism� : B → Cb (P). We will show now that �(B) = A. For all H ∈ M, let �H

be the restriction of � on the components of B, C0(E/H).If dim H � n − 2 we have P ↪→ E/H since P ∩ H = {0}. This implies that

the image of C0(E/H) by �H in Cb (P) is C0(P).If we denote by Hi = P{1,...,n}\{i}, i = 1, ..., n the hyperplans of E and H =

Hi, i = 1, ..., n then for any i, E/Hi is homeomorphic to P/δi since δi = P ∩ Hi

for i = 1, ..., n and we have an isomorphism �Hi : C0(E/Hi) → C0(P/δi).

Denote by P the closure of P in Tn and consider C(P) as a subalgebra of

Cb (P), then � is the restriction of C(Tn) to P . Thus Sp(A) = P . We will showthat it is homeomorphic to the torus Tg with g holes where g = E

( n2

)which is

pinched, i.e. we force the identification of two of its points, when n is odd.Denote by Q the closure of P in (R)n. Then the closure P of P in T

n is equalto ρ(Q) where ρ is the canonical surjection (R)n → T

n.With our first assumptions, the plane P will have the following form:

P = {(x1, ..., xn) ∈ R

n; xk = Aijkxi + Bijkx j},

where i, k, j ∈ {1, ..., n}, i, j are fixed, k /∈ {i, j} the real coefficients Aijk, Bijk

are non zero and they satisfy Bijk Aij� = Aijk Bij� for all k, � /∈ {i, j} with k = �.This form of P implies that the only possible limits in Q are vectors ofwhom a coefficient is finite and the others are infinite i.e. of the form(∞, ..., ∞, ai, ∞, ..., ∞) with i ∈ {1, ..., n}. We will call them the infinite linesand since i covers the set {1, ..., n} they are 2n. We will show that the numberof intersection points of these lines, which we will call infinite points, is also 2n.In consequence Q will have the form of a polygon with 2n sides.

We can see this by choosing in the first place an orientation of P and a pointA ∈ P\ ∪n

i=1 δi. We then choose to call the first line that we meet, counter-clockwise, δ1, the second one δ2 until the nth δn. We choose now the n linearforms fi, i = 1, ..., n so as to have fi(A) = 1 for all i. Then, inside the sector

394 A. Mageira

formed by the lines δ1 and δn which contain the point A, the signs of theinfinite point will all be positive. If we pass the line δ1 to the sector formedby δ1 and δ2 above the point A, the values of f1 become negative, thus the firstsign will change to negative i.e. the infinite point will be (−∞, +∞, ..., +∞).With the same procedure we find only 2n possibilities of signs that have theform (+, ..., +, −, ..., −) and (−, ..., −, +, ..., +).

If we pass now from (R)n to Tn, we identify +∞ and −∞, which means that

a) we identify each side with the opposite oneb) we identify all vertices.

The first identification, gives rise to a Riemann surface Yg of whom thecanonical form is α1α2...αnα

−11 α−1

2 ...α−1n (cf. for example [11, 13]) which is

homeomorphic to

α1α2α−11 α−1

2 ...αn−1αnα−1n−1α

−1n

if n is even and

α1α2α−11 α−1

2 ...αn−2αn−1α−1n−2α

−1n−1αnα

−1n

if n is odd. It will be then a torus with g holes where g = E( n

2

).

We index the vertices by Z/2nZ. By doing the identification of the edges wealso identify the jth vertex with the j + n − 1th one for all j ∈ Z/2nZ. In otherwords two vertices are identified if and only if they are in the same orbit forthe addition of (n − 1) in Z/2nZ. But if n is even GCD(n − 1, 2n) = 1 and if nis odd GCD(n − 1, 2n) = 2.


Therefore the map Yg → Sp(A) is a homeomorphism if n is even.If n is odd, we get from Yg to Sp(A) by identifying those two points, the

spectrum Sp(A) is then homeomorphic to a pinched surface of genus g.Note that if we take the lattice F minus a line, for example let us choose

F ′ = F\{δn}, then for the corresponding graded C∗-algebras AF and AF ′ wehave AF ′ ⊂ AF and therefore we have a map � : Sp(AF ) → Sp(AF ′). If wedenote by AH = C0(P/H) for all H ∈ F , since F and F ′ are finite, theyare good lattices and Proposition 3.4 implies that Sp(AF ) = ⋃

H∈FSp(AH) and

Sp(AF ′) = ⋃

K∈F ′Sp(AK) .

Take χ ∈ Sp(AF ). There exists a unique H ∈ F such that χ = χH ∈Sp(AH). If we apply Remark 3.5 we get that: If H ∈ F ′, the image of χH by themap � is itself. If H ∈ F\F ′ i.e. H = δn, then inf{K ∈ F ′; δn ⊆ K} = P and theimage of χδn by � is in Sp(AP) i.e. it is the point at infinity.

In other words, one obtains the spectrum of AF ′ from the spectrum of AFby contracting the line P/δn to one point: the point at infinity.

6 Example of a C∗-Algebra Associated to a Semi Lattice

Let L be a semilattice and (Ai)i∈L the family of C∗-algebras defined by Ai =C for all i ∈ L. For i, j ∈ L such that i ≤ j set ϕi, j = IdC. These morphismssatisfy properties a) and b) of proposition 1.16 in [12], so there exists upto isomorphism a unique L-graded C∗-algebra (A, (Ai)i∈L) whose structuremorphism are the ϕi, j’s ([12], theorem 2.2). We want to compute its spectrumSp(A).

We will show that the set of nonempty final subsemilattices of L is inbijection with the set of characters of A.

First of all, denote by ei, i ∈ L the image of 1 ∈ Ai in A. It is easy to seethat ei is an idempotent for all i ∈ L therefore if χ is a character of A we haveχ(ei) ∈ {0, 1}. Moreover, by the definition of the product in A we have eie j =ei∧ j for all i, j ∈ L.

Let χ be a character of A. Set Mχ = {i ∈ L; χ(ei) = 1}. It is a subsemilatticeof L since for i, j ∈ Mχ we have χ(ei∧ j) = χ(eie j) = χ(ei)χ(e j) = 1. Moreoverit is a final subsemilattice because for i ∈ Mχ and j ∈ L such that i ≤ j we haveχ(e j) = χ(ei)χ(e j) = χ(eie j) = χ(ei) = 1 i.e. j ∈ Mχ .

Let M be a non zero final subsemilattice of L. There exists a unique

homomorphism χM : A → C satisfying χM(λei) ={

λ, i ∈ M0, i /∈ M for all λ ∈ C and

all i ∈ L. Indeed, since M is a final subsemilattice we have χM(λei)χM(μe j) =χM(λμei∧ j) then apply proposition 1.10 from [12]. Since M = ∅, χM is acharacter of A.

Note that since each character is determined by its value on ei (i ∈ L) themaps χ �→ Mχ and M �→ χM give the wanted bijection.

Finally, observe that since Ai = C, the spectrum Sp(Ai) has only one point

IdC. We have ψi(IdC)(e j) ={

1, i � j0, i � j i.e. ψi(IdC) = χLi . This will mean that a

396 A. Mageira

character χ of A is in⋃

i∈Lψi(Sp(Ai)) if and only if Mχ has the form of Li i.e. if

and only if it has a least element. In other words Sp(A) =⋃

i∈Lψi(Sp(Ai)) if and

only if L is a good semilattice.We will study now a particular case of this situation.

Example 6.1 Let us consider the set Q with its order. Note that Q being totallyordered, it is a semilattice which is not a good semilattice. We will study thespectrum of the associated graded C∗-algebra (A, (Ai)i∈Q).

Denote by Bb (R) the C∗-algebra of Borel bounded complex functionsdefined on R.

For any i ∈ Q we define a map τi : Ai → Bb (R) by τi(λei) = λ1]−∞,i] where1 is the characteristic function. On can see that τi is an injective morphism ofC∗-algebras satisfying the equality τi∧ j(λeiμe j) = τi(λei)τ j(μe j) for all i, j ∈ Q

and λ, μ ∈ C. Therefore there exists a unique morphism τ : A → Bb (R) whoserestriction to Ai is τi for all i ∈ Q.

To see that τ is injective we need to show that τF , the restriction of τ toAF , is injective for any finite subsemilattice F of Q ([12], proposition 1.9).Let F ∈ FQ. Assume that F has n elements. Since it is totally ordered we canalso assume that F = {i1, ..., in} with i1 < i2 < ... < in. Take x ∈ ker τF . Thenx =

∑

i∈Fλiei and

∑

i∈Fλi1]−∞,i](s) = 0 for all s ∈ R. If s is such that in−1 < s � in

we find λn = 0. If s is taken such that in−2 < s ≤ in−1 we have λn−1 + λn = 0therefore λn−1 = 0. In the same way we show that λi = 0 for all i ∈ F and theassertion follows.

We identify now A with its image by τ and we will show that τ(A) = Awhere A is the space of bounded Borel complex functions defined on R thatare regulated, left-continuous at all points of R, continuous on R\Q, zero at+∞ and that have a limit at −∞.

Note that A is a C∗-subalgebra of Bb (R) since it is closed. This impliestogether with the fact that AL is dense in A and τ(AL) ⊂ A that τ(A) ⊂ A.

Take f ∈ A. Assume that the limit of f at −∞ is equal to λ, then let ε > 0,there exists a ∈ Q such that for all x � a we have | f (x) − λ| < ε. On the otherhand f is zero at +∞ therefore there exists A ∈ Q such that for all x ≥ Awe have | f (x)| < ε. This means that for all x ∈] − ∞, a]∪]A, +∞[ we have| f (x) − λ1]−∞,a](x)| < ε.

Since f is a regulated function, it is uniformly approximated on [a, A] by astep function g. There exists then a subdivision i0, i1, ..., is of [a, A] such thati0 = a and is = A and g is equal to a constant cr over ]ir−1, ir[ for all r ∈ {1, ..., s}.

By assumption f is left-continuous at ir for all r ∈ {0, ..., s} then | f (ir) − cr| =lim

t→ir−| f (t) − g(t)| � ε.

If ir /∈ Q, the function f is continuous at ir therefore there exists jr ∈]ir, ir+1[∩Q such that for all t ∈ [ir, jr] we have | f (t) − f (ir)| < ε. Then fort ∈ [ir, jr] we have | f (t) − cr| � | f (t) − f (ir)| + | f (ir) − cr| < 2ε.


If ir ∈ Q, set jr = ir.We have now a new subdivision j0, ..., js of [a, A]. Let g be the function

defined by

g(t) =

⎧⎪⎪⎨

⎪⎪⎩

0, t > A

λ, t � a

cr, t ∈] jr−1, jr] with r ∈ {1, ..., s}.

In other words g = λ1]−∞,a] +s∑

r=1

cr1] jr−1, jr] = λτ(ea) +s∑

r=1

crτ(e jr − e jr−1) ∈τ(AF ) where F = { j0, ..., js} ∈ FQ. Since || f − g||∞ = sup

t∈R

| f (t) − g(t)| < 2ε

and τ(A) is closed then f ∈ τ(A).We have Sp(A) = Sp(τ (A)) = Sp(A). Since there is a bijection between

the characters χ of A and the nonempty final subsemilattices M of Q whichhave the only possible forms: M = Q or M = Q ∩ [t, +∞[≡ Lt or M =Q∩]t, +∞[≡ Mt with t ∈ Q then we have a natural bijection between χ and{Q} ∪ {Lt; t ∈ R} ∪ {Mt; t ∈ Q}.

Identify now the character which corresponds to each of these final subsemi-lattices through the isomorphism τ : A → A described as above. We find thefollowing cases:

• χQ(a) = lims→−∞ τ(a)(s),

• χLt(a) = τ(a)(t) for all t ∈ R,• χMt(a) = lim

s→t+ τ(a)(s) for all t ∈ Q,

(we only have to check it on the generators ei (i ∈ L)) that give the bijection ofthe spectrum of A with the set R

∐Q

∐{−∞}.

Acknowledgements I am much indebted to Georges Skandalis for his valuable advice and usefulcomments during the preparation of the paper.

References

1. Bourbaki, N.: Eléments des mathématiques, Topologie Générale, Chapitres 1 à 4, Hermann,Paris (1971)

2. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗-algebras in the N-body problem.J. Math. Phys. 32, 3101–3110 (1991)

3. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗-algebras and many-body pertur-bation theory. I. The N-body problem. C. R. Acad. Sci. Paris Sér. I Math. 312(6), 477–482(1991)

4. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗-algebras and many-body perturba-tion theory. II. The Mourre estimate. Astérisque 6–7(210), 75–96 (1992)

5. Boutet de Monvel-Berthier, A., Georgescu, V.: Graded C∗-algebras associated to symplecticspaces and spectral analysis of many channel Hamiltonians. In: Dynamics of Complex andIrregular Systems (Bielefeld 1991), Bielefeld Encount. Math. Phys., vol. 8, pp. 22–66. WorldScience, River Edge (1993)

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6. Damak, M., Georgescu, V.: C∗-cross products and a generalized mechanical N-body problem.Mathematical physics and quantum field theory. Electron. J. Differential Equations 4, 51–69(2000)

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8. Georgescu, V., Iftimovici, A.: C∗-algebras of energy observables. I. General theory and bumpsalgebras, preprint 00-521. http://www.ma.utexas.edu/mparc/

9. Georgescu, V., Iftimovici, A.: C∗-algebras of energy observables. II. Graded symplectic alge-bras and magnetic Hamiltonians, preprint 01-99. http://www.ma.utexas.edu/mparc/

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11. Massey, W.S.: Algebraic Topology: An Introduction. Springer, New York (1967)12. Mageira, A.: Graded C∗-algebras. J. Funct. Anal. 254, 1683–1701 (2008)13. Springer, G.: Introduction to Riemann Surfaces. Addison-Wesley, Reading (1957)

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