mathematical physics, analysis and geometry - volume 10

Math Phys Anal Geom (2007) 10:1–41DOI 10.1007/s11040-007-9019-2

Groupoids, von Neumann Algebrasand the Integrated Density of States

Daniel Lenz · Norbert Peyerimhoff · Ivan Veselic

Received: 9 March 2006 / Accepted: 12 March 2007 /Published online: 17 May 2007© D. Lenz, N. Peyerimhoff and I. Veselic 2007

Abstract We study spectral properties of random operators in the generalsetting of groupoids and von Neumann algebras. In particular, we establish anexplicit formula for the canonical trace of the von Neumann algebra of randomoperators and define an abstract density of states. While the treatment appliesto a general framework we lay special emphasis on three particular examples:random Schrödinger operators on manifolds, quantum percolation and quasi–crystal Hamiltonians. For these examples we show that the distribution func-tion of the abstract density of states coincides with the integrated density ofstates defined via an exhaustion procedure.

Keywords Groupoids · Von Neumann algebras · Integrated density of states ·Random operators · Schrödinger operators on manifolds · Trace formula

Mathematics Subject Classifications (2000) 46L10 · 35J10 · 46L51 · 82B44

1 Introduction

The aim of this paper is to review and present a unified treatment of basicfeatures of random (Schrödinger) operators using techniques from Connes’

D. Lenz (B) · I. VeselicFakultät für Mathematik, D-09107 TU Chemnitz, Germanye-mail: [email protected]:www.tu-chemnitz.de/mathematik/mathematische_physik/URL:www.tu-chemnitz.de/mathematik/schroedinger/

N. PeyerimhoffDepartment of Mathematical Sciences, Durham University, Durham, UKURL:http://www.maths.du.ac.uk/˜dma0np/

©2007 by D. Lenz, N. Peyerimhoff and I. Veselic. Reproduction, by any means, of the entirearticle for non-commercial purposes is permitted without charge.

2 D. Lenz et al.

noncommutative integration theory and von Neumann algebras [22]. Partic-ular emphasis will be laid on an application of the general setting to theexample of

– a group action on a manifold proposed by two of the authors [76].

This example merges and extends two situations, viz periodic operatorson manifolds as studied first by Adachi/Sunada [1] and random Schrödingeroperators on R

d or Zd as studied by various people (s. below) starting with the

work of Pastur [71]. In the first situation a key role is played by the geometry ofthe underlying manifold. In the second situation, the crucial ingredient is therandomness of the corresponding potential.

We also apply our discussion to two more examples:

– Random operators on tilings and Delone sets whose mathematically rigor-ous study goes back to Hof [38] and Kellendonk [47].

– Random operators on site-percolation graphs, see e.g. [14, 17, 25, 90].

As for the above three examples let us already point out the followingdifferences: in the first example the underlying geometric space is continuousand the group acting on it is discrete; in the second example the underlyinggeometric space is discrete and the group acting on it is continuous; finally, inthe third example both the underlying geometric space and the group actingon it are discrete.

The use of von Neumann algebras in the treatment of special random oper-ators is not new. It goes back at least to the seminal work of Šubin on almostperiodic operators [83]. These points will be discussed in more detail next.

Random Schrödinger operators arise in the quantum mechanical treatmentof disordered solids. This includes, in particular, periodic operators, almostperiodic operators and Anderson type operators on Z

d or Rd (cf. the textbooks

[20, 24, 51, 75, 85]). In all these cases one is given a family (Hω) of selfadjointoperators Hω acting on a Hilbert space Hω, indexed by ω in a measure space(�, μ) and satisfying an equivariance condition with respect to a certain set ofunitary operators (Ui)i∈I .

While specific examples of these cases exhibit very special spectral features,there are certain characteristics shared by all models. These properties are asfollows. (In parentheses we give a reference where the corresponding propertyis established.)

(P1) Almost sure constancy of the spectral properties of Hω givensome ergodicity condition. In particular, the spectrum � is nonrandom(Theorem 5.1).

(P2) Absence of discrete spectrum (Corollary 5.9) and, in fact, a dichotomy(between zero and infinity) for the values of the dimensions of spectralprojections.

(P3) A naturally arising von Neumann algebra (Section 3) with a canonicaltrace τ , to which the random operators are affiliated (Section 4).

Groupoids, von Neumann algebras and the IDS 3

(P4) A measure ρ, called the density of states, governing global features ofthe family (Hω), in particular, having � as its support (Proposition 5.2).This measure is related to the trace of the von Neumann algebra.

Let us furthermore single out the following point, which we show for thethree abovementioned examples:

(P5) A local procedure to calculate ρ via an exhaustion given some amenabil-ity condition. This is known as Pastur–Šubin trace formula. It impliesthe self-averaging property of the density of states (discussed for theexamples mentioned above in Sections 6, 7, 8).

Let us now discuss these facts for earlier studied models. The interest inproperty (P5) arouse from the physics of disordered media. First mathemat-ically rigorous results on the (integrated) density of states are due to Pastur[71–73], Fukushima, Nakao and Nagai [32–35, 70], Kotani [56], and Kirsch andMartinelli [52, 54]. In these papers two different methods for constructing theintegrated density of states (IDS) can be found (property (P5)). Either oneuses the Laplace transform to conclude the convergence of certain normalizedeigenvalue counting functions, or one analyzes the counting functions directlyvia the so called Dirichlet–Neumann bracketing. In our setting the Laplacetransform method seems to be of better use, since the pointwise superadditiveergodic theorem [2] used in the Dirichlet–Neumann bracketing approach [52]has no counterpart in the (nonabelian) generality we are aiming at.

For the more recent development in the study of the IDS of alloy typeand related models, as well as the results on its regularity and asymptoticbehaviour, see [20, 75, 85, 88] and the references cited there.

For almost periodic differential operators on Rd and the associated von

Neumann algebras, a thorough study of the above features (and many more)has been carried out in the seminal papers by Coburn, Moyer and Singer [21]and Šubin [83]. Almost periodic Schrödinger operators on Z

d and Rd were then

studied by Avron and Simon [4, 5].An abstract C∗-algebraic framework for the treatment of almost periodic

operators was then proposed and studied by Bellissard [6, 7] and Bellissard,Lima and Testard [9]. While these works focus on K-theory and the so calledgap-labeling, they also show (P1)–(P5) for almost periodic Schrödinger typeoperators on R

d and Zd. Let us emphasize that large parts of this C∗-algebraic

treatment are not confined to almost periodic operators. In fact, (P1)–(P4) areestablished there for crossed products arising from arbitrary actions of locallycompact abelian groups on locally compact spaces X.

After the work of Aubry/André [3] and the short announcement ofBellissard/Testard in [10], investigations in this framework, centered aroundso called spectral duality, were carried out by Kaminker and Xia [40] andChojnacki [18]. A special one-dimensional version of spectral duality basedon [37] can also be found in [59].

An operator algebraic framework of crossed-products (involving vonNeumann crossed products) can also be used in the study of general randomoperators if one considers R

d actions together with operators on L2(Rd)

4 D. Lenz et al.

(cf. [59]). However, certain of these models rather use actions of Zd together

with operators on L2(Rd), like the thoroughly studied alloy or continuousAnderson type models. This presents a difficulty which was overcome in a workby Kirsch [50] introducing a so called suspension construction, see also [9] forrelated material. This allows to “amplify” these Z

d actions to Rd actions and

thus reduce the treatment of (P1)–(P4) in the Zd case to the R

d case.In recent years three more classes of examples have been considered.

These are random operators on manifolds [60, 61, 76, 86, 87], discrete randomoperators on tilings [8, 38, 39, 47, 48, 62, 64], and random Hamiltoniansgenerated by percolation processes [14, 55, 89, 90]. In these cases the algebraicframework developed earlier could not be used to establish (P1)–(P5). Note,however, that partial results concerning, e.g., (P1) or restricted versions of(P5) are still available. Note also that continuous operators associated withtilings as discussed in [8, 13] fall within the C∗-algebraic framework of [6, 7]. Amore detailed analysis of the point spectrum of discrete operators associatedto tilings and percolation graphs will be carried out in [65].

The model considered in [76] includes periodic operators on manifolds. Infact, it was motivated by work of Adachi and Sunada [1], who establish anexhaustion construction for the IDS as well as a representation as a �-tracein the periodic case. For further investigations related to the IDS of periodicoperators in both discrete and continuous geometric settings, see e.g. [29–31,65, 67, 68].

More precisely, our first example concerned with Random Schrödingeroperators on Manifolds (RSM) can be described as follows,see [60, 76]:

Example (RSM) Let (X, g0) be the Riemannian covering of a compactRiemannian manifold M = X/�. We assume that there exists a family (gω)ω∈�

of Riemannian metrics on X which are parameterized by the elements of aprobability space (�,B�, P) and which are uniformly bounded by g0, i.e., thereexists a constant A � 1 such that

1

Ag0(v, v) � gω(v, v) � Ag0(v, v) for all v ∈ T X and ω ∈ �.

Let λω denote the Riemannian volume form corresponding to the metricgω. We assume that � acts ergodically on � by measure preserving trans-formations. The metrics are compatible in the sense that for all γ ∈ � thecorresponding deck transformations

γ : (X, gω) → (X, gγω)

are isometries. Then the induced maps U(ω,γ ) : L2(X, λγ −1ω) → L2(X, λω),(U(ω,γ ) f )(x) = f (γ −1x) are unitary operators. Based on this geometric setting,we consider a family (Hω : ω ∈ �), Hω = ω + Vω, of Schrödinger operatorssatisfying the following equivariance condition

Hω = U(ω,γ ) Hγ −1ωU∗(ω,γ ), (1)


for all γ ∈ � and ω ∈ �. We also assume some kind of weak measurability inω, namely, we will assume that

ω �→ 〈 f (ω, ·), F(Hω) f (ω, ·)〉ω :=∫

Xf (ω, x) [F(Hω) f ](ω, x) dλω(x) (2)

is measurable for every measurable f on � × X with f (ω, ·) ∈ L2(X, λω), ω ∈�, and every function F on R which is uniformly bounded on the spectra ofthe Hω. Note that L2(X, λω) considered as a set of functions (disregardingthe scalar product) is independent of ω. The expectation with respect to themeasure P will be denoted by E.

This example covers the following two particular cases:

(a) A family of Schrödinger operators ( + Vω)ω∈� on a fixed Riemannianmanifold (X, g0) with random potentials, see [76]. In this case theequivariance condition (1) transforms into the following property of thepotentials

Vγω(x) = Vω(γ −1x).

(b) A family of Laplacians ω on a manifold X with random metrics (gω)ω∈�

satisfying some additional assumptions [60].

By the properties of X and M in (RSM), the group � is discrete, finitelygenerated and acts cocompactly, freely and properly discontinuously on X.

In the physical literature the equivariance condition (1) is denoted eitheras equivariance condition, see e.g. [6], or as ergodicity of operators [51, 74],where it is assumed that the measure preserving transformations are ergodic.From the probabilistic point of view this property is simply the stationarity ofan operator valued stochastic process.

It is our aim here to present a groupoid based approach to (P1)–(P4)covering all examples studied so far. This includes, in particular, the case ofrandom operators on manifolds, the tiling case and the percolation case.

Our framework applies also to Schrödinger operators on hyperbolic space(e.g. the Poissonian model considered in [86]). However, our proof of (P5)does not apply to this setting because of the lack of amenability of the isometrygroup.

For the example (RSM) with amenable group action �, we will prove (P5)in Section 6. Note that case (a) of (RSM) includes the models treated earlierby the suspension construction. Thus, as a by product of our approach, weget an algebraic treatment of (P5) for these models. As mentioned already,our results can also be applied to further examples. Application to tilingsis discussed in [62, 64]. There, a uniform ergodic type theorem for tilingsalong with a strong version of Pastur–Šubin-formula (P5) is given. The resultsalso apply to random operators on percolation graphs. In particular theyprovide complementary information to the results in [55, 89, 90], where theintegrated density of states was defined rigorously for site and edge percolationHamiltonians. A basic discussion of these examples and the connection to our

6 D. Lenz et al.

study here is given in Sections 7 and 8, respectively. This will in particular showthat (P5) remains valid for these examples. For further details we refer to thecited literature. The results also apply to random operators on foliations (see[57] for related results).

Our approach is based on groupoids and Connes theory of noncommutativeintegration [22]. Thus, let us conclude this section by sketching the mainaspects of the groupoid framework used in this article. The work [22] onnoncommutative integration theory consists of three parts. In the first partan abstract version of integration on quotients is presented. This is then usedto introduce certain von Neumann algebras (viz. von Neumann algebras ofrandom operators) and to classify their semifinite normal weights. Finally,Connes studies an index type formula for foliations. We will be only concernedwith the first two parts of [22].

The starting point of the noncommutative integration is the fact that certainquotients spaces (e.g., those coming from ergodic actions) do not admit anontrivial measure theory, i.e., there do not exist many invariant measurablefunctions. To overcome this difficulty the inaccessible quotient is replaced bya nicer object, a groupoid. Groupoids admit many transverse functions, re-placing the invariant functions on the quotient. In fact, the notion of invariantfunction can be further generalized yielding the notion of random variable inthe sense of [22]. Such a random variable consists of a suitable bundle togetherwith a family of measures admitting an equivariant action of the groupoid. Thissituation gives rise to the so called von Neumann algebra of random operatorsand it turns out that the random operators of the form (Hω) introduced aboveare naturally affiliated to this von Neumann algebra. Moreover, each family(Hω) of random operators gives naturally rise to many random variables inthe sense of Connes. Integration of these random variables in the sense ofConnes yields quite general proofs for main features of random operators. Inparticular, an abstract version of the integrated density of states is induced bythe trace on the von Neumann algebra.

2 Abstract Setting for Basic Geometric Objects: Groupoidsand Random Variables

In this section we discuss an abstract generalization for the geometric sit-uation given in example (RSM). The motivation for this generalization isthat it covers many different settings at once, such as tilings, percolation,foliations, equivalence relations and our concrete situation, a group acting on ametric space.

Let us first introduce some basic general notations which are frequentlyused in this paper. For a given measurable space (S,B) we denote the set ofmeasures by M(S) and the corresponding set of measurable functions by F(S).The symbol F+(S) stands for the subset of nonnegative measurable functions.M f denotes the operator of multiplication with a function f .

We begin our abstract setting with a generalization of the action of the group� on the measurable space (�,B�). This generalization, given by G = � × � in


the case at hand, is called a groupoid. The main reason to consider it is the factthat it serves as a useful substitute of the quotient space �/�, which often is avery unpleasant space (e.g., in the case when � acts ergodically). The generaldefinition of a groupoid is as follows [79].

Definition 2.1 A triple (G, ·,−1 ) consisting of a set G, a partially definedassociative multiplication ·, and an inverse operation −1 : G → G is called agroupoid if the following conditions are satisfied:

– (g−1)−1 = g for all g ∈ G,– If g1 · g2 and g2 · g3 exist, then g1 · g2 · g3 exists as well,– g−1 · g exists always and g−1 · g · h = h, whenever g · h exists,– h · h−1 exists always and g · h · h−1 = g, whenever g · h exists.

A given groupoid G comes along with the following standard objects. Thesubset G0 = {g · g−1 | g ∈ G} is called the set of units. For g ∈ G we define itsrange r(g) by r(g) = g · g−1 and its source by s(g) = g−1 · g. Moreover, we setGω = r−1({ω}) for any unit ω ∈ G0. One easily checks that g · h exists if and onlyif r(h) = s(g).

The groupoids under consideration will always be measurable, i.e., theyposses a σ -algebra B such that all relevant maps are measurable. More pre-cisely, we require that · : G(2) → G, −1 : G → G, s, r : G → G0 are measurable,where

G(2) := {(g1, g2) | s(g1) = r(g2)} ⊂ G2

and G0 ⊂ G are equipped with the induced σ -algebras. Analogously, Gω ⊂ Gare measurable spaces with the induced σ -algebras.

As mentioned above, the groupoid associated with (RSM) is simply G =� × � and the corresponding operations are defined as

(ω, γ )−1 = (γ −1ω, γ −1), (3)

(ω1, γ1) · (ω2, γ2) = (ω1, γ1γ2), (4)

where the left hand side of (4) is only defined if ω1 = γ1ω2. It is very useful toconsider the elements (ω, γ ) of this groupoid as the set of arrows γ −1ω

γ�−→ ω.This yields a nice visualization of the operation · as concatenation of arrowsand of the operation −1 as reversing the arrow. The units G0 = {(ω, ε) | ω ∈ �}can canonically be identified with the elements of the probability space �.Via this identification, the maps s and r assign to each arrow its origin andits destination. Our groupoid can be seen as a bundle over the base space � ofunits with the fibers Gω = {(ω, γ ) | γ ∈ �} ∼= �. For simplicity, we henceforthrefer to the set of units as � also in the setting of an abstract groupoid. Thenotions associated with the groupoid are illustrated in Fig. 1 for both theabstract case and the concrete case of (RSM).

Next, we introduce an appropriate abstract object which corresponds to theRiemannian manifold X in (RSM).

8 D. Lenz et al.

Fig. 1 Notations of the groupoid G = � × � in (RSM)

Definition 2.2 Let G be a measurable groupoid with the previously introducednotations. A triple (X , π, J) is called a (measurable) G-space if the followingproperties are satisfied: X is a measurable space with associated σ -algebra BX .The map π : X → � is measurable. Moreover, with X ω = π−1({ω}), the mapJ assigns, to every g ∈ G, an isomorphism J(g) : X s(g) → X r(g) of measurablespaces with the properties J(g−1) = J(g)−1 and J(g1 · g2) = J(g1) ◦ J(g2) ifs(g1) = r(g2).

Note that a picture similar to Fig. 1 exists for a G-space X .An easy observation is that every groupoid G itself is a G-space with π = r

and J(g)h = g · h.The G-space in (RSM) is given by X = � × X together with the maps

π(ω, x) = ω and

J(ω, γ ) : X γ −1ω → X ω, J(ω, γ )(γ −1ω, x) = (ω, γ x).

Similarly to the groupoid G, an arbitrary G-space can be viewed as a bundleover the base � with fibers X ω.

Our next aim is to exhibit natural measures on these objects. We firstintroduce families of measures on the fibers Gω. In the case of (RSM), thiscan be viewed as an appropriate generalization of the Haar measure on �.

Definition 2.3 Let G be a measurable groupoid and the notation be given asabove.

(a) A kernel of G is a map ν : � → M(G) with the following properties:

– the map ω �→ νω( f ) is measurable for every f ∈ F+(G),– νω is supported on Gω, i.e., νω(G − Gω) = 0.


(b) A transverse function ν of G is a kernel satisfying the following invariancecondition ∫

Gs(g)

f (g · h)dνs(g)(h) =∫Gr(g)

f (k)dνr(g)(k)

for all g ∈ G and f ∈ F+(Gr(g)).

In (RSM) the discreteness of � implies that any kernel ν can be identifiedwith a function L ∈ F+(� × �) via νω = ∑

γ∈� L(ω, γ )δ(ω,γ ). For an arbitraryunimodular group �, the Haar measure m� induces a transverse function ν byνω = m� for all ω ∈ � on the groupoid � × � via the identification Gω ∼= �.

In the next definition we introduce appropriate measures on the base space� of an abstract groupoid G.

Definition 2.4 Let G be a measurable groupoid with a transverse function ν. Ameasure μ on the base space (�,B�) of units is called ν-invariant (or simplyinvariant, if there is no ambiguity in the choice of ν) if

μ ◦ ν = (μ ◦ ν)∼,

where (μ ◦ ν)( f )=∫�

νω( f )dμ(ω) and (μ ◦ ν)∼( f )=(μ ◦ ν)( f ) with f (g)=f (g−1).

In (RSM) it can easily be checked that, with the above choice ν ≡ m� , ameasure μ on (�,B�) is ν-invariant if and only if μ is �-invariant in theclassical sense, see [22, Cor. II.7] as well. Thus a canonical choice for am�-invariant measure on � is P.

Analogously to transverse functions on the groupoid, we introduce a cor-responding fiberwise consistent family α of measures on the G-space, see thenext definition. We refer to the resulting object (X , α) as a random variablein the sense of Connes. These random variables are useful substitutes formeasurable functions on the quotient space �/� with values in X. Measurablefunctions on �/� can be identified with �-invariant measurable functions on�. Note that, in the case of an ergodic action of � on �, there are no nontrivial�-invariant measurable functions, whereas there are usually lots of randomvariables in the sense of Connes (see below for examples).

Definition 2.5 Let G be a measurable groupoid and X be a G-space. A choice ofmeasures α : � → M(X ) is called a random variable (in the sense of Connes)with values in X if it has the following properties

– the map ω �→ αω( f ) is measurable for every f ∈ F+(X ),– αω is supported on X ω, i.e., αω(X − X ω) = 0,– α satisfies the following invariance condition∫

X s(g)

f (J(g)p)dαs(g)(p) =∫X r(g)

f (q)dαr(g)(q)

for all g ∈ G and f ∈ F+(X r(g)).

10 D. Lenz et al.

To simplify notation, we write gh, respectively, gp for g · h, respectively,J(g)p.

The general setting for the sequel consists of a groupoid G equipped witha fixed transverse function ν and an ν-invariant measure μ on �, and a fixedrandom variable (X , α). We use the following notation for the ‘averaging’ of au ∈ F+(X ) with respect to ν

ν ∗ u0(p) :=∫Gπ(p)

u0(g−1 p)dνπ(p)(g) for p ∈ X .

We will need the following further assumptions in order to apply theintegration theory developed in [22].

Definition 2.6 Let (G, ν, μ) be a measurable groupoid and (X , α) be a randomvariable on the associated G-space X satisfying the following two conditions

(6) The σ -algebras BX and B� are generated by a countable family of sets,all of which have finite measure, w.r.t. μ ◦ α (respectively w.r.t. μ).

(7) There exists a strictly positive function u0 ∈ F+(X ) satisfying ν ∗ u0(p) =1 for all p ∈ X .

Then we call the tupel (G, ν, μ,X , α, u0) an admissible setting.

Before continuing our investigation let us shortly discuss the relevance ofthe above conditions: Condition (6) is a strong type of separability conditionfor the Hilbert space L2(X , μ ◦ α). It enables us to use the techniques fromdirect integral theory discussed in Appendix A which are crucial to theconsiderations in Section 3.

Condition (7) is important to apply Connes’ non-commutative integrationtheory. Namely, it says that (X , J) is proper in the sense of Lemma III.2 andDefinition III.3 of [22]. Therefore (X , J) is square integrable by PropositionIV.12 of [22]. This square integrability in turn is a key condition for theapplications of [22] we give in Sections 3, 4 and 5.

On an intuitive level, (7) can be understood as providing an ‘embedding’of G into X . Namely, every u ∈ F(X ) with ν ∗ u ≡ 1 gives rise to the fibrewisedefined map q = qu : F(G) → F(X ) by

q( f )(p) :=∫Gπ(p)

u(g−1 p) f (g)dνπ(p)(g) (8)

for all p ∈ X . Note that the convolution property of u implies that the map(8) satisfies q(1G) = 1X . Moreover, q can be used to obtain new functions w ∈F(X ) satisfying ν ∗ w ≡ 1. This is the statement of the next proposition.

Proposition 2.7 Let u ∈ F+(X ) with ν ∗ u ≡ 1 be given, and q be as above. Forany function f ∈ F(G) with ν( f ) ≡ 1 on � we have ν ∗ q( f ) ≡ 1.


Proof The proof is given by the following direct calculation with ω=π(p)∈ �:

(ν ∗ q( f ))(p) =∫Gω

q( f )(g−1 p)dνω(g)

=∫Gω

∫Gs(g)

u(h−1g−1 p) f (h)dνs(g)(h)dνω(g)

(ν transverse function) =∫Gω

∫Gω

u(k−1 p) f (g−1k)dνω(k)dνω(g)

(Fubini) =∫Gω

u(k−1 p)

∫Gω

f (g−1k)dνω(g)dνω(k)

=∫Gω

u(k−1 p)

∫Gω

f (k−1g)dνω(g)dνω(k)

(ν( f ) ≡ 1) =∫Gω

u(k−1 p)dνω(k)

= 1.

Note that in the above calculation the integration variable g has the propertyr(g) = π(p) = ω. This finishes the proof. �

Remark 2.8 We consider examples of admissible settings in the case (RSM).Recall that we will identify Gω with � and X ω with X for all ω ∈ �. A transversefunction of G is given by copies of the Haar measure: νω = m� for all ω ∈ �. Xtogether with the Riemannian volume forms λω (corresponding to the metricsgω) on the fibers X ω ≡ X is an example of a random variable. This will beshown next by discussing validity of conditions (7) and (6).

Condition (7): Let D be a fundamental domain for the � action on X suchthat

⋃γ∈� γD = X is a disjoint union, c.f. [77, §6.5]. Then every function v ∈

F(�) with∑

γ∈� v(γ ) = 1 gives rise to a u0 satisfying ν ∗ u0 ≡ 1 by

u0(ω, x) = v(γ )

where γ ∈ � is the unique element with x ∈ γD. Note that this constructionassigns to a strictly positive v, again a strictly positive u0 on X . Hence thesetting (� × �, m�, P, � × X, λ, u0) satisfies condition (7).

Condition (6): This condition is clearly satisfied if the σ -algebra of �

is countably generated. In the case (RSM) (a) this countability conditioncan always be achieved by passing to an equivalent version of the definingstochastic process. Namely, given a random potential V : � × X → R, weconstruct a stochastic process V : � × X → R with the same finite dimensionaldistributions such that (6) is satisfied, c.f. e.g. [36, 75].

Assume at first that the random potential V : � × X → R can be written as

Vω(x) =∑γ∈�

fγ (ω, γ −1x). (9)

12 D. Lenz et al.

Here ( fγ )γ∈� is a sequence of measurable functions on the probability space �

with values in a separable Banach space B of functions on X. Such models havebeen studied by Kirsch in [49]. Note that the Borel-σ -algebra BB is generatedby a countable set E ⊂ BB. In this case � can be chosen to be the canonicalprobability space B� . Its σ -algebra is generated by a countable family ofcylinder sets of the form

{ω ∈ B�| ∀γ ∈ Hn : ω(γ ) ∈ Mγ }where H denotes a finite set of generators of �, n ∈ N and Mγ ∈ E for each γ .An appropriate choice for the (separable) Banach space B is:

�1(Lp)(X) :={

f : X → C

∣∣∣∣∫

�

(∫γD

| f (x)|pdλ(x)

)1/p

dm�(γ ) < ∞}

orLp(�1)(X) :={

f : X → C

∣∣∣∣(∫

D

(∫�

| f (γ −1x)|dm�(γ )

)p

dλ(x)

)1/p

< ∞}

.

Here D denotes an arbitrary �-fundamental domain in X. We have theinclusion

Lp(X) ⊂ �1(Lp(X)) ⊂ Lp(�1(X))

and Lp(�1(X)) is separable, cf. [49, 51]. More correctly, one could use thenotation �1(�, Lp(D)) for �1(Lp)(X) and Lp(D, �1(�)) for Lp(�1)(X).

Now we show that actually all models which are of the type (RSM) (a) canbe written in the way (9). Let u : X → R

+ be a bounded measurable functionsuch that

∑γ∈� u(γ −1x) = 1 for all x ∈ X. Let φ = Vu. Then

V(ω, x) =∑γ∈�

V(ω, x)u(γ −1x) =∑γ∈�

V(γ −1ω, γ −1x)u(γ −1x)

=∑γ∈�

φ(γ −1ω, γ −1x).

Setting fγ (ω, x) = φ(γ −1ω, x) we have a representation as in (9).The regularity assumptions on V in (RSM) imply that V(ω, ·) ∈ Lp

loc,unif (X)

for all ω ∈ � and all p ∈ [1, ∞]. Here Lploc,unif (X) = �∞(Lp)(X) denotes the

set of locally Lp-integrable functions f such that the Lp-norm of fχγD isuniformly bounded in γ ∈ �. The choice u = χD yields φ(ω, ·) ∈ Lp(X). Thusthe functions fγ (ω, ·) are in the Banach space B = Lp(X).

Summarizing the above considerations, we conclude that (RSM) (a) satisfiesboth condition (6) and (7) (after a suitable modification of the underlyingprobability space). The case (RSM) (b) can be treated similarly.

A crucial fact about the integration of random variables is given in thefollowing lemma, essentially contained in (the proof of) Lemma III.1 in [22].We include a proof for the convenience of the reader.


Lemma 2.9 Let G be a groupoid with transverse function ν and ν-invariantmeasure μ. Let furthermore X be a G-space.

(a) For a given transverse function φ on G, the integral∫�

φω( f ) dμ(ω) doesnot depend on f ∈ F+(G), provided f satisfies ν( f ) ≡ 1.

(b) For a given random variable α with values in X the integral∫�

αω(u) dμ(ω)

does not depend on u, provided u ∈ F+(X ) satisfies ν ∗ u ≡ 1.

Proof We prove first part (b). So let (X , α) and u, u′ ∈ F(X ) with ν ∗ u′ ≡ ν ∗u ≡ 1 be given. Inserting 1 ≡ ν ∗ u′, we calculate∫

�

αω(u) dμ(ω) =∫

�

∫X ω

u(p) · 1 · dαω(p) dμ(ω)

(Fubini) =∫

�

∫Gω

∫X ω

u′(g−1 p) u(p) dαω(p) dνω(g) dμ(ω)

(μ inv.) =∫

�

∫Gω

∫X s(g)

u′(gp′) u(p′) dαs(g)(p′) dνω(g) dμ(ω)

(α inv.) =∫

�

∫Gω

∫X ω

u′(p) u(g−1 p) dαω(p) dνω(g) dμ(ω).

Another application of Fubini and use of ν ∗ u ≡ 1 then gives∫�

αω(u)dμ(ω) =∫�

αω(u′)dμ(ω) and the proof of (b) is finished.Note that every groupoid G is a G-space X . Furthermore, in this case the

convolution condition ν ∗ u ≡ 1 translates into ν( f ) ≡ 1. This proves (a). �

Finally, we introduce, for a given G-space X , a new G-space X ×� X . Itconsists of the set

X ×� X := {(p, q) | π(p) = π(q)} ⊂ X × X .

equipped with the induced σ -algebra. The corresponding maps π and J ofX ×� X are defined in an obvious way. If X is actually a random variable withmeasures (αω)ω, then X ×� X inherits the structure of a random variable bysetting

(α ×� α)ω := αω ⊗ αω.

3 The von Neumann Algebra N (G, X )

In this section we discuss how a von Neumann Algebra arises naturally foran admissible setting (G, ν, μ,X , α, u0), cf. [22]. The random operators weare interested in are affiliated to this von Neumann algebra. Recall that aselfadjoint operator is called affiliated to a von Neumann algebra if its spectralfamily is contained in the von Neumann algebra.

A G-space X is, by definition, a bundle over � = G(0) via π : X −→ �. Thisbundle structure of a random variable (X , α) induces a bundle structure of

14 D. Lenz et al.

L2(X , μ ◦ α): Using the Fubini Theorem, we can associate with f ∈ L2(X , μ ◦α) a family ( fω)ω∈� of functions

fω ∈ L2(X ω, αω) such that f (x) = fπ(x)(x) (10)

for μ-almost all ω ∈ �. This way of decomposing will be a key tool in thesequel. On the technical level, this is very conveniently expressed using directintegral theory (see e.g. [28]) and the fact that there is a canonical isomorphism

L2(X , μ ◦ α) �∫ ⊕

�

L2(X ω, αω) dμ(ω). (11)

As direct integral theory tends to be rather technical, we try to avoid it in thesequel and rather give direct arguments which, however, are inspired by thegeneral theory (cf. Appendix A).

A special role will be played by those operators which respect the bundlestructure of L2(X , μ ◦ α) given by (10). Namely, we say that the (not neces-sarily bounded) operator A : L2(X , μ ◦ α) −→ L2(X , μ ◦ α) is decomposableif there exist operators Aω : L2(X ω, αω) −→ L2(X ω, αω) such that (Af )(x) =(Aω fω)(x), for almost every ω ∈ �. We then write A = ∫ ⊕

�Aω dμ(ω).

Let the groupoid (G, μ, ν) and the random variable (X , α) be as in the lastsection. For g ∈ G, let the unitary operator Ug be given by

Ug : L2(X s(g), αs(g)) −→ L2(X r(g), αr(g)), Ug f (p) := f (g−1 p).

A family (Aω)ω∈� of bounded operators Aω : L2(X ω, αω) → L2(X ω, αω) iscalled a bounded random operator if it satisfies :

– ω �→ 〈gω, Aω fω〉 is measurable for arbitrary f, g ∈ L2(X , μ ◦ α).– There exists a C � 0 with ‖Aω‖ � C for almost all ω ∈ �.– The equivariance condition Ar(g) = Ug As(g)U∗

g for all g ∈ G is satisfied.

Two bounded random operators (Aω), (Bω) are called equivalent, (Aω) ∼(Bω) if Aω = Bω for μ-almost every ω ∈ �. Each equivalence class of boundedrandom operators (Aω) gives rise to a bounded operator A on L2(X , μ ◦ α) byAf (p) := Aπ(p) fπ(p) (cf. Appendix A). This allows us to identify the class of(Aω) with the bounded operator A. The following is the main theorem on thestructure of the space of random operators.

Theorem 3.1 The set N (G,X ) of classes of bounded random operators is a vonNeumann algebra.

Proof This follows immediately from [22, Thm. V.2]. �

To get some insight we give a proof of Theorem 3.1 for the particular case of(RSM). In this example we can canonically identify L2(X ω, αω) with L2(X, λω)

and L2(X , μ ◦ α) with L2(� × X, P ◦ λ). Under this identification a boundedrandom operator is just a family of uniformly bounded operators

Aω : L2(X, λω) → L2(X, λω), ω ∈ �


with (ω, x) �→ (Aω fω)(x) measurable and ω �→ Aω fω ∈ L2(X, λω) for everyf ∈ L2(� × X, P ◦ λ), satisfying the equivariance condition

Aω = U(ω,γ ) Aγ −1ωU∗(ω,γ ), (12)

where we write Ug as U(ω,γ ) due to the product structure of G = � × �.In this case we can also associate with (Aω) an operator A on the Hilbert

space L2(� × X, P ◦ λ) by setting (Af )(ω, x) ≡ (Aω fω)(x). Now, (12) implies

AUγ = Uγ A, (13)

where Uγ is the unitary operator on L2(� × X, P ◦ λ) defined by

(Uγ f )ω(x) = fγ −1ω(γ −1x).

Conversely, one can show for a decomposable operator A that (13) implies(12) P-almost everywhere. Now, a suitable averaging procedure and a changeof the fibers Aω on a set of P-measure zero yields (12) for all ω (cf. [59] or[22, p. 88]).

It is well known that a bounded operator A on L2(� × X) is decomposable,i.e. A = ∫ ⊕

� Aω dP(ω) for a suitable family (Aω), if and only if A commuteswith the multiplication operators Mh◦π for every h ∈ L∞(�, P) (see, e.g., [28,Thm. 1 in II.2.5]). Summarizing these considerations, we conclude

N (� × �, � × X) = {Uγ | γ ∈ �}′ ∩ {Mh◦π | h ∈ L∞(�, P)}′,

where S′ denotes the commutant of a set S of operators. Obviously, {Uγ | γ ∈�} and {Mh◦π | h ∈ L∞(�, P)} are closed under taking adjoints. Thus, theircommutants are von Neumann algebras. This reasoning shows directly, in thecase of (RSM), that the space of random operators is a von Neumann algebra.

4 The Canonical Trace on N (G, X )

In this section we start with an admissible setting (G, ν, μ,X , α, u0) and itsassociated von Neumann algebra N (G,X ) of bounded operators on L2(X , μ ◦α). Let N+(G,X ) denote the set of non negative selfadjoint operators inN (G,X ). We will show that every operator A ∈ N+(G,X ) gives rise to a newrandom variable (X , βA). Integrating this random variable, we obtain a weighton N (G,X ). Under certain (mild) assumptions this weight can be shown to bea trace.

We start by associating a transversal function as well as a random variablewith each element in N+(G,X ). In the following, qω( f ) denotes the restrictionof q( f ) as defined in (8) to the fiber X ω.

16 D. Lenz et al.

Lemma 4.1 Let A ∈ N+(G,X ).

(a) Then φA, given by φωA( f ) := tr(Aω Mqω( f )), f ∈ F(Gω), defines a transverse

function.(b) Then βA, given by βω

A( f ) := tr(Aω Mf ), f ∈ F(X ω), is a random variable.

Proof This follows by direct calculation using the equivariance properties ofthe family Aω and the qω. �

Let us recall the following definitions. A weight on a von Neumann algebraN is a map τ : N+ → [0, ∞] satisfying τ(A + B) = τ(A) + τ(B) and τ(λA) =λτ(A) for arbitrary A, B ∈ N+ and λ � 0. The weight is called normal if τ(An)

converges to τ(A) whenever An is an increasing sequence of operators (i.e.An � An+1, n ∈ N) converging strongly to A. It is called faithful if τ(A) = 0implies A = 0. It is called semifinite if τ(A) = sup{τ(B) : B � A, τ (B) < ∞}.If a weight τ satisfies τ(CC∗) = τ(C∗C) for arbitrary C ∈ N (or equivalentlyτ(U AU∗) = τ(A) for arbitrary A ∈ N+ and unitary U ∈ N , cf. [28, Cor. 1 inI.6.1]), it is called a trace.

Theorem 4.2 For A ∈ N+(G,X ), the expression

τ(A) :=∫

�

tr(A12ω Muω

A12ω)dμ(ω) =

∫�

tr(M12uω

Aω M12uω

)dμ(ω)

does not depend on u ∈ F+(X ) provided ν ∗ u ≡ 1.

(a) The map τ : N+(G,X ) −→ [0, ∞] is a faithful, semifinite normal weighton N (G,X ).

(b) If the groupoid G satisfies the freeness condition

r−1(ω) ∩ s−1(ω) = {ω} for μ-almost all ω ∈ �, (14)

then τ is a trace.

Proof That τ is independent of u follows easily from Lemma 2.9 andLemma 4.1.

(a) The proof that τ is a normal weight is straightforward. The proof of thesemifiniteness of τ is not simple and we refer the reader to Theorem VI.2in [22]. To show that τ is faithful, assume τ(A) = 0 for A ∈ N+(G,X ).This implies tr(Muω

Aω) = 0 for almost every ω. As the trace tr is faithful,this implies Muω

Aω = 0 for almost every ω. Choosing a strictly positivefunction u (e.g. u = u0), we infer Aω = 0 for almost every ω and we seethat τ is faithful.

(b) The freeness condition (14) easily shows that Corollary VI.7 of [22] isapplicable and the statement follows. �


Remark 4.3

(a) Property (14) can be slightly relaxed (cf. [22]). However, some conditionof this form is necessary to prove the trace property in the generalityaddressed in the theorem. It will turn out to be dispensable in manyconcrete cases. Indeed, we will see below that τ is a trace in the case of(RSM) without any condition of this form. However, when consideringfactorial type properties this condition again plays an important role.

(b) The motivation to call property (14) a freeness condition is easily under-stood in the context of (RSM). Namely, in this case (14) is equivalent to

P(ω | γ −1ω = ω) = 0 for all γ ∈ �\{ε}. (15)

(c) Given condition (14) and some ergodicity assumptions (see the nextsection), it follows from [22] that the von Neumann algebra is actuallya factor. In this case the trace τ is unique (up to a scalar factor).

We are now heading towards an alternative direct proof of part (b) ofTheorem 4.2 in the particular case (RSM). This proof will be based on a studyof τ for certain operators with a kernel. It will in fact be more general, in thatit does not use the freeness condition (14) on the action of � on �. Along ourway, we will also find an instructive formula for τ on these operators. Let usemphasize that the approach to prove that τ is a trace given below is in no wayrestricted to (RSM) (cf. [62, 64] for related material).

An operator K on L2(X , μ ◦ α) is called a Carleman operator (cf. [91] forfurther details) if there exists a k ∈ F(X ×� X ) with

k(p, ·) ∈ L2(X π(p), απ(p)) for all p ∈ X

such that

K f (p) =∫X π(p)

k(p, q) f (q)dαπ(p)(q) =: Kπ(p) fπ(p)(p).

This k is called the kernel of K: Obviously, K = ∫ ⊕�

Kωdμ. Let K be the set ofall Carleman operators satisfying for all g ∈ G

k(gp, gq) = k(p, q) for απ(g) × απ(g) almost all p, q . (16)

Proposition 4.4 K is a right ideal in N (G,X ).

Proof Obviously, K ∈ K is decomposable. Thus, to show that K is a subset ofN (G,X ), it suffices to show the equivariance condition. But this is immediatefrom (16). Thus, it remains to show that K is a right ideal. This follows from astandard calculation in the theory of Carleman operators:

K Af (p) = Kπ(p) Aπ(p) fπ(p)(p) = 〈k(p, ·), Aπ(p) fπ(p)〉π(p),

18 D. Lenz et al.

where 〈 f1, f2〉ω = ∫X ω f1(q) f2(q)dαω(q). This shows that K A has a kernel kK A

given by

kK A(p, q) =(

A∗π(p)k(p, ·)

)(q),

which finishes the proof. �

For a Carleman operator K the expressions τ(KK∗) and τ(K∗K) candirectly be calculated.

Proposition 4.5 Let K ∈ K be given. Then we have

τ(K∗K) =∫

�

∫X ω

∫X ω

u(q)|k(p, q)|2dαω(p)dαω(q)dμ(ω) = τ(KK∗),

for any u ∈ F+(X ) satisfying ν ∗ u ≡ 1.

Proof We use the well known formula tr(A∗ A) = ∫S×S |a(x, y)|2dλ(x)dλ(y)

valid for any bounded operator A with kernel a. Note that this formula holdsfor both a ∈ L2(S × S, λ ⊗ λ) and a /∈ L2(S × S, λ ⊗ λ). In the latter case bothsides of the formula are just infinity. Now, we can calculate as follows:

τ(K∗K) =∫

�

tr(M1/2uω

K∗ω Kω M1/2

uω)dμ(ω)

=∫

�

∫X ω

∫X ω

u(q)|k(p, q)|2dαω(p)dαω(q)dμ(ω).

This proves the first equation. To show the second equation, we insert ν ∗ u ≡ 1in the first equation, finding

τ(K∗K)=∫

�

∫X ω

∫X ω

(∫Gω

u(g−1 p)dνω(g)

)u(q)|k(p, q)|2dαω(q)dαω(p)dμ(ω).

By Fubini and the fact that K is invariant, we infer

· · · =∫G

∫X r(g)

∫X r(g)

u(g−1 p)u(q)|k(g−1 p, g−1q)|2dαr(g)(p)dαr(g)(q)d(μ ◦ ν)(g).

Invoking invariance of μ, i.e. (μ ◦ ν)∼ = μ ◦ ν, we finally obtain

· · · =∫G

∫X s(g)

∫X s(g)

u(gp)u(q)|k(gp, gq)|2dαs(g)(p)dαs(g)(q)d(μ ◦ ν)(g)

=∫G

∫X r(g)

∫X r(g)

u(p′)u(g−1q′)|k(p′, q′)|2dαr(g)(p′)dαr(g)(q′)d(μ ◦ ν)(g).

Now, the desired equality follows by reversing the steps from the precedingcalculation. This finishes the proof. �

The proposition shows that τ has the trace property τ(AA∗) = τ(A∗ A) forall Carleman operators A. To show that τ is actually a trace, we need two moresteps. In the first step, we show that τ must be a trace if it satisfies this trace


property for sufficiently many operators. In the second step, we show that theset of Carleman operators is large.

Lemma 4.6 Let I be a right ideal in a von Neumann algebra N acting on aseparable Hilbert space H with I∗I(H) = H. Let τ be a normal weight on Nsatisfying τ(A∗ A) = τ(AA∗) for all A ∈ I. Then τ is a trace.

Proof It suffices to show τ(U AU∗) = τ(A) for arbitrary A ∈ N+ and unitaryU ∈ N . Let C(I) be the norm closure of I∗I. Obviously, I∗I is a dense ideal inC(I). By [27, Prop. 1.7.2], there exists then an increasing approximate identity(Iλ) for C(I) in I. As H is separable, we can choose (Iλ) to be a sequence (In).By I∗I(H) = H, we infer that In converges monotonously to the identity Id onH. By polarization, every In ∈ I∗I can be written as

In =∑

D∗i,n Di,n −

∑C∗

j,nC j,n, (17)

for suitable, finitely many Di,n, C j,n ∈ I. Now, let an arbitrary D ∈ I be given.Using that I is a right ideal, we can calculate

τ(U A12 D∗ DA

12 U∗)=τ(DA

12 U∗U A

12 D∗) = τ(DA

12 A

12 D∗) = τ(A

12 D∗ DA

12 ).

(18)

Combining (18), (17) and the fact that τ is normal, one can concludethe proof of the Lemma as follows: τ(U AU∗) = limn→∞ τ(U A

12 In A

12 U∗) =

limn→∞ τ(A12 In A

12 ) = τ(A). �

Proposition 4.7 Consider (RSM) and define, for t � 0, the operator S(t) :L2(� × X) → L2(� × X), by S(t) fω(x) := (e−tω fω)(x). Then, S(t) is a self-adjoint bounded random operator for each t � 0. For t > 0, the operator S(t)belongs to K. The family t �→ S(t) is a strongly continuous semigroup.

Proof Property (2) gives immediately the necessary measurability of S(t).Moreover, e−tω is a selfadjoint contraction for every ω ∈ �. Therefore,Fubini easily shows ‖S(t) f‖ � ‖ f‖ for every f ∈ L2(� × X). Thus, S(t) is adecomposable operator. As its fibres are selfadjoint, it is selfadjoint as well(cf. [78, Thm. XIII.85] and [27, Appendix A.78]). The transformation formulaω = U(ω,γ )γ −1ωU∗

(ω,γ ) shows that S(t) satisfies the equivariance property.Thus, S(t) is indeed a random operator. Using that t �→ e−tω is a stronglycontinuous semigroup of operators with norm not exceeding 1 for every ω ∈ �,we can directly calculate that S(t) is a strongly continuous semigroup, as well.

For t > 0, every S(t) has a kernel ktω (see, e.g., [16] or [80]). Using selfad-

jointness and the semigroup property, we can calculate∫X

|ktω(x, y)|2dλω(y) =

∫X

ktω(x, y)kt

ω(y, x)dλω(y) = k2tω (x, x) < ∞.

This shows that S(t) belongs to K. �

20 D. Lenz et al.

Now, the following theorem is not hard to prove.

Theorem 4.8 In the example (RSM), the map τ is a normal trace.

Proof By Proposition 4.5 and Lemma 4.6, it suffices to show that K∗KL2(X ×�) is dense in L2(� × X). But, this follows from the foregoing proposition,which shows that S(t) f converges to f for every f ∈ L2(� × X), where S(t) ∈K∗K by the semigroup property. �

5 Fundamental Results for Random Operators

In this section we present a comprehensive treatment of the basic features(P1)–(P4) of random operators mentioned in the introduction. This unifies andextends the corresponding known results about random Schrödinger operatorsin Euclidean space and random operators induced by tilings. As in the previoussection, we always assume that (G, ν, μ,X , α, u0) is an admissible setting.

A function f on � is called invariant if f ◦ r = f ◦ s. The groupoid G is saidto be ergodic (with respect to μ) if every invariant measurable function f isμ-almost everywhere constant. This translates, in the particular case (RSM), toan ergodic action of � on �. We will be mostly concerned with decomposableselfadjoint operators on L2(X , μ ◦ α).

For a selfadjoint operator H, we denote by σdisc(H), σess(H), σac(H), σsc(H)

and σpp(H) the discrete, essential, absolutely continuous, singular continuous,and pure point part of its spectrum, respectively.

Theorem 5.1 Let G be an ergodic groupoid. Let H = ∫ ⊕�

Hωdμ(ω) be a self-adjoint operator affiliated to N (G,X ). There exist �′ ⊂ � of full measure and�, �• ⊂ R, • = disc, ess, ac, sc, pp, such that

σ(Hω) = �, σ•(Hω) = �• for all ω ∈ �′

for • = disc, ess, ac, sc, pp. Moreover, σ(H) = �.

Note that σpp denotes the closure of the set of eigenvalues.

Proof The proof is essentially a variant of well known arguments (cf. e.g. [19,20, 24, 53, 58]). However, as our stetting is different and, technically speaking,involves direct integrals with non constant fibres, we sketch a proof.

Let J be the family of finite unions of open intervals in R, all of whoseendpoints are rational. Let J k consist of those elements of J which are unionsof exactly k intervals. Denote the spectral family of a selfadjoint operator Hby EH . It is not hard to see that ω �→ trEHω

(B) is an invariant measurablefunction for every B ∈ J (and, in fact, for every Borel measurable B ⊂ R).


Thus, by ergodicity, this map is almost surely constant. Denote this almost surevalue by fB. As J is countable, we find �′ ⊂ � of full measure, such that forevery ω ∈ �′, we have trEHω

(B) = fB for every B ∈ J . By

σ(H) = {λ ∈ R : EH(B) �= 0, for all B ∈ J with λ ∈ B }, (19)

and as tr is faithful, we infer constancy of σ(Hω) on �′. By a completelyanalogous argument, using σess(H) = {λ ∈ R : trEH(B) = ∞ for all B ∈ I withλ ∈ B}, we infer almost sure constancy of σess(Hω) and thus also of σdisc(Hω).

To show constancy of the remaining spectral parts, it suffices to showmeasurability of

ω �→ 〈gω, EppHω

(B)gω〉ω, and ω �→ 〈gω, EsingHω

(B)gω〉ω (20)

for every g ∈ L2(X, μ ◦ α) and every B ∈ J . Here, of course, EppH and Esing

Hdenote the restrictions of the spectral family to the pure point and singular partof the underlying Hilbert space, respectively. To show these measurabilities,recall that for an arbitrary measure μ on R with pure point part μpp andsingular part μsing we have

μsing(B)= limn→∞ sup

J∈J ,|J|�n−1

μ(B ∩ J), μpp(B)= limk→∞

limn→∞ sup

J∈J k,|J|�n−1

μ(B ∩ J).

Here, the first equation was proven by Carmona (see [19, 20, 24]), and the sec-ond follows by a similar argument. As this latter reasoning does not seem to bein the literature, we include a discussion in Appendix B. Given these equalities,(20) is an immediate consequence of measurability of ω �→ 〈gω, EHω

(B)gω〉ω,(which holds by assumption on H). (Note that instead of considering μpp asabove, one could have considered the continuous part μc of μ by a methodgiven in [20].)

It remains to show the last statement. Obviously, EH(B) = 0 if and onlyif EHω

(B) = 0 for almost every ω ∈ �. Using this, (19), and almost sureconstancy of trEHω

(B), infer that λ ∈ σ(H) if and only if EHω(B) �= 0 for every

B ∈ J with λ ∈ B and almost every ω ∈ �. This proves the last statement.Alternatively, one could follow the proof of [78, Thm.XIII.85] which is valid inthe case of non-constant fibres, too. �

Recall that a measure φ on R is a spectral measure for a selfadjoint operatorH with spectral family EH if, for Borel measurable B ⊂ R, φ(B) = 0 ⇔EH(B) = 0.

Proposition 5.2 For a Borel measurable B in R and a selfadjoint H affiliated toN (G,X ), let ρH(B) be defined by ρH(B) := τ(EH(B)). Then ρH is a spectralmeasure for H. Moreover, for a bounded measurable function F : R −→[0, ∞), the equality τ(F(H)) = ρH(F) := ∫

F(x)dρH(x) holds.

22 D. Lenz et al.

Proof As τ is a normal weight, ρH is a measure. As τ is faithful, ρH is a spectralmeasure. The last statement is then immediate for linear combinations ofcharacteristic functions with non negative coefficients. For arbitrary functionsit then follows after taking suitable monotone limits and using normality of τ .

�

Definition 5.3 The measure ρH is called (abstract) density of states.

Corollary 5.4 The topological support supp(ρH) = {λ ∈ R | ρH(]λ − ε, λ +ε[) > 0 for all ε > 0} coincides with the spectrum σ(H) of H for every selfad-joint H affiliated to N (G,X ). If G is, furthermore, ergodic this gives supp(ρH) =σ(Hω) for almost every ω ∈ �.

Proof The first statement is immediate as ρH is a spectral measure. The secondfollows from Theorem 5.1. �

Remark 5.5 In general, ρH is not the spectral measure of the fibre Hω. Actu-ally, there are several examples where the set

�′ := {ω ∈ �| ρH is a spectral measure for the operator Hω}has measure zero. We shortly discuss two classes of them (see [5] as well forrelated material).

Firstly we consider the case where (Hω)ω exhibits localization in an energyinterval I. This means that for a set �loc ⊂ � of full measure

σ(Hω) ∩ I �= ∅ and σc(Hω) ∩ I = ∅for all ω ∈ �loc. Examples of such ergodic, random operators on L2(Rd) and�2(Zd) can be found e.g. in the textbooks [20, 75, 85]. They particularly includerandom Schrödinger operators in one- and higher dimensional configurationspace. If Hω is an ergodic family of Schrödinger operators on �2(Zd) or L2(R)

one knows moreover for all energy values E ∈ R

P{�(E)} = 0, where �(E) := {ω ∈ �| E is an eigenvalue of Hω}.Assume that ρH is a spectral measure of Hω for some ω ∈ �loc. Then thereexists an eigenvalue E ∈ I of Hω and consequently ρH({E}) > 0. Thus ρH canonly be a spectral measure for Hω′ if ω′ ∈ �(E), but this set has measure zero.

Similarly, there are one-dimensional discrete random Schrödinger operatorswhich have purely singular continuous spectrum almost surely. An exampleis the almost Mathieu operator, cf. [5], for a certain range of parameters.Moreover, in [37] it is proven that the singular continuous components of thespectral measures of these models are almost surely pairwise orthogonal. Thusagain, ρH can be a spectral measure only for a set of ω of measure zero.


Lemma 5.6 Let G be ergodic with respect to μ with μ(�) < ∞ such that thefollowing exhaustion property holds:

There exists a sequence ( fn) in F+(G) with G =⋃

n{g : fn(g) > 0},

‖ fn‖∞ → 0 as n → ∞ and ν( fn) ≡ 1 for all n ∈ N. (21)

Then, for every transverse function φ, either φω(1) ≡ 0 almost surely or φω(1) ≡∞ almost surely.

Proof By ergodicity, φω(1) is constant almost surely. Assume φω(1) = c < ∞for almost every ω. Lemma 2.9 (a) implies that

∫�

φω( f )dμ(ω) is independentof f ∈ F+ with ν( f ) ≡ 1. For such f , we infer

Constant =∫

�

φω( f )dμ � ‖ f‖∞ c μ(�).

As this is in particular valid for every fn, n ∈ N, we infer Constant = 0. Thisshows φω( fn) = 0 for almost every ω and every n ∈ N. By G = ⋃

n{g : fn(g) >

0}, this gives φω(1) = 0 for almost every ω ∈ �. �

Remark 5.7 Let (G, ν, μ) be a measurable groupoid, (X , α) be an associatedrandom variable and u ∈ F+(X ) such that property (6) holds and ν ∗ u ≡ 1.Let fn be a sequence satisfying condition (21). Then (G, ν, μ,X , α, u0) withu0 = qu(

∑∞n=1

12n fn) is an admissible setting.

Remark 5.8 The exhaustion property (21) can easily be seen to hold in the case(RSM), if � is infinite. Namely, we can choose

fn(g) = fn(ω, γ ) = 1

|In|χIn(γ ).

Here, In is an exhaustion of the group �.

The lemma has an interesting spectral consequence.

Corollary 5.9 Let the assumptions of Lemma 5.6 be satisfied and the selfadjointH = ∫ ⊕

�Hωdμ(ω) be affiliated to N (G,X ). Then Hω has almost surely no

discrete spectrum.

Proof We already know that the discrete spectrum is constant almost surely.Let B be an arbitrary Borel measurable subset of R. Then, the mapω �→ tr(EHω

(B)Mqω(·)) is a transverse function, by Lemma 4.1 (a). Thus,tr(EHω

(B)Mqω(1)) = tr(EHω(B)) equals almost surely zero or infinity. As B is

arbitrary, this easily yields the statement. �

Remark 5.10 Corollary 5.9 is well known for certain classes of random opera-tors. However, our proof of the key ingredient, Lemma 5.6, is new. It seems tobe more general and more conceptual. Moreover, as discussed in the proof of

24 D. Lenz et al.

Theorem 5.11 below, Lemma 5.6 essentially implies that N (G,X ) is not type I.Thus, the above considerations establish a connection between the absence ofdiscrete spectrum and the type of the von Neumann algebra N (G,X ).

Let us finish this section by discussing factorial and type properties ofN (G,X ). By [22, Cor. V.8] (cf. Cor. V.7 of [22], as well), the von Neumannalgebra N is a factor (i.e. satisfies N ∩ N ′ = C Id) if G is ergodic with respectto μ and the freeness condition (14) holds.

There are three different types of factors. These types can be introduced invarious ways. We will focus on an approach centered around traces (cf. [23] forfurther discussion and references).

A factor is said to be of type I I I, if it does not admit a semifinite normaltrace. If a factor admits such a trace, then this trace must be unique (up toa multiplicative constant) and there are two cases. Namely, either, this traceassumes only a discrete set of values on the projections, or the range of thetrace on the projections is an interval of the form [0, a] with 0 < a � ∞. Inthe first case, the factor is said to be of type I. It must then be isomorphic tothe von Neumann algebra of bounded operators on a Hilbert space. In thesecond case, the factor is said to be of type I I.

Theorem 5.11 Let the assumptions of Lemma 5.6 and condition (14) be satis-fied, then N (G,X ) is a factor of type I I.

Proof By Lemma 5.6, there does not exist a bounded transversal function φ

whose support {ω : φω(1) �= 0} has positive μ measure. By [22, Cor. V.9], weinfer that N (G,X ) is not type I. On the other hand, as τ is a semifinite normaltrace on N by Theorem 4.2, it is not type I I I. �

Remark 5.12

(a) In the case (RSM), under the countability and freeness assumptions (6)and (14), we know that N (� × �, � × X) is actually a factor of type I I∞:Since the identity on an infinite dimensional Hilbert space has trace equalto infinity, we conclude

trL2(X)(MχD ) = trL2(D)(Id) = ∞,

where D is a fundamental domain as in Remark 2.8. Now, choosinguω(x) ≡ χD(x), we have ν ∗ u ≡ 1 and, consequently,

τ(Id) = E{tr(MχD )} = ∞.

(b) In the case of tiling groupoids and percolation models one finds factorsof type I I with a finite value of the canonical trace on the identity. Thisfinite value is determined by geometric features of the underlying tiling,respectively, the percolation process, cf. Sections 7 and 8.


6 The Pastur–Šubin-Trace Formula for (RSM)

The aim of this section is to give an explicit exhaustion construction for theabstract density of states (cf. Section 5) of the model (RSM). The constructionshows in particular that the IDS, the distribution function of the density ofstates, is self-averaging. This means that it can be expressed by a macroscopiclimit which is ω-independent, although one did note take the expectation overthe randomness.

We recall in this section the relevant definitions and results of [60] concern-ing the example (RSM) and outline the main steps of the proofs.

In the following we assume that the group � is amenable to be able toapply an appropriate ergodic theorem. The exhaustion procedure yields alimiting distribution function which coincides, at all continuity points, with thedistribution function of the abstract density of states. For the calculation of thisdistribution function the Laplace transformation has proved useful [71, 84]. Werefer to the second reference for a detailed description of the general strategy.

In Section 3 of [60] the operators Hω of (RSM) are defined via quadraticforms, the measurability of the latter is established, and the validity of the mea-surability condition (2) is deduced under the following additional hypotheses

(22) The map � × T X → R, (ω, v) �→ gω(v, v) is jointly measurable.(23) There is a Cg ∈ ]0, ∞[ such that

C−1g g0(v, v) � gω(v, v) � Cg g0(v, v) for all v ∈ T X.

(24) There is a Cρ ∈ ]0, ∞[ such that

|∇0 ρω(x)|0 � Cρ for all x ∈ X,

where ∇0 denotes the gradient with respect to g0, ρω is the uniquesmooth density of λ0 with respect to λω, and |v|20 = g0(v, v).

(25) There is a uniform lower bound K ∈ R for the Ricci curvatures of allRiemannian manifolds (X, gω). Explicitly, Ric(gω) � Kgω for all ω ∈ �

and on the whole of X.(26) Let V : � × X → R be a jointly measurable mapping such that for

all ω ∈ � the potential Vω := V(ω, ·) � 0 is in L1(A) for any compactA ⊂ X.

A key technique is an ergodic theorem by Lindenstrauss [66], valid foramenable groups � acting ergodically by measure preserving transformationson �. This theorem relies on suitable sequences (In)n, In ⊂ �, so calledtempered Følner sequences introduced by Shulman [82]. For an appropriatefundamental domain D (cf. §3 in [1]), the sequence (In)n induces a sequence(An)n, An ⊂ X by An = int(

⋃γ∈In

γD). The ergodic theorem in [66] together

26 D. Lenz et al.

with the equivariance property (1) imply that the following limits hold point-wise almost surely and in L1(�, P) sense

limn→∞

1

|In| tr(χAn e−tHω ) = E{tr(χDe−tH•)} (27)

limn→∞

1

|In|λω(An) = E(λ•(D)). (28)

Moreover the L∞(�, P) norms of the sequences(

tr(χAn e−tHω )

|In|)

n

,

(λω(An)

|In|)

n

,

( |In|λω(An)

)n

are uniformly bounded in the variable n ∈ N.Another important ingredient in the proof of (30) below is the following

heat kernel lemma (see [60, Lemma 7.2]): Let (An)n be as above. Then wehave

limn→∞ sup

ω∈�

1

λω(An)

∣∣tr(χAn e−tHω ) − tr(e−tHnω )

∣∣ = 0. (29)

The sets An together with the random family (Hω) of Schrödinger operatorson X are used to introduce the normalized eigenvalue counting functions

Nnω(λ) = |{i | λi(Hn

ω) < λ}|λω(An)

,

where Hnω denotes the restriction of Hω = + Vω to the domain An with

Dirichlet boundary conditions and λi(Hnω) denotes the ith eigenvalue of Hn

ω

counted with multiplicities. The cardinality of a set is denoted by | · |. Notethat tr(e−tHn

ω ) = ∫e−tλdNn

ω(λ). Using (27), (28) and (29) it is shown in [60] thatthere exists a distribution function NH : R → [0, ∞[, i.e., NH is left continuousand monotone increasing, such that the following P-almost sure pointwise andL1-convergence of Laplace-transforms holds true: for all t > 0

limn→∞ Nn

ω(t) = limn→∞

∫ ∞

−∞e−tλdNn

ω(λ) =∫ ∞

−∞e−tλdNH(λ) = NH(t), (30)

and that NH can be identified with the explicit expression

NH(t) = E(tr(χDe−tH•))

E(λ•(D))= τ(e−tH)

E(λ•(D)). (31)

This implies by the Pastur–Šubin-Lemma [71, 83] the following convergence(for P-almost all ω ∈ �)

limn→∞ Nn

ω(λ) = NH(λ) (32)

at all continuity points of NH . Note that NH does not depend on ω. Moreover,NH does not depend on the sequence (An)n as long as (An)n is chosen in theabove way. The function NH is called the integrated density of states.


Now, our trace formula reads as follows.

Theorem 6.1 Let the measure ρH be the abstract density of states introduced inSection 5. Then we have

NH(λ) = ρH (] − ∞, λ[)E(λ•(D))

at all continuity points λ of NH.

Remark 6.2 The theorem implies in particular

NH(λ) = 1

E(λ•(D))E

{tr

(χDEHω

(] − ∞, λ[))} ,

where D is a fundamental domain of � as in Remark 2.8. This alternativelocalized formula for the IDS is well-known in the Euclidean case. Note that itdoesn’t rely on a choice of boundary condition.

Proof By the uniqueness lemma for the Laplace transform (see Lemma C.1 inthe Appendix) it suffices to show that for all t > 0

E(λ•(D))

∫e−tλdNH(λ) =

∫e−tλdρH(λ). (33)

To this end we observe that by (31) τ(e−tH) = E(λ•(D))∫

e−tλdNH(λ) whichleaves to prove that

τ(e−tH) = ρH(e−tλ). (34)

To do so, we will use that the operator H is bounded below, say H � C,with a suitable C ∈ R. Define F : R −→ [0, ∞[ by F(λ) = e−tλ if λ � C and byF(λ) = 0 otherwise. By spectral calculus, we infer e−tH = F(H) and, in par-ticular, τ(e−tH) = τ(F(H)). By Proposition 5.2, this implies ρH(F) = τ(e−tH).As, again by Proposition 5.2, ρH is a spectral measure for H, its support iscontained in [C, ∞[ and we easily find ρH(e−tλ) = ρH(F). Combining theseequalities, we end up with the desired equality (34). �

7 Quasicrystal Models

In this section we shortly discuss how to use the above framework to study ran-dom operators associated to quasicrystals. Quasicrystals are usually modelledby tilings or Delone sets and these two approaches are essentially equivalent.Here, we work with Delone sets and follow [62–64] to which we refer forfurther details. The investigation of quasicrystals via groupoids goes back toKellendonk [47, 48] and his study of K-theory and gap labelling in this context(see [8, 62–64] for further discussion of quasicrystal groupoids and [11–13, 41]for recent work proving the so-called gap-labelling conjecture).

28 D. Lenz et al.

A subset ω of Rd is called Delone if there exist 0 < r � R such that r � ‖x −

y‖ whenever x, y ∈ ω with x �= y and ω ∩ {y : ‖y − x‖ � R} �= ∅ for all x ∈ Rd.

Here, ‖ · ‖ denotes the Euclidean norm on Rd.

There is a natural action T of Rd on the set of all Delone sets by translation

(i.e. Ttω = t + ω). Moreover, there is a topology (called the natural topologyby some authors) such that T is continuous. Then, (�, T) is called a Delonedynamical system if � is a compact T-invariant set of Delone sets.

In this case G(�, T) := � × Rd is clearly a groupoid with transversal func-

tion ν with νω = Lebesgue measure for all ω ∈ �. If μ is a T-invariant measureon �, it is an invariant measure on G(�, T) in the sense discussed above. Bythe compactness of � there exists at least one such non trivial μ by the Krylov–Bogolyubov theorem. In fact, in the prominent examples for quasicrystals,there is a unique such probability measure; these systems are called uniquelyergodic. This notation comes from the fact that this unique T-invarant measureis necessarily ergodic.

We now assume that (�, T) with invariant measure μ is given. Then, thereis a natural space X given by

X = {(ω, x) ∈ G(�, T) : x ∈ ω} ⊂ G(�, T).

Then, X inherits a topology from G(�, T) and is in fact a closed subset. Thespace X is fibred over � with fibre map

π : X −→ �, (ω, x) �→ ω.

Thus, the fibre X ω can naturally be identified with ω. In particular, everyg = (ω, x) ∈ G(�, T) gives rise to a isomorphism J(g) : X s(g) → X r(g), J(g)(ω −x, p) = (ω, p + x) and J(g1g2) = J(g1)J(g2) and J(g−1) = J(g)−1. Each fibre ω

carries the discrete measure αω giving the weight one to each point of ω. Then,(X , α) is a random variable.

Let furthermore u � 0 be a continuous function on Rd with

∫u(t)dt = 1.

Then, u gives rise to a function u0 on X via

u0(ω, x) = u(x)

and ∫G(�,T)π(p)

u0(γ−1 p)dνπ(p)(γ ) =

∫Rd

u(t)dt = 1.

Therefore, we are in an admissible setting.The freeness condition that γ −1ω �= ω whenever γ �= (ε, ω) is known as

aperiodicity.The associated operators are given by families Aω : �2(ω) −→ �2(ω), ω ∈ �

satisfying a measurability and boundedness assumption as well as the equivari-ance condition

Aω = U(ω,t) Aω−tU∗(ω,t)

for ω ∈ � and t ∈ Rd, where U(ω,t) : �2(ω − t) −→ �2(ω) is the unitary operator

induced by translation.


There is a canonical trace on these operators given by

τ(A) :=∫

�

tr(Mu0 Aω)dμ(ω).

It is not hard to see that τ(Id) < ∞ as there exists r > 0 with ‖x − y‖ � rwhenever x, y ∈ ω for ω ∈ � and x �= y. If μ is ergodic, then τ(Id) is just thedensity of points of almost all ω ∈ �. In this case, we can conclude from thediscussion in Section 5 that the discrete spectrum is absent. The necessarysequence fn is defined by

fn(ω, x) = 1

vol(Bn)χBn(x),

where Bn is the ball in Rd around the origin with radius n, χ denotes the

characteristic function and vol stands for Lebesgue measure.In fact, assuming ergodicity of μ together with aperiodicity, i.e. freeness, we

can even conclude from Section 5 that the von Neumann algebra of randomoperators is a factor of type I I1.

Of course, in the ergodic case the results of Section 5 can be applied. Theygive almost sure constancy of the spectral components and the possibility toexpress the spectrum of a random operator A with the help of the measure ρA

defined there by ρA(ϕ) = τ(ϕ(A)) for continuous ϕ on R with compact support.

Theorem 7.1 Let (Aω) be a selfadjoint random operator in the setting discussedin this section and assume that μ is ergodic. Then there exists �′ ⊂ � of full mea-sure and subsets of the real numbers � and �•, where • ∈ {disc, ess, ac, sc, pp},such that for all ω ∈ �′

σ(Aω) = � and σ•(Aω) = �•

for any • = disc, ess, ac, sc, pp. Moreover, �disc = ∅ and � coincides with thetopological support of ρA.

In the situation of the theorem it is also possible to calculate the distributionfunction of ρA by a limiting procedure. Details are discussed in the literaturecited above, see [63, 64]. Here, we mention the following results for so calledfinite range operators (Aω): Denote by | · | the number of elements of a set andby Aω|Bn the restriction of Aω to ω ∩ Bn. Define the measures μn

ω on R by

C0(R) � ϕ �→ μnω(ϕ) := 1

|Bn ∩ ω| tr ϕ(Aω|Bn).

These are just the measures associated to the eigenvalue counting functionsstudied so far, i.e.

Nnω(λ) := μn

ω(] − ∞, λ[) = |{i : λi(Aω|Bn) < λ}||Bn ∩ ω|

30 D. Lenz et al.

with the (ordered) eigenvalues λi(Aω|Bn) of Aω|Bn . Set D := τ(Id). Then, themeasures μn

ω converge for n → ∞ vaguely to the measure

ϕ �→ 1

Dτ(ϕ(A)) = 1

DρA(ϕ)

for almost every ω ∈ �. If (�, T) is uniquely ergodic, the convergence holdsfor all ω ∈ �. Assuming further regularity, one can even show convergence ofthe corresponding distribution functions with respect to the supremum norm.In any case there exists a distribution function NA such that limn→∞ Nn

ω(λ) =NA(λ) = 1

DρA(] − ∞, λ[) exists almost surely at all continuity points of NA.Note that amenability is not an issue here since the group R

d is abelian. Infact, instead of the balls Bn we could also consider rather general van Hovesequences.

Let us finish this section by giving an explicit example of what may be calleda nearest neighbour Laplacian in the context of a Delone set ω. For x ∈ ω,define the Voronoi cell V(x) of x by

V(x) := {p ∈ Rd : ‖p − x‖ � ‖p − y‖ for all y ∈ ω }.

Then, it is not hard to see that V(x) is a convex polytope for every x ∈ ω. Theoperator Aω is then defined via its matrix elements by

Aω(x, y) = 1 if V(x) and V(y) share a (d − 1)-dimensional face

and Aω(x, y) = 0 otherwise.

8 Percolation Models

In this section we shortly discuss how to fit percolation operators, more pre-cisely site percolation operators, in our framework. In fact, edge percolationor mixed percolation could be treated along the same lines.

Theoretical physicists have been interested in Laplacians on percolationgraphs as quantum mechanical Hamiltonians for quite a while [17, 25, 26, 46].Somewhat later several computational physics papers where devoted to thenumerical analysis of spectral properties of percolation Hamiltonians, seee.g [42–45, 81]. More recently there was a series of rigorous mathematicalresults on percolation models [14, 55, 69, 89, 90].

We have to identify the abstract quantities introduced in the abstract settingin the context of percolation.

A graph G may be equivalently defined by its vertex set X and its edgeset E, or by its vertex set X and the distance function d : X × X → {0} ∪ N.We choose the second option and tacitly identify the graph with its vertexset. In particular, each graph X gives naturally rise to the so called adjacencymatrix A(X) : X × X −→ {0, 1} defined by A(X)(x, y) = 1 if d(x, y) = 1 andA(X)(x, y) = 0 otherwise. This adjacency matrix can be considered to be anoperator on �2(X).


We assume that the vertex set of the graph is countable. Let � be a groupacting freely on X such that the quotient X/� is a finite graph, i.e. the �-actionis quasi-transitive. The associated probability space is given by � = {0, 1}X ,with the σ -field defined by the finite-dimensional cylinder sets. For simplicitylet us consider only independent, identically distributed percolation on thevertices of X. The statements hold analogously for correlated site or bondpercolation processes under appropriate ergodicity assumptions. Thus we aregiven a sequence of i.i.d. random variables ωx : � → {0, 1}, for x ∈ X, withdistribution measure pδ1 + (1 − p)δ0. The measure P on the probability space� is given by the product P = ⊗

X(pδ1 + (1 − p)δ0).The groupoid is given by G := � × � � g = (ω, γ ) and the G-space by

X := � × X.

The operation of the groupoid is the same as defined in (3) and (4), theprojection π is again given by π(ω, x) := ω for all (ω, x) ∈ � × X, and the mapJ is the same as described on page 7 for the model (RSM), where x now standsfor a vertex of the graph X.

Define for ω ∈ � the random subset

X(ω) := {x ∈ X | ωx = 1}of X. Defining α by αω := δX(ω) for all ω ∈ � we obtain a random variable(X , α).

Similarly as in the case of random Schrödinger operators on manifoldswe define the transverse function ν on the groupoid by setting νω equal tothe counting measure m� on the group for every ω ∈ � and the ν-invariantmeasure μ equal to P.

We next show that we are in an admissible setting according toDefinition 2.6. As for the countability condition in this definition, we proceedas follows: By the countability of the vertex set of the graph X the σ -field BX iscountably generated. Since B� is generated by the finite dimensional cylindersets, it is countably generated as well.

Denote by D a fundamental domain of the covering graph X and setu0(ω, x) := χD(x). Then clearly

∫Gπ(p)

u0(g−1 p)dνπ(p)(g) =∑γ∈�

u0(ω, γ x) = 1 for all ω ∈ �, x ∈ X.

This shows the second condition of Definition 2.6.The corresponding random operators are given by families Hω : �2

(X, αω) → �2(X, αω), ω ∈ �, satisfying a measurability and a boundednessassumption as well as the equivariance condition

Hω = U(ω,γ ) Hγ −1ωU∗(ω,γ )

for ω ∈ � and γ ∈ �, where U(ω,γ ) : �2(X, αγ −1ω) → �2(X, αω) is the unitaryoperator mapping φ to φ(γ −1 ·). A special example of such a random operator

32 D. Lenz et al.

is given by the family (Aω)ω, where each Aω is the adjacency matrix of theinduced subgraph of X generated by the vertex set X(ω).

We next show that the functional

τ(H) := E{tr(Mu0 Hω)}defined in Theorem 4.2 is a trace. As (a) of this Theorem holds by generalarguments, we just have to show that the freeness condition (14) is satisfied.This can be shown as follows: Since � acts freely, γ x �= x for all x ∈ X, γ ∈� \ {ε}. Thus the set {ω | ωγ x = ωx} has measure smaller than one for all x ∈ X.If the graph is infinite, this poses infinitely many independent conditions on theelements in

�γ := {ω | γω = ω}.Therefore, �γ has measure zero for each γ ∈ � \ {ε}. As � is countable,⋃

γ∈�\{ε} �γ has P-measure zero as well. Thus the desired freeness statementfollows.

To show that an associated operator family ω �→ Hω has almost surely nodiscrete spectrum for infinite �, we define the sequence of functions fn by

fn(ω, γ ) := 1

|In|χIn(γ ).

Here, In is an exhaustion of the infinite group �. The sequence ( fn) satisfiesproperty (21). Thus, Lemma 5.6 and Corollary 5.9 hold.

In fact, the above exhaustion and freeness properties allow us to applyTheorem 5.11 and conclude that the von Neumann algebra is of type I I. Moreprecisely

τ(Id) = E{tr(χD)} = p|D| < ∞shows that the type is I I1.

Besides the discrete, essential, absolutely continuous, singular continuous,and pure point spectrum, σdisc, σess, σac, σsc, σpp, the set σ f in consisting ofeigenvalues which posses an eigenfunction with finite support is a quantitywhich may be associated with the whole family Hω, ω ∈ �.

The following two theorems hold for a class of percolation Hamiltonians(Hω)ω which are obtained from a deterministic finite hopping range operatorby a percolation process, cf. [89]. In particular the class contains the adjacencyoperator (Aω)ω introduced above.

Recall from Section 5 that the measure ρH on R is given by ρH(ϕ) = τ(ϕ(H))

for continuous ϕ on R with compact support.

Theorem 8.1 There exists an �′ ⊂ � of full measure and subsets of the realnumbers � and �•, where • ∈ {disc, ess, ac, sc, pp, f in}, such that for all ω ∈ �′

σ(Hω) = � and σ•(Hω) = �•


for any • = disc, ess, ac, sc, pp, f in. Moreover, the almost-sure spectrum �

coincides with the topological support of ρH. If � is infinite, �disc = ∅.

If the group � acting on X is amenable it was shown in [89] that property(P5), too, holds for the family (Hω)ω. More precisely, it is possible to constructthe measure ρH by an exhaustion procedure using the finite volume eigenvaluecounting functions Nn

ω(λ). These are defined by the formula

Nnω(λ) := |{i ∈ N | λi(Hn

ω) < λ}||In| · |D| ,

where In is a tempered Følner sequence, An = ⋃γ∈In

γD, and Hnω is the

restriction of the operator Hω to the space �2(X(ω) ∩ An).

Theorem 8.2 Let � be amenable and {In} be a tempered Følner sequence of �.Then there exists a subset �′ ⊂ � of full measure and a distribution functionNH, called integrated density of states, such that for all ω ∈ �′

limn→∞ Nn

ω(λ) = NH(λ), (35)

at all continuity points of NH. NH is related to the measure ρH via the followingtrace formula

NH(λ) = ρH(] − ∞, λ[)|D| . (36)

Appendix A: Some Direct Integral Theory

The aim of this appendix is to prove the following lemma and to discusssome of its consequences. By standard direct integral theory [28], the lemmais essentially equivalent to the statement that L2(X , μ ◦ α) is canonicallyisomorphic to

∫ ⊕�

L2(X ω, αω) dμ(ω).Throughout this section let a measurable groupoid (G, ν, μ) and a random

variable (X , α) satisfying condition (6) on the associated G-space be given.

Lemma A.1 There exists a measurable function N : � −→ N0 ∪ {∞}, a se-quence (g(n))n of measurable functions on X and a subset �′ of � of full measuresatisfying the following:

– (g(n)ω : 1 � n � N(ω)) is an orthonormal basis of L2(X ω, αω) for every

ω ∈ �′.– g(n)

ω = 0 for n > N(ω), ω ∈ �′.– g(n)

ω = 0 for ω /∈ �′.

Proof Let D be a countable generator of the σ -algebra of X such that μ ◦α(D) < ∞ for every D ∈ D. Such a D exists by condition (6). Let ( f (n))n∈N be

34 D. Lenz et al.

the family of characteristic functions of sets in D. By assumption on D, we inferthat ( f (n)) are total in L2(X , μ ◦ α).

By the Fubini Theorem, there exist a set �′ of full measure, such that f (n)ω

belongs to L2(X ω, αω) for every n ∈ N and every ω ∈ �′. As D generates theσ -algebra of X and X ω is equipped with the induced σ -algebra, we infer that( f (n)

ω )n∈N is total in L2(X , αω) for every ω ∈ �′.Now, define for n ∈ N the function h(n) ∈ L2(X , μ ◦ α) by setting h(n)(p) =

f (n)(p) if π(p) ∈ �′ and h(n)(p) = 0, otherwise.Applying the Gram–Schmidt-orthogonalization procedure to (h(n)

ω )n∈N si-multaneously for all ω ∈ �′, we find N : � −→ N0 ∪ {∞} and g(n)

ω as desired.(This simultaneous Gram-Schmidt procedure is a standard tool in directintegral theory, see [28] for details.) The proof of the Lemma is finished. �

Proposition A.2 Let (Aω) be a family of bounded operators Aω :L2(X ω, αω) −→ L2(X ω, αω) such that

ω �→ 〈 fω, Aω gω〉ω is measurable for arbitrary f, g ∈ L2(X , μ ◦ α). (37)

Then, Ah : X −→ C, (Ah)(p) ≡ (Aπ(p)hπ(p))(p) is measurable for every h :X −→ C measurable with hω ∈ L2(X ω, αω) for every ω ∈ �.

Proof Let h be as in the assumption. Invoking suitable cutoff procedures, wecan assume without loss of generality

h ∈ L2(X , μ ◦ α) as well as ‖Aω‖ � C for all ω ∈ �, (38)

for a suitable C independent of ω. Obviously, (37) implies that ω �→〈 fω, Aωgω〉ω is measurable for every f, g : X −→ C measurable with fω, gω ∈L2(X ω, αω) for every ω ∈ �. Thus, with g(n), n ∈ N, as in the previous lemma,we see that

ω �→ 〈g(n)ω , Aωhω〉 is measurable for every n ∈ N.

In particular,

X −→ C, p �→ 〈g(n)

π(p), Aπ(p)hπ(p)〉π(p) g(n)(p) is measurable for every n ∈ N.

As (g(n)ω : 1 � n � N(ω)) is an orthonormal basis in L2(X ω, αω) for every ω ∈

�′ and g(n)ω = 0 for n > N(ω) and ω ∈ �′, we have

(Aωhω)(p) =∞∑

n=1

〈g(n)ω , (Aωhω)〉ω g(n)(p)

for almost every ω ∈ �. Note that the last equality holds in the L2(X ω,

αω)-sense. By (38) and the Fubini Theorem, this implies (Aωhω)(p) =∑∞n=1〈g(n)

ω , (Aωhω)〉ω g(n)(p) in the sense of L2(X , μ ◦ α) and the desired mea-surability follows. �


Proposition A.3 Let C > 0 and (Aω) be a family of operators Aω : L2(X ω,

αω) −→ L2(X ω, αω) with ‖Aω‖ � C for every ω ∈ � and p �→ (Aπ(p) fπ(p))(p)

measurable for every f ∈ L2(X , μ ◦ α). Let A : L2(X , μ ◦ α) −→ L2(X , μ ◦α), (Af )(p) = (Aπ(p) fπ(p))(p) be the associated operator. Then, Aω = 0 forμ-almost every ω ∈ � if A = 0.

Proof Choose g(n), n ∈ N, as in Lemma A.1 and let E be a countable densesubset of L2(�, μ). For f ∈ L2(X , μ ◦ α) with f (p) = g(n)(p)ψ(π(p)) for n ∈N and ψ ∈ E , we can then calculate

0 = Af = (p �→ ψ(π(p))(Aπ(p)g(n)

π(p))(p)).

As E is dense and countable, we infer, for μ-almost all ω ∈ �,

Aωg(n)ω = 0 for all n ∈ N.

This proves the statement, as (g(n)ω : n ∈ N) is total in L2(X ω, αω) for almost

every ω ∈ �. �

Corollary A.4 Let (Aω) and (Bω) be random operators with associated opera-tors A and B, respectively. Then A = B implies (Aω) ∼ (Bω).

Proof This is immediate from the foregoing proposition. �

Appendix B: A Proposition from Measure Theory

In this appendix we give a way to calculate the point part of a finite measure onR. Recall that a measure is called continuous if it does not have a point part.

We start with the following Lemma.

Lemma B.1 Let μ be a continuous finite measure on R. Then limn→∞ μ(In) = 0for every sequence (In) of open intervals whose lengths tend to zero.

Proof Assume the contrary. Then there exists a sequence of open intervals(In) with |In| → 0 and a δ > 0 with μ(In) � δ, n ∈ N. For each n ∈ N choose anarbitrary xn ∈ In.

If the sequence (xn) were unbounded, one could find a subsequence (Ink)k∈N

of (In) consisting of pairwise disjoint intervals. This would imply the contradic-tion μ(R) �

∑∞k=1 μ(Ink) �

∑∞k=1 δ = ∞.

Thus, the sequence (xn) is bounded and therefore contains a convergingsubsequence. Without loss of generality we assume that xn → x for n → ∞.For every open interval I containing x we then have μ(I) � μ(In) for n largeenough. This gives μ(I) � δ for every such interval. From Lebesgue Theorem,we then infer μ({x}) � δ, contradicting the continuity of μ. �

36 D. Lenz et al.

Proposition B.2 Let μ be a finite measure on R with point part μpp andcontinuous part μc. Then, for every B ⊂ R, μpp(B) is given by

μpp(B) = limk→∞

limn→∞ sup

|J|�n−1,J∈J k

μ(B ∩ J).

Proof Obviously, the limits on the right hand side of the formula make sense.We show two inequalities:

“�”: Let {xi} be a countable subset of R with μpp = ∑i μ({xi})δxi , where

δx denotes the point measure with mass one at x. Then, we have μpp(B) =∑xi∈B μ({xi}). For every ε > 0, we can then find a finite subset Bε of {xi : xi ∈

B} with number of elements #Bε and μpp(B) � ε + ∑x∈Bε

μ({x}). This easilygives

μpp(B) � ε +∑x∈Bε

μ({x}) � ε + μ(B ∩ I)

for suitable J ∈ J #Bε of arbitrary small Lebesgue measure. As ε is arbitrary,the desired inequality follows.

“�”: By the foregoing lemma, we easily conclude for every k ∈ N thatlimn→∞ supJ∈J k,|J|�n−1 μc(B ∩ J) = 0. Combining this with the obvious in-equality μpp(B) � μpp(B ∩ J) valid for arbitrary measurable B, J ⊂ R, weinfer

μpp(B) � limk→∞

limn→∞ sup

|J|�n−1,J∈J k

μpp(B ∩ J)

= limk→∞

limn→∞ sup

|J|�n−1,J∈J k

μpp(B ∩ J) + limk→∞

limn→∞ sup

|J|�n−1,J∈J k

μc(B ∩ J)

= limk→∞

limn→∞ sup

|J|�n−1,J∈J k

μ(B ∩ J).

This finishes the proof. �

Appendix C: Uniqueness Lemma for the Laplace Transform

The results in [76] use heavily the Laplace transform techniques developed inpapers by Pastur and Šubin [71, 84]. In the present paper only the uniquenesslemma is used. In the literature the uniqueness lemma for the Laplace trans-form is stated mostly for finite measures (e.g., Theorem 22.2 in [15]). For theconvenience of the reader we show how to adapt the uniqueness result to ourcase, where the distribution function NH is unbounded.

Lemma C.1 Let f1, f2 : ]0, ∞[→ R be monotonously increasing function withlimλ↘0 f1(λ) = limλ↘0 f2(λ) = 0. Let the integrals

∫ ∞

0e−tλdf j(λ), j = 1, 2, t > 0 (39)


be finite and moreover ∫ ∞

0e−tλdf1(λ) =

∫ ∞

0e−tλdf2(λ) (40)

for all positive t.Then the sets of continuity points of f1 and f2 coincide and for λ0 in this set

we have f1(λ0) = f2(λ0).

Proof Choose s > 0 arbitrary. The measures

μ j(g) :=∫ ∞

0g(λ) e−sλ df j(λ) (41)

are finite, since μ j(1) = ∫ ∞0 e−sλ df j(λ) < ∞ by assumption. Since

μ j(e−t·) =∫ ∞

0e−(t+s)λ df j(λ) = f j(t + s), (42)

we have, by assumption, that the Laplace transforms of the measures μ j

coincide for all t > 0:

μ1(e−t·) = μ2(e−t·). (43)

As we are dealing with finite measures, the Theorem 22.2 in [15] implies μ1 =μ2. We consider

μ j([0, E]) =∫ E

0e−sλ df j(λ) (44)

as a sequence of integrals depending on the parameter s → 0. Since

e−s· : [0, λ0] → [0, 1] (45)

converges uniformly and monotonously to the constant function 1, we con-clude by Beppo Levi’s theorem

lims↘0

μ j([0, λ0]) =∫ λ0

0df j(λ). (46)

For a continuity point λ0 of f1 we have∫ λ0

0df1(λ) = f1(λ0), (47)

which implies f1(λ0) = f2(λ0). �

Corollary C.2 Under the assumptions of the Lemma C.1 we have∫ ∞

0g(λ) df1(λ) =

∫ ∞

0g(λ) df2(λ) (48)

for all continuous functions g with compact support.

38 D. Lenz et al.

Acknowledgements It is a pleasure to thank W. Kirsch and P. Stollmann for various stimulatingdiscussions on random operators and hospitality at Ruhr-Universität Bochum, respectively, TUChemnitz. This work was supported in part by the DFG through the SFB 237, the Schwerpunkt-programm 1033, and grants Ve 253/1 & Ve 253/2 within the Emmy-Noether Programme.

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Well-posedness for Semi-relativistic HartreeEquations of Critical Type

Enno Lenzmann

Received: 1 April 2006 / Accepted: 1 December 2006 /Published online: 26 May 2007© Springer Science + Business Media B.V. 2007

Abstract We prove local and global well-posedness for semi-relativistic, non-linear Schrödinger equations i∂tu = √−� + m2u + F(u) with initial data inHs(R3), s � 1/2. Here F(u) is a critical Hartree nonlinearity that correspondsto Coulomb or Yukawa type self-interactions. For focusing F(u), which arisein the quantum theory of boson stars, we derive global-in-time existence forsmall initial data, where the smallness condition is expressed in terms of theL2-norm of solitary wave ground states. Our proof of well-posedness does notrely on Strichartz type estimates. As a major benefit from this, our methodenables us to consider external potentials of a quite general class.

Keywords Well-posedness · Cauchy problem · Semi-relativisticHartree equation · Boson stars

Mathematics Subject Classifications (2000) 35Q40 · 35Q55 · 47J35

1 Introduction

In this paper we study the Cauchy problem for nonlinear Schrödinger equa-tions with kinetic energy part originating from special relativity. That is, weconsider the initial value problem for

i∂tu =√

−� + m2 u + F(u), (t, x) ∈ R1+3, (1)

E. Lenzmann (B)Department of Mathematics, Massachusetts Institute of TechnologyBuilding 2, Room 230, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USAe-mail: [email protected]

44 E. Lenzmann

where u(t, x) is complex-valued, m � 0 denotes a given mass parameter, andF(u) is some nonlinearity. Here the operator

√−� + m2 is defined via itssymbol

√ξ 2 + m2 in Fourier space.

Such “semi-relativistic” equations have (though not Lorentz covariant ingeneral) interesting applications in the quantum theory for large systems ofself-interacting, relativistic bosons. Equation (1) arises, for instance, as aneffective description of boson stars, see, e. g., [4, 5, 12], where F(u) is a focusingHartree nonlinearity given by

F(u) =(

λ

|x| ∗ |u|2)

u, (2)

with some constant λ < 0 and ∗ as convolution. Motivated by this physical ex-ample with focusing self-interaction of Coulomb type, we address the Cauchyproblem for (1) and a class of Hartree nonlinearities including (2). In fact,we shall prove well-posedness for initial data u(0, x) = u0(x) in Hs = Hs(R3),s � 1/2; see Theorems 1–3 below.

Let us briefly point out a decisive feature of the example cited in (2)above. Apart from its physical relevance, the nonlinearity given by (2) leadsto an L2-critical equation as indicated by the fact that the coupling constantλ has to be dimensionless. In consequence of this, L2-smallness of the initialdatum enters as a sufficient condition for global-in-time solutions. More pre-cisely, we derive for u0 ∈ Hs, s � 1/2, the following criterion implying globalwell-posedness

∫

R3|u0(x)|2 dx <

∫

R3|Q(x)|2 dx. (3)

This condition holds irrespectively of the parameter m � 0 in (1); seeTheorem 2 below. Here Q ∈ H1/2 is a positive solution (ground state) for thenonlinear equation

√−� Q +(

λ

|x| ∗ |Q|2)

Q = −Q, (4)

which gives rise to solitary wave solutions, u(t, x) = eit Q(x), for (1) with m = 0.In fact, it can be shown that criterion (3) guaranteeing global-in-time solutionsin the focusing case is optimal in the sense that there exist solutions, u(t), with‖u0‖2

2 > ‖Q‖22, which blow up within finite time; see [7]. Physically, this blow-

up phenomenon indicates “gravitational collapse” of a boson star whose massexceeds a critical value.

Furthermore, criterion (3) can be linked with established results as follows.First, it is reminiscent to a well-known condition derived in [16] for globalwell-posedness of nonrelativistic Schrödinger equations with focusing, localnonlinearity (see also [13] for Hartree nonlinearities). Second, criterion (3) isin accordance with a sufficient stability condition proved in [12] for the relatedtime-independent problem (i. e., a static boson star); see [6] for more detailsconcerning known results on Hartree equations.

We now give an outline of our methods. The proof of well-posednesspresented below does not rely on Strichartz (i. e., space-time) estimates for the

Well-posedness for semi-relativistic Hartree equations of critical type 45

propagator, e−it√−�+m2 , but it employs sharp estimates (e. g., Kato’s inequality

(17) below) to derive local Lipschitz continuity of L2-critical nonlinearitiesof Hartree type. Local well-posedness then follows by standard methods forabstract evolution equations. Furthermore, global well-posedness is derivedby means of a priori estimates and conservation of charge and energy whoseproof requires a regularization method.

This paper is organized as follows.

– In Section 2 we introduce a class of critical Hartree nonlinearities including(2). First, we state Theorems 1 and 2 that establish local and globalwell-posedness in energy space H1/2 for this class of nonlinearities. InTheorem 3 we extend these results to Hs, for every s � 1/2. Finally,external potentials are included, i. e., we consider

i∂tu =(√

−� + m2 + V)

u + F(u), (5)

where V : R3 → R is given. In Theorem 4 we state local and global well-

posedness for (5) with initial datum u(0, x) = u0(x) in the appropriateenergy space. Assumption 1 imposed below on V is considerably weakand implies that

√−� + m2 + V defines a self-adjoint operator via itsform sum.

– The main results (i. e., Theorems 1–4) are proved in Section 3.– Appendices A and B contain some useful facts about fractional derivatives,

a discussion of ground states, and some details of the proofs.

Notation

Throughout this text, the symbol ∗ stands for convolution on R3, i. e.,

( f ∗ g)(x) :=∫

R3f (x − y)g(y) dy,

and Lp(R3), with norm ‖ · ‖p and 1 � p � ∞, denotes the usual LebesgueLp-space of complex-valued functions on R

3. Moreover, L2(R3) is associatedwith the scalar product defined by

〈u, v〉 :=∫

R3u(x)v(x) dx.

For s ∈ R and 1 � p � ∞, we introduce fractional Sobolev spaces (see, e. g.,[1]) with their corresponding norms according to

Hs,p(R3) := {u ∈ S ′(R3) : ‖u‖Hs,p := ‖F−1[(1 + ξ 2)s/2Fu]‖p < ∞}

,

where F denotes the Fourier transform in S ′(R3) (space of tempered distribu-tions). In our analysis, the Sobolev spaces

Hs(R3) := Hs,2(R3),

with norms ‖ · ‖Hs := ‖ · ‖Hs,2 , will play an important role.

46 E. Lenzmann

In addition to the common Lp-spaces, we also make use of local Lp-space,Lp

loc(R3), with 1 � p � ∞, and weak (or Lorentz) spaces, Lp

w(R3), with 1 <

p < ∞ and corresponding norms given by

‖u‖p,w := sup�

|�|−1/p′∫

�

|u(x)| dx,

where 1/p + 1/p′ = 1 and � denotes an arbitrary measurable set withLebesgue measure |�| < ∞; see, e. g., [11] for this definition of Lp

w-norms.Note that Lp(R3) � Lp

w(R3), for 1 < p < ∞.The symbol � = ∑3

i=1 ∂2xi

stands for the usual Laplacian on R3, and√−� + m2 is defined via its symbol

√ξ 2 + m2 in Fourier space. Besides the

operator√−� + m2 , we also employ Riesz and Bessel potentials of order

s ∈ R, which we denote by (−�)s/2 and (1 − �)s/2, respectively; see alsoAppendix A.

Except for theorems and lemmas, we often use the abbreviations Lp =Lp(R3), Lp

w = Lpw(R3), and Hs = Hs(R3). In what follows, a � b always de-

notes an inequality a � cb , where c is an appropriate positive constant thatcan depend on fixed parameters.

2 Main Results

We consider the following initial value problem

⎧⎨

⎩i∂tu =

√−� + m2 u +

(λe−μ|x|

|x| ∗ |u|2)

u,

u(0, x) = u0(x), u : [0, T) × R3 → C,

(6)

where m � 0, λ ∈ R, and μ � 0 are given parameters. Note that |λ| could beabsorbed in the normalization of u(t, x), but we shall keep λ explicit in thefollowing; see also [4] for this convention.

Our particular choice of the Hartree type nonlinearities in (6) is motivatedby the fact that (6) can be rewritten as the following system of equations

{i∂tu = √−� + m2 u + �u,

(μ2 − �)� = 4πλ|u|2, u(0, x) = u0(x),(7)

where � = �(t, x) is real-valued and �(t, x) → 0 as |x| → ∞. This reformu-lation stems from the observation that e−μ|x|/4π |x| is the Green’s function of(μ2 − �) in R

3; see Appendix A. System (7) now reveals the physical intuitionbehind (6), i. e., the function u(t, x) corresponds to a “positive energy wave”with instantaneous self-interaction that is either of Coulomb or Yukawa typedepending on whether μ = 0 or μ > 0, respectively. To prove well-posednesswe shall, however, use formulation (6) instead, and we refer to facts frompotential theory only when estimating the nonlinearity.


2.1 Local Well-posedness

Let us begin with well-posedness in energy space, i. e., we assume that u0 ∈H1/2 holds in (6). The following Theorem 1 establishes local well-posednessin the strong sense, i. e., we have existence and uniqueness of solutions,their continuous dependence on initial data, and the blow-up alternative. Theprecise statements is as follows.

Theorem 1 Let m � 0, λ ∈ R, and μ � 0. Then initial value problem (6) islocally well-posed in H1/2(R3). This means that, for every u0 ∈ H1/2(R3), thereexist a unique solution

u ∈ C0([0, T); H1/2(R3)

) ∩ C1([0, T); H−1/2(R3)

),

and it depends continuously on u0. Here T ∈ (0, ∞] is the maximal time ofexistence, where we have that either T = ∞ or T < ∞ and limt↑T ‖u(t)‖H1/2 = ∞holds.

Remark Continuous dependence means that the map u0 → u ∈ C0(I; H1/2) iscontinuous for every compact interval I ⊂ [0, T).

2.2 Global Well-posedness

The local-in-time solutions derived in Theorem 1 extend to all times, by vir-tue of Theorem 2 below, provided that either λ � 0 holds (correspondingto a repulsive nonlinearity) or λ < 0 and the initial datum is sufficiently smallin L2.

Theorem 2 The solution of (6) derived in Theorem 1 is global in time, i. e., wehave that T = ∞ holds, provided that one of the following conditions is met.

(1) λ � 0.(2) λ < 0 and ‖u0‖2

2 < ‖Q‖22, where Q ∈ H1/2(R3) is a strictly positive solution

(ground state) of

√−� Q +(

λ

|x| ∗ |Q|2)

Q = −Q. (8)

Moreover, we have the estimate ‖Q‖22 > 4

π |λ| .

Remarks

(1) Notice that condition (2) implies global well-posedness for (6) irrespec-tively of m � 0.

48 E. Lenzmann

(2) For a > 0, the function Qa(x) = a3/2 Q(ax) yields another ground statewith ‖Qa‖2 = ‖Q‖2 that satisfies

√−� Qa +(

λ

|x| ∗ |Qa|2)

Qa = −aQa. (9)

We refer to Appendix B for a discussion of Q ∈ H1/2.(3) Condition (2) resembles a well-known criterion derived in [16] for global-

in-time existence for L2-critical nonlinear (nonrelativistic) Schrödingerequations.

(4) It is shown in [7] that criterion (3) for having global-in-time solutions inthe focusing case is optimal in the sense that there exist solutions, u(t),with ‖u0‖2

2 > ‖Q‖22, which blow up within finite time.

2.3 Higher Regularity

We now turn to well-posedness of (6) in Hs, for s � 1/2, which is settled by thefollowing result.

Theorem 3 For every s � 1/2, the conclusions of Theorems 1 and 2 hold, whereH1/2(R3) and H−1/2(R3) in Theorem 1 are replaced by Hs(R3) and Hs−1(R3),respectively.

Remark For s = 1, this result is needed in [4] for a rigorous derivation of(6) with Coulomb type self-interaction (i. e., μ = 0) from many-body quantummechanics.

2.4 External Potentials

Now we consider the following extension of (6) that arises by adding anexternal potential:

⎧⎨

⎩i∂tu = (√−� + m2 + V

)u +

(λe−μ|x|

|x| ∗ |u|20

)u,

u(0, x) = u0(x), u : [0, T) × R3 → C,

(10)

where m � 0, λ ∈ R, μ � 0 are given parameters, and V : R3 → R denotes a

preassigned function that meets the following condition.

Assumption 1 Suppose that V = V+ + V− holds, where V+ and V− are real-valued, measurable functions with the following properties.

(1) V+ ∈ L1loc(R

3) and V+ � 0.(2) V− is

√−�-form bounded with relative bound less than 1, i. e., there existconstants 0 � a < 1 and 0 � b < ∞, such that

|〈u, V−u〉| � a〈u,√−� u〉 + b〈u, u〉

holds for all u ∈ H1/2(R3).


We mention that Assumption 1 implies that√−� + m2 + V leads to a

self-adjoint operator on L2 via its form sum. Furthermore, the energy spacegiven by

X :={

u ∈ H1/2(R3) :∫

R3V(x) |u(x)|2 dx < ∞

}(11)

is complete with norm ‖ · ‖X , and its dual space is denoted by X∗. We refer toSection 3.4 for more details on

√−� + m2 + V and X.After this preparing discussion, the extension of Theorems 1 and 2 for the

initial value problem (10) can be now stated as follows.

Theorem 4 Let m � 0, λ ∈ R, μ � 0, and suppose that V satisfies Assumption 1.Then (10) is locally well-posed in the following sense. For every u0 ∈ X, thereexists a unique solution

u ∈ C0([0, T); X) ∩ C1([0, T); X∗),

and it depends continuously on u0. Here T ∈ (0, ∞] is the maximal time ofexistence such that either T = ∞ or T < ∞ and limt↑T ‖u(t)‖X = ∞ holds.Moreover, we have that T = ∞ holds, if one of the following conditionsis satisfied.

(1) λ � 0.(2) λ < 0 and ‖u0‖2

2 < (1 − a)‖Q‖22, where Q is the ground state mentioned

in Theorem 2 and 0 � a < 1 denotes the relative bound introduced inAssumption 1.

Remarks

(1) To meet Assumption 1 for V+, we can choose, for example, V+(x) = |x|β ,with β � 0; or even super-polynomial growth such as V+(x) = e|x|. Notethat Assumption 1 for V− is satisfied (by virtue of Sobolev inequalities), if

|V−(x)| � c|x|1−ε

+ d

holds for some 0 < ε � 1 and constants 0 � c, d < ∞. In fact, we can evenadmit ε = 0 provided that c < 2/π holds, as can be seen from inequality(17) below.

(2) Since we avoid using Strichartz estimates in our well-posedness proof be-low, we only need that V+ belongs to L1

loc. In contrast to this, compare, forinstance, the conditions on V in [17] for deriving Strichartz type estimatesfor e−it(−�+V) in order to prove local well-posedness for (nonrelativistic)nonlinear Schrödinger equations with external potentials.

50 E. Lenzmann

3 Proof of the Main Results

In this section we prove Theorems 1–4. Although Theorem 4 generalizesTheorems 1 and 2, we postpone the proof of Theorem 4 to the final part ofthis section.

3.1 Proof of Theorem 1 (Local Well-posedness)

Let u0 ∈ H1/2 be fixed. In view of (6) we put

A :=√

−� + m2 and F(u) :=(

λe−μ|x|

|x| ∗ |u|2)

u, (12)

and we consider the integral equation

u(t) = e−itAu0 − i∫ t

0e−i(t−τ)A F(u(τ )) dτ. (13)

Here u(t) is supposed to belong to the Banach space

YT := C0([0, T); H1/2(R3)

), (14)

with some T >0 and norm ‖u‖YT :=supt∈[0,T) ‖u(t)‖H1/2 . The proof of Theorem 1is now organized in two steps as follows.

Step 1: Estimating the Nonlinearity

We show that the nonlinearity F(u) is locally Lipschitz continuous from H1/2

into itself. This is main point of our argument for local well-posedness and itreads as follows.

Lemma 1 For μ � 0, the map J(u) := ( e−μ|x||x| ∗ |u|2)u is locally Lipschitz contin-

uous from H1/2(R3) into itself with

‖J(u) − J(v)‖H1/2 � (‖u‖2H1/2 + ‖v‖2

H1/2)‖u − v‖H1/2 ,

for all u, v ∈ H1/2(R3).

Proof (Of Lemma 1) We prove the claim for μ = 0 and μ > 0 in a commonway, so let μ � 0 be fixed. For s ∈ R, it is convenient to introduce

Ds := (μ2 − �)s/2.

Note that due to the equivalence

‖u‖2 + ‖D1/2u‖2 � ‖u‖H1/2 � ‖u‖2 + ‖D1/2u‖2,

it is sufficient to estimate the quantities

I := ‖J(u) − J(v)‖2 and I I := ‖D1/2[J(u) − J(v)]‖2,


where I is needed only if μ = 0. Using now the identity

J(u) − J(v) = 1/2

[(e−μ|x|

|x| ∗ (|u|2 − |v|2))

(u + v) +

+(

e−μ|x|

|x| ∗ (|u|2 + |v|2))

(u − v)

]

together with Hölder’s inequality (which we tacitly apply from now on), wefind that

I �∥∥∥∥

(e−μ|x|

|x| ∗ (|u|2 − |v|2))

(u + v)

∥∥∥∥2

+

+∥∥∥∥

(e−μ|x|

|x| ∗ (|u|2 + |v|2))

(u − v)

∥∥∥∥2

�∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 − |v|2)∥∥∥∥

6

‖u + v‖3 +

+∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 + |v|2)∥∥∥∥

∞‖u − v‖2. (15)

Observing that e−μ|x||x|−1 ∈ L3w holds, the first term of right-hand side of (15)

can be bounded by means of the weak Young inequality (see, e. g., [11])as follows

∥∥∥∥e−μ|x|

|x| ∗ (|u|2 − |v|2)∥∥∥∥

6

�∥∥∥∥

e−μ|x|

|x|∥∥∥∥

3,w

‖|u|2 − |v|2‖6/5

� ‖u + v‖3‖u − v‖2. (16)

The second term in (15) can be estimated by noting that

∥∥∥∥e−μ|x|

|x| ∗ |u|2∥∥∥∥

∞� sup

y∈R3

∫

R3

|u(x)|2|x − y| dx � ‖(−�)1/4u‖2

2, (17)

which follows from the operator inequality |x − y|−1 � π2 (−�x−y)

1/2 (see, e. g.,[10, Section V.5.4]) and translational invariance, i. e., we use that �x−y = �x

holds for all y ∈ R3. Combining now (16) and (17) we find that

I � ‖u + v‖23‖u − v‖2 + (‖u‖2

H1/2 + ‖v‖2H1/2

) ‖u − v‖2

�(‖u‖2

H1/2 + ‖v‖2H1/2

) ‖u − v‖H1/2 ,

where we make use of the Sobolev inequality ‖u‖3 � ‖u‖H1/2 in R3.

52 E. Lenzmann

It remains to estimate I I. To do so, we appeal to the generalized (orfractional) Leibniz rule (see Appendix A) leading to

I I �∥∥∥∥D

1/2

[(e−μ|x|

|x| ∗ (|u|2 − |v|2))

(u + v)

] ∥∥∥∥2

+

+∥∥∥∥D

1/2

[(e−μ|x|

|x| ∗ (|u|2 + |v|2))

(u − v)

] ∥∥∥∥2

�∥∥∥∥D

1/2

(e−μ|x|

|x| ∗ (|u|2 − |v|2)) ∥∥∥∥

6

‖u + v‖3 +

+∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 − |v|2)∥∥∥∥

∞‖D1/2(u + v)‖2 +

+∥∥∥∥D

1/2

[(e−μ|x|

|x| ∗ (|u|2 + |v|2)] ∥∥∥∥

6

‖u − v‖3 +

+∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 + |v|2)∥∥∥∥

∞‖D1/2(u − v)‖2. (18)

By referring to Appendix A, we notice that e−μ|x|4π |x| ∗ f can be expressed as

D−2 f = (μ2 − �)−1 f in R3 (here f ∈ S(R3) is initially assumed, but our argu-

ments follow by density). Thus, the first term of the right-hand side of (18)is found to be

∥∥∥∥D1/2

(e−μ|x|

|x| ∗ (|u|2 − |v|2)) ∥∥∥∥

6

� ‖D−3/2(|u|2 − |v|2)‖6

�∥∥Gμ

3/2 ∗ (|u|2 − |v|2)∥∥6

�∥∥Gμ

3/2

∥∥2,w

‖|u|2 − |v|2‖3/2

� ‖u + v‖3‖u − v‖3, (19)

where we use weak Young’s inequality together with the fact that D−3/2 fcorresponds to Gμ

3/2 ∗ f with some Gμ

3/2 ∈ L2w(R3); see (42). The ‖ · ‖∞-part

of the second term occurring in (18) can be estimated by using the Cauchy-Schwarz inequality and (17) once again:

∥∥∥∥e−μ|x|

|x| ∗ (|u|2 − |v|2)∥∥∥∥

∞�

∥∥∥∥1

|x| ∗ (|u|2 − |v|2)∥∥∥∥

∞

� supy∈R3

∣∣∣∫

R3

|u(x)|2 − |v(x)|2|x − y| dx

∣∣∣

� supy∈R3

∣∣∣∣

⟨(u(x) + v(x)),

1

|x − y| (u(x) − v(x))

⟩ ∣∣∣∣


� ‖(−�)1/4(u + v)‖2‖(−�)1/4(u − v)‖2

� (‖u‖H1/2 + ‖v‖H1/2)‖u − v‖H1/2 (20)

The remaining terms in (18) deserve no further comment, since they canbe estimated in a similar fashion to all estimates derived so far. Thus, weconclude that

‖J(u) − J(v)‖H1/2 � I + I I � (‖u‖2H1/2 + ‖v‖2

H1/2)‖u − v‖H1/2

and the proof of Lemma 1 is now complete. ��

Remarks

(1) The proof of Lemma 1 relies on (17) in a crucial way. Employing just theSobolev embedding H1/2 ⊂ L2 ∩ L3 (in R

3) together with the (non weak)Young inequality is not sufficient to conclude that ‖ e−μ|x|

|x| ∗ |u|2‖∞ < ∞whenever u ∈ H1/2.

(2) The proof of Lemma 1 fails for “super-critical” Hartree nonlinearitiesJ(u) = (|x|−α ∗ |u|2)u, where 1 < α < 3. Thus, the choice α = 1 representsa borderline case when deriving local Lipschitz continuity in energyspace H1/2.

Step 2: Conclusion

Returning to the proof of Theorem 1, we note that A defined in (12) gives riseto a self-adjoint operator L2 with domain H1. Moreover, its extension to H1/2,which we denote by A : H1/2 → H−1/2, generates a C0-group of isometries,{e−itA}t∈R, acting on H1/2. Local well-posedness in the sense of Theorem 1 nowfollows by standard methods for evolution equations with locally Lipschitznonlinearities. That is, existence and uniqueness of a solution u ∈ YT forthe integral equation (13) is deduced by a fixed point argument, for T > 0sufficiently small. The equivalence of the integral formulation (13) and theinitial value problem (6), with u0 ∈ H1/2, as well as the blow-up alternative canalso be deduced by standard arguments; see, e. g., [3, 14] for general theory onsemilinear evolution equations. Finally, note that u ∈ C1([0, T); H−1) followsby (6) itself. The proof of Theorem 1 is now accomplished.

3.2 Proof of Theorem 2 (Global Well-posedness)

The first step taken in the proof of Theorem 2 settles conservation of energyand charge that are given by

E[u] := 1/2∫

R3u(x)

√−� + m2 u(x) dx +

+ 1/4∫

R3

(λe−μ|x|

|x| ∗ |u|2)

(x) |u(x)|2 dx, (21)

N[u] :=∫

R3|u(x)|2 dx, (22)

54 E. Lenzmann

respectively. After deriving the corresponding conservation laws (where prov-ing energy conservation requires a regularization), we discuss how to obtaina-priori bounds on the energy norm of the solution.

Step 1: Conservation Laws

Lemma 2 The local-in-time solutions of Theorem 1 obey conservation of energyand charge, i. e.,

E[u(t)] = E[u0] and N[u(t)] = N[u0],for all t ∈ [0, T).

Proof (Of Lemma 2) Let u be a local-in-time solution derived in Theorem 1,and let T be its maximal time of existence. Since u(t) ∈ H1/2 holds, we canmultiply (6) by iu(t) and integrate over R

3. Taking then real parts yields

ddt

N[u(t)] = 0 for t ∈ [0, T), (23)

which shows conservation of charge.At a formal level, conservation of energy follows by multiplying (6) with

˙u(t) ∈ H−1/2 and integrating over space, but the paring of two elementsof H−1/2 is not well-defined. Thus, we have to introduce a regularizationprocedure as follows; see also, e. g., [2, 8] for other regularization methodsfor nonlinear (nonrelativistic) Schrödinger equations. Let us define the familyof operators

Mε := (εA + 1)−1, for ε > 0, (24)

where the operator A = √−� + m2 � 0 is taken from (12). Consider thesequences of embedded spaces

. . . H3/2 ↪→ H1/2 ↪→ H−1/2 ↪→ H−3/2 . . .

It is easy to see (by using functional calculus) that the following propertieshold.

(a) For ε > 0 and s ∈ R, we have that Mε is a bounded map from Hs intoHs+1.

(b) ‖Mεu‖Hs � ‖u‖Hs whenever u ∈ Hs and s ∈ R.(c) For u ∈ Hs and s ∈ R, we have that Mεu → u strongly in Hs as ε ↓ 0.

We shall use tacitly properties (a)–(c) in the following analysis.By means of Mε and noting that E ∈ C1(H1/2; R), we can compute in a well-

defined way for t1, t2 ∈ [0, T) as follows

E[Mεu(t2)]−E[Mεu(t1)] =∫ t2

t1〈E′(Mεu),Mεu〉 dt

=∫ t2

t1Re 〈AMεu+F(Mεu),−iMε(Au+F(u))〉 dt


=∫ t2

t1Im

[〈AMεu,Mε Au〉+〈F(Mεu),Mε Au〉+

+〈AMεu,Mε F(u)〉+〈F(Mεu),Mε F(u)〉]

dt

=:∫ t2

t1fε(t) dt, (25)

where we write u = u(t) for brevity and recall the definition of F from (12).We observe that the first term in fε(t) is the “most singular” part, i. e., if ε = 0we would have pairing of two H−1/2-elements. But for ε > 0 we can use theobvious fact that Mε A = AMε holds and conclude that

Im 〈AMεu,Mε Au〉 = Im 〈AMεu, AMεu〉 = 0.

Notice that this manipulation is well-defined, since AMεu and Mε Au are inH1/2 whenever u ∈ H1/2. After some simple calculations, we find fε(t) to be ofthe form

fε(t) = Im[〈F(Mεu),Mε Au〉 + 〈AMεu,Mε F(u)〉 +

+ 〈F(Mεu),Mε F(u)〉]

= Im[〈A1/2 F(Mεu), A1/2Mεu〉 + 〈A1/2Mεu, A1/2Mε F(u)〉 +

+ 〈F(Mεu),Mε F(u)〉],

Since Mεu → u strongly in H1/2 as ε ↓ 0, we can infer, by Lemma 1, that

limε↓0

fε(t) = Im[〈A1/2 F(u), A1/2u〉 + 〈A1/2u, A1/2 F(u)〉 + 〈F(u), F(u)〉]

= Im (Real Number) = 0.

To interchange the ε-limit with the t-integration in (25), we appeal to thedominated convergence theorem. That is, we seek for a uniform bound onfε(t). In fact, by using the Cauchy–Schwarz inequality and Lemma 1 again wefind the following estimate

| fε(t)| � |〈A1/2 F(Mεu), A1/2Mεu〉| + |〈A1/2Mεu, A1/2Mε F(u)〉| ++ |〈F(Mεu),Mε F(u)〉|

� ‖A1/2 F(Mεu)‖2‖A1/2Mεu‖2 ++ ‖A1/2Mεu‖2‖A1/2Mε F(u)‖2 + ‖F(Mεu)‖2‖Mε F(u)‖2

� ‖u‖4H1/2 + ‖u‖6

H1/2 ,

56 E. Lenzmann

for all ε > 0. Putting now all together leads to conservation of energy, i. e., wefind for all t1, t2 ∈ [0, T) that

E[u(t2)] − E[u(t1)] = limε↓0

(E[Mεu(t2)] − E[Mεu(t1)]

)

= limε↓0

∫ t2

t1fε(t) dt =

∫ t2

t1limε↓0

fε(t) dt = 0.

This completes the proof of Lemma 2. ��

Step 2: A Priori Bounds

To fill the last gap towards the global well-posedness result of Theorem 2, wenow discuss how to obtain a priori bounds on the energy norm. By the blow-up alternative of Theorem 1, global-in-time existence follows from an a prioribound of the form

‖u(t)‖H1/2 � C(u0). (26)

First, let us assume that λ � 0 holds. Then, for all t ∈ [0, T), we find fromLemma 2 and (22) that

‖(−�)1/4u(t)‖2 � E[u(t)] = E[u0].This implies together with charge conservation derived in Lemma 2, i. e.,

‖u(t)‖22 = N[u(t)] = N[u0] (27)

an a priori estimate (26). Therefore condition (1) in Theorem 2 is sufficient forglobal existence.

Suppose now a focusing nonlinearity, i. e., λ < 0 holds, and without lossof generality we assume that λ = −1 is true (the general case follows byrescaling). Now we can estimate as follows.

E[u] = 1/2‖(−� + m2)1/4u‖22 − 1/4

∫

R3

(e−μ|x|

|x| ∗ |u|2)

(x) |u(x)|2 dx

� 1/2‖(−� + m2)1/4u‖22 − 1/4

∫

R3

(1

|x| ∗ |u|2)

(x) |u(x)|2 dx

� 1/2‖(−�)1/4u‖22 − 1

4K‖(−�)1/4u‖2

2‖u‖22

=(

1/2 − 1

4K‖u‖2

2

)‖(−�)1/4u‖2

2, (28)

where K > 0 is the best constant taken from Appendix B. Thus, energyconservation leads to an a priori bound on the H1/2-norm of the solution, if

‖u0‖22 < 2K (29)

holds. In fact, the constant K satisfies

K = ‖Q‖22

2>

2

π,


where Q(x) is a strictly positive (ground state) solution of

√−� Q −(

1

|x| ∗ |Q|2)

Q = −Q; (30)

see Appendix B. Going back to (29), we find that

‖u0‖22 < ‖Q‖2

2 (31)

is sufficient for global existence for λ = −1. The assertion of Theorem 2for all λ < 0 now follows by simple rescaling. The proof of Theorem 2 isnow complete.

3.3 Proof of Theorem 3 (Higher Regularity)

To prove Theorem 3, we need the following generalization of Lemma 1, whoseproof is a careful but straightforward generalization of the proof of Lemma 1.We defer the details to Appendix A.1.

Lemma 3 For μ � 0 and s � 1/2, the map J(u) := ( e−μ|x||x| ∗ |u|2)u is locally

Lipschitz continuous from Hs(R3) into itself with

‖J(u) − J(v)‖Hs � (‖u‖2Hs + ‖v‖2

Hs)‖u − v‖Hs

for all u, v ∈ Hs(R3). Moreover, we have that

‖J(u)‖Hs � ‖u‖2Hr‖u‖Hs

holds for all u ∈ Hs(R3), where r = max{s − 1, 1/2}.

Local well-posedness of (10) in Hs, for s > 1/2, can be shown now as follows.We note that {e−itA}t∈R, with A = √−� + m2 , is a C0-group of isometrieson Hs. Moreover, since the nonlinearity defined in (12), is locally Lipschitzcontinuous from Hs into itself, local well-posedness in Hs follows similarly asexplained in the proof of Theorem 1 for H1/2. To show global well-posednessin Hs, we prove by induction and Lemma 3 that an a priori bound on the H1/2-norm of solution implies uniform bounds on the Hs-norm on any compactinterval [0, T∗] ⊂ [0, T). This claim follows from (13) and the second inequalitystated in Lemma 3 by noting that

‖u(t)‖Hs � ‖e−itAu0‖Hs +∫ t

0‖e−i(t−τ)A F(u(τ ))‖Hs dτ

� ‖u0‖Hs +∫ t

0‖F(u(τ ))‖Hs dτ

� C1 + C2

∫ t

0‖u(τ )‖Hs dτ,

holds, provided that ‖u(t)‖Hr � 1 for r = max{s − 1, 1/2} < s. InvokingGronwall’s inequality we conclude that

‖u(t)‖Hs � eC2T∗ , for t ∈ [0, T∗] ⊂ [0, T).

58 E. Lenzmann

Induction now implies that an a-priori bound on ‖u(t)‖H1/2 guarantees uniformbounds ‖u(t)‖Hs on any compact interval I ⊂ [0, T). Thus, the maximal timeof existence of an Hs-valued solution coincides with the maximal time of exis-tence when viewed as an H1/2-valued solution. Therefore sufficient conditionsfor global existence for H1/2-valued solutions imply global-in-time Hs-valuedsolutions. This completes the proof of Theorem 3.

3.4 Proof of Theorem 4 (External Potentials)

Let V = V+ + V− satisfy Assumption 1 in Section 2. We introduce the quad-ratic form

Q(u, v) := 〈u,√

−� + m2 v〉 + 〈u, V−v〉 + 〈u, V+v〉, (32)

which is well-defined on the set (energy space)

X := {u ∈ L2(R3) : Q(u, u) < ∞}

. (33)

Note that Assumption 1 also guarantees that C∞0 (R3) ⊂ X. It easy to show

that our assumption on V implies that the quadratic form (32) is boundedfrom below, i. e., we have Q(u, u) � −M〈u, u〉 holds for all u ∈ X and someconstant M � 0. By the semi-boundedness of Q, we can assume from now on(and without loss of generality) that

Q(u, u) � 0 (34)

holds for all u ∈ X. Since Q(·, ·) is closed (it is a sum of closed forms), theenergy space X equipped with its norm

‖u‖X := √〈u, u〉 + Q(u, u) (35)

is complete, and we have the equivalence

‖u‖H1/2 + ‖V1/2+ u‖2 � ‖u‖X � ‖u‖H1/2 + ‖V1/2

+ u‖2. (36)

Furthermore, there exists a nonnegative, self-adjoint operator

A : D(A) ⊂ L2 → L2 (37)

with X = D(A1/2), such that

〈u, Av〉 = Q(u, v) (38)

holds for all u ∈ X and v ∈ D(A); see, e. g., [10]. This operator can be extendedto a bounded operator, still denoted by A : X → X∗, where X∗ is the dualspace of X.

To prove now the assertion about local well-posedness in Theorem 4, wehave to generalize Lemma 1 to the following statement.


Lemma 4 Suppose μ � 0 and let V satisfy Assumption 1. Then the map J(u) :=( e−μ|x|

|x| ∗ |u|2)u is locally Lipschitz continuous from X into itself with

‖J(u) − J(v)‖X � (‖u‖2X + ‖v‖2

X)‖u − v‖X

for all u, v ∈ X.

Proof (Of Lemma 4) By (36), it suffices to estimate ‖J(u) − J(v)‖H1/2 and‖V1/2

+ [J(u) − J(v)]‖2 separately. By Lemma 1, we know that

‖J(u) − J(v)‖H1/2 � (‖u‖2H1/2 + ‖v‖2

H1/2)‖u − v‖H1/2

� (‖u‖2X + ‖v‖2

X)‖u − v‖X .

It remains to estimate ‖V1/2+ [J(u) − J(v)]‖2, which can be achieved by recalling

(20) and proceeding as follows.

‖V1/2+ [J(u) − J(v)]‖2 �

∥∥∥∥V1/2+

[(e−μ|x|

x∗ (|u|2 − |v|2)

)(u + v)

]∥∥∥∥2

+

+∥∥∥∥V1/2

+

[(e−μ|x|

x∗ (|u|2 + |v|2)

)(u − v)

]∥∥∥∥2

�∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 − |v|2)∥∥∥∥∞

‖V1/2+ (u + v)‖2 +

+∥∥∥∥

e−μ|x|

|x| ∗ (|u|2 + |v|2)∥∥∥∥∞

‖V1/2+ (u − v)‖2

�∥∥u + v‖H1/2‖u − v‖H1/2‖V1/2

+ (u + v)‖2 ++ (‖u‖2

H1/2 + ‖v‖2H1/2)‖V1/2

+ (u − v)‖2

�(‖u‖2

X + ‖v‖2X)‖u − v‖X .

This completes the proof of Lemma 4. ��

Returning to the proof of Theorem 4, we simply note that {e−itA}t∈R is aC0-group of isometries on X, where A = √−� + m2 + V is defined in theform sense (see above). By Lemma 4, the nonlinearity is locally Lipschitz onX. Thus, local well-posedness now follows in the same way as for Theorem 1.

To establish global well-posedness we have to prove conservation of charge,N[u], and energy, E[u], which is for (10) given by

E[u] := 1/2∫

R3u(x)

√−� + m2 u(x) dx + 1/2

∫

R3V(x)|u(x)|2 dx +

+ 1/4∫

R3

(λe−μ|x|

|x| ∗ |u|2)

(x) |u(x)|2 dx. (39)

As done in Section 3.2, we have to employ a regularization method using theclass of operators

Mε := (εA + 1)−1, for ε > 0, (40)

60 E. Lenzmann

where we assume without loss of generality that A � 0 holds. The mappingMε acts on the sequence of embedded spaces

. . . X+2 ↪→ X+1 ↪→ X−1 ↪→ X−2 . . . , (41)

with corresponding norms given by ‖u‖Xs := ‖(1 + A)s/2u‖2. Note that X =X+1 (with equivalent norms) and that its dual space obeys X∗ = X−1. Byusing functional calculus, it is easy to show that Mε exhibits properties thatare analog to (a)–(c) in Section 3.2.

The rest of the argument for proving conservation of energy carries overfrom Section 3.2 without major modifications. Finally, we mention that de-riving a priori bounds on ‖u(t)‖X leads to a similar discussion as presentedin Section 3.2, while noting that we have to take care that V− has a relative(−�)1/2-form bound, 0 � a < 1, introduced in Assumption 1. This completesthe proof of Theorem 4.

Acknowledgements The author is grateful to Demetrios Christodoulou, Jürg Fröhlich, LarsJonsson, and Simon Schwarz for many valuable and inspiring discussions.

Appendix A: Fractional Calculus

The following result (generalized Leibniz rule) is proved in [9] for Riesz andBessel potentials of order s ∈ R, which are denoted by (−�)s/2 and (1 − �)s/2,respectively. But as a direct consequence of the Milhin multiplier theorem [1],the cited result holds for Ds := (μ2 − �)s/2, where μ � 0 is a fixed constant.

Lemma 5 (Generalized Leibniz Rule) Suppose that 1 < p < ∞, s � 0, α � 0,β � 0, and 1/pi + 1/qi = 1/p with i = 1, 2, 1 < qi � ∞, 1 < pi � ∞. Then

‖Ds( fg)‖p � c(‖Ds+α f‖p1‖D−αg‖q1 + ‖D−β f‖p2‖Ds+βg‖q2),

where the constant c depends on all of the parameters but not on f and g.

A second fact we use in the proof of our main result is as follows. For 0 <

α < 3 and μ � 0, the potential operator D−α = (μ2 − �)−α/2 corresponds tof → Gμ

α ∗ f , with f ∈ S(R3), and we have that

Gμα ∈ L3/(3−α)

w (R3). (42)

To see this, we refer to the inequality and the exact formula

0 � Gμα (x) � G0

α(x) = cα

|x|3−α, for μ � 0 and 0 < α < 3, (43)

with some constant cα ; these facts can be derived from [15, Section V.3.1]. Now(42) follows from |x|−σ ∈ L3/σ

w (R3) whenever 0 < σ < 3. Another observationused in Section 2 is the well-known explicit formula

Gμ

2 (x) = e−μ|x|

4π |x| . (44)


That is, (μ2 − �) in R3 has the Green’s function e−μ|x|

4π |x| with vanishing boundaryconditions.

A.1 Proof of Lemma 3

Proof (Of Lemma 3) We only show the second inequality derived in Lemma 3,since the first one can be proved in a similar way.

Let μ � 0 and s � 1/2. We put Dα := (μ2 − �)α/2 for α ∈ R. By the gener-alized Leibniz rule and (17), we have that

‖Ds J(u)‖2 � ‖Ds[(D−2|u|2)u]‖2

� ‖Ds−2|u|2‖p1‖u‖q1 + ‖D−2|u|2‖∞‖Dsu‖2

� ‖Ds−2|u|2‖p1‖u‖q1 + ‖u‖2H1/2‖u‖Hs , (45)

where 1/p1 + 1/q1 = 1/2 with 1 < p1, q1 � ∞. The first term of the right-hand side of (45) can be controlled as follows, where we introduce r =max{s − 1, 1/2}.(1) For 1/2 � s < 3/2, we choose p1 = 3/s and q1 = 6/(3 − 2s) which leads to

‖Ds−2|u|2‖3/s‖u‖6/(3−2s) � ‖Gμ

2−s‖3/(1+s),w‖|u|2‖3/2‖u‖Hs

� ‖u‖2H1/2‖u‖Hs � ‖u‖2

Hr‖u‖Hs ,

where we use the weak Young inequality, as well as Sobolev’s inequality‖u‖6/(3−2s) � ‖u‖Hs in R

3, and (42) once again.(2) For s � 3/2, we choose p1 = 6 and q1 = 3. This yields

‖Ds−2|u|2‖6‖u‖3 � ‖Ds−1|u|2‖2‖u‖3 � ‖Ds−1u‖6‖u‖23

� ‖Dsu‖2‖u‖23 � ‖u‖Hs‖u‖2

Hr ,

while using twice Sobolev’s inequality ‖ f‖6 � ‖D f‖2 in R3.

Putting now all together, we conclude that

‖J(u)‖Hs � ‖J(u)‖2 + ‖Ds J(u)‖2

� ‖u‖2H1/2‖u‖2 + ‖u‖2

Hr‖u‖Hs � ‖u‖2Hr ‖u‖Hs .

��

Appendix B: Ground States

We consider the functional (see also [12])

K[u] := ‖(−�)1/4u‖22‖u‖2

2∫R3(|x|−1 ∗ |u|2)(x) |u(x)|2 dx

, (46)

62 E. Lenzmann

which is well-defined for all u ∈ H1/2 with u �≡ 0. Note that by using (17) wecan estimate the denominator in K[u] as follows.

∫

R3

(1

|x| ∗ |u|2)

(x) |u(x)|2 dx �∥∥∥∥

1

|x| ∗ |u|2∥∥∥∥∞

‖u‖22 � π

2‖(−�)1/4u‖2

2‖u‖22,

(47)which leads to the bound

2

π� K[u] < ∞. (48)

Indeed, we will see that the estimate from below is a strict inequality. Withrespect to the related variational problem

K := inf{

K[u] : u ∈ H1/2(R3), u �≡ 0}

(49)

we can state the following result.

Lemma 6 (Ground States) There exists a minimizer, Q ∈ H1/2(R3), for (49),and we have the following properties.

(1) Q(x) is a smooth function that can be chosen to be real-valued, strictlypositive, and spherically symmetric with respect to the origin. It satisfies

√−� Q −(

1

|x| ∗ |Q|2)

Q = −Q, (50)

and it is nonincreasing, i. e., we have that Q(x) � Q(y) whenever |x| � |y|.(2) The infimum satisfies K = ‖Q‖2

2/2 and K > 2/π .

Proof (Sketch of Proof) We present the main ideas for the proof of thepreceding lemma. That (49) is attained at some real-valued, radial, nonneg-ative and nonincreasing function Q(x) � 0 can be proved by direct methodsof variational calculus and rearrangement inequalities; see also [16] for asimilar variational problem for nonrelativistic Schrödinger equations withlocal nonlinearities. Furthermore, any minimizer, Q ∈ H1/2, has to satisfy thecorresponding Euler–Lagrange equation that reads

√−� Q −(

λ

|x| ∗ |Q|2)

Q = −Q, (51)

after a suitable rescaling Q(x) → aQ(b x) with some a, b > 0.Let us make some comments about the properties of Q. Using an bootstrap

argument and Lemma 3 for the nonlinearity, it follows that Q belongs to Hs,for all s � 1/2. Hence it is a smooth function. To see that Q(x) � 0 is strictlypositive, i. e., Q(x) > 0, we rewrite (51) such that

Q = (√−� + 1)−1

W, (52)

where W := (|x|−1 ∗ |Q|2)Q. By functional calculus, we have that

(√−� + 1)−1 =

∫ ∞

0e−te−t

√−� dt. (53)


Next, we notice by the explicit formula for the kernel (in R3)

e−t√−�(x, y) = F−1

(e−t|ξ |)(x − y) = C · t

[t2 + |x − y|2]2,

with some constant C > 0; see, e. g., [11]. This explicit formula shows thate−t

√−� is positivity improving. This means that if f � 0 with f �≡ 0 thene−t

√−� f > 0 almost everywhere. Hence (√−� + 1)−1 is also positivity improv-

ing, by (53), and we conclude that Q(x) > 0 holds almost everywhere, thanks to(52) and W � 0. Moreover, we know that Q(x) is a nonincreasing, continuousfunction. Therefore Q(x) > 0 holds in the strong sense, i. e., for every x ∈ R

3.Finally, to see that (2) holds, we consider the variational problem

IN := inf{

E[u] : u ∈ H1/2(R3), ‖u‖22 = N

}, (54)

where N > 0 is a given parameter and

E[u] = 1/2‖(−�)1/4u‖22 − 1/4

∫

R3

(1

|x| ∗ |u|2)

|u(x)|2 dx.

Due to the scaling behavior E[α3/2u(α·)] = αE[u], we have that either IN = 0or IN = −∞ holds. By noting that

E[u] �(

1

2− N

4K

)‖(−�)1/4u‖2

2,

and the fact that equality holds if and only if u minimizes K[u], we find thatIN = 0 holds if and only if N � Nc := 2K. Moreover, IN = 0 is attained ifand only if N = Nc. Let Q be such a minimizer with ‖Q‖2

2 = Nc. Thanks tothe proof of part (1), we can assume without loss of generality that Q is real-valued, radial, and strictly positive. Calculating the Euler–Lagrange equationfor (54), with N = Nc, yields

√−� Q −(

1

|x| ∗ |Q|2)

Q = −θ Q,

for some multiplier θ , where it is easy to show that θ > 0 holds. Setting nowQ(x) = θ−3/2 Q(θ−1x), which conserves the L2-norm, leads to a ground stateQ(x) satisfying (51). Thus, we have that

K = ‖Q‖22/2 = ‖Q‖2

2/2.

To prove that K > 2/π holds, let us assume K = 2/π . This implies that thefirst inequality in (47) is an equality for u(x) = Q(x) > 0. But this leads to(|x|−1 ∗ |Q|2)(x) = const., which is impossible.

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Feynman’s Operational Calculi: Spectral Theoryfor Noncommuting Self-adjoint Operators

Brian Jefferies · Gerald W. Johnson ·Lance Nielsen

Received: 3 November 2006 / Accepted: 16 April 2007 /Published online: 29 June 2007© Springer Science + Business Media B.V. 2007

Abstract The spectral theorem for commuting self-adjoint operators alongwith the associated functional (or operational) calculus is among the most use-ful and beautiful results of analysis. It is well known that forming a functionalcalculus for noncommuting self-adjoint operators is far more problematic.The central result of this paper establishes a rich functional calculus for anyfinite number of noncommuting (i.e. not necessarily commuting) bounded,self-adjoint operators A1, . . . , An and associated continuous Borel probabilitymeasures μ1, · · · , μn on [0, 1]. Fix A1, . . . , An. Then each choice of an n-tuple(μ1, . . . , μn) of measures determines one of Feynman’s operational calculiacting on a certain Banach algebra of analytic functions even when A1, . . . , An

are just bounded linear operators on a Banach space. The Hilbert space settingalong with self-adjointness allows us to extend the operational calculi wellbeyond the analytic functions. Using results and ideas drawn largely from theproof of our main theorem, we also establish a family of Trotter product typeformulas suitable for Feynman’s operational calculi.

B. JefferiesSchool of Mathematics, The University of New South Wales,Sydney 2052, Australiae-mail: [email protected]

G. W. JohnsonDepartment of Mathematics, 333 Avery Hall, The University of Nebraska,Lincoln, Lincoln, NE 68588-0130, USAe-mail: [email protected]

L. Nielsen (B)Department of Mathematics, Creighton University,Omaha, NE 68178, USAe-mail: [email protected]

66 B. Jefferies et al.

Keywords Noncommuting self-adjoint operators · Spectral theories ·Feynman’s operational calculi · Disentangling

Mathematics Subject Classifications (2000) Primary 47A13 · 47A60 ·Secondary 46J15

1 Introduction

Let X be a Banach space and suppose that A1, . . . , An are noncommutingelements in L(X), the space of bounded linear operators on X. Further, foreach i ∈ {1, . . . , n}, let μi be a continuous probability measure defined onB([0, 1]), the Borel class of [0, 1]. (Recall that a measure μ is continuousprovided that μ({s}) = 0 for every single point set {s}.) Such measures de-termine an operational calculus or ‘disentangling map’ Tμ1,...,μn from a com-mutative Banach algebra D(A1, . . . , An), called the ‘disentangling algebra’ ofanalytic functions into the noncommutative Banach algebra L(X). (See [4] orDefinition 1.1 below.) It is natural to seek conditions under which suchan operational calculus can be extended beyond the analytic functions inD(A1, . . . , An). Theorem 2.2, the main result of this paper will show, inconjunction with results from [6], that when X = H is a Hilbert space andA1, . . . , An are self-adjoint, the domain of each of the operational calculi ismuch richer than D(A1, . . . , An).

Feynman developed ‘rules’ for his operational calculus for noncommutingoperators while discovering the famous perturbation series and Feynmangraphs of quantum electrodynamics. By the time he wrote [2] he realizedthat this operational calculus could be developed into a widely applicablemathematical technique. Feynman was aware that his work was far from beingmathematically rigorous (see page 108 of [2]), especially with regard to the‘disentangling’ process, the central operation of his functional calculus. Heregarded his operational calculus as a kind of generalized path integral. (SeeSection 14.3 of [10].)

We now give a brief description of Feynman’s heuristic rules:

(a) Attach time indices to the operators to keep track of the order of theoperators in products. Operators with smaller (or earlier) time indicesare to act before operators with larger (or later) time indices no matterhow they are ordered on the page.

(b) With time indices attached, functions of the operators are formed just asif they were commuting.

(c) Finally, the operator expressions are to be restored to their natural order;this is the so-called disentangling process. This final step is often difficult;it consists roughly of manipulating the operator expressions until theirorder on the page is consistent with the time ordering.

How does one accomplish (a)–(c)? There have been several quite variedapproaches to this subject. Many of the references can be found in one of

Feynman’s operational calculi: spectral theory 67

the books [10, 13]. We also call attention to the recent monograph by B.Jefferies [3] and the 1968 paper by Taylor [18]. Both of these are essentiallyconcerned with the Weyl calculus for a finite number of noncommutingbounded operators. The paper [18] focused on operators which are also self-adjoint. The work begun by Maslov [12] and pursued by him and by severalothers is the furthest developed. See especially the book [13] by Nazaikinskii,Shatalov, and Sternin.

We will follow the approach initiated recently by Jefferies and Johnson([4–7]) and further developed by them, Nielsen and others ([8, 9, 11, 14]). Alarge family of operational calculi is defined at one time in this approach. Thisallows us to study a variety of operational calculi within one framework. Italso permits us to solve a wide variety of evolution equations using variousexponential functions of sums of noncommuting operators. (This was carriedout in [8], Section 4, and we hope to pursue this further in later work.) Finally,one can sometimes get information about one (or one type of) operationalcalculus by showing that it is the limit of simpler operational calculi. Indeed,the main theorem of this paper will rest in large part on such an argument.

Johnson and Nielsen established a stability theorem for Feynman’s opera-tional calculi [11] which will supply one of the central facts that we will need forour main result. We state that theorem now even though the precise definitionsof the disentangling algebra and the disentangling map will be postponed untilfurther on in this section.

Theorem 1.1 For each i = 1, . . . , n, let μi and μik, k = 1, 2, . . . be continu-ous probability measures on B([0, 1]) and suppose that the sequence (μik)

converges weakly to μi (denoted μik ⇀ μi) as k → ∞. Then for every f ∈D(A1, . . . , An), Tμ1k,...,μnk f (A1, . . . , An) → Tμ1,...,μn f (A1, . . . , An) in the oper-ator norm on L(X) as k → ∞.

Note:

(a) The weak convergence above is meant in the probabilist’s sense (see [1],p. 229).

(b) We can alternatively describe the conclusion of Theorem 1.1 as follows:The sequence of operational calculi specified by the sequence of n-tuples{(μ1k, . . . , μnk) : k = 1, . . . , ∞} converges as k → ∞ to the operationalcalculus specified by the n-tuple (μ1, . . . , μn).

We finish this introduction by briefly outlining the essential definitions andsome basic facts of the approach to Feynman’s operational calculi initiated in[4, 5]. A discussion of the heuristic ideas behind these operational calculi canbe found in Chapter 14 of [10].

Let X be a Banach space and let A1, . . . , An be nonzero bounded linearoperators on X. Except for the numbers ‖A1‖, . . . , ‖An‖, which will serveas weights, we ignore for the present the nature of A1, . . . , An as oper-ators and introduce a commutative Banach algebra consisting of ‘analytic


functions’ f (A1, . . . , An), where A1, . . . , An are treated as purely formalcommuting objects.

Consider the collection D = D(A1, . . . , An) of all expressions of the form

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mn Am11 · · · Amn

n (1.1)

where cm1,...,mn ∈ C for all m1, . . . , mn = 0, 1, . . . , and

‖ f (A1, . . . , An)‖ = ‖ f (A1, . . . , An)‖D(A1,...,An),

:=∞∑

m1,...,mn=0

|cm1,...,mn |‖A1‖m1 · · · ‖An‖mn < ∞. (1.2)

As pointed out in [4] the function on D(A1, . . . , An) defined by (1.2) makesD(A1, . . . , An) into a commutative Banach algebra under pointwise operations([4], Proposition 1.1). We refer to D(A1, . . . , An) as the disentangling algebraassociated with the n-tuple (A1, . . . , An) of bounded linear operators actingon X. This commutative Banach algebra will provide us with a frameworkwhere we can apply Feynman’s ‘rule’ (b) above rigorously rather than justheuristically.

Let μ1, . . . , μn be continuous probability measures defined at least onB([0, 1]), the Borel class of [0, 1]. The idea is to replace the operatorsA1, . . . , An with the elements A1, . . . , An from D and then form the desiredfunction of A1, . . . , An. Still working in D, we time order the expression forthe function and then pass to L(X) simply by removing the tildes.

Given nonnegative integers m1, . . . , mn, we let m = m1 + · · · + mn and

Pm1,...,mn(z1, . . . , zn) = zm11 · · · zmn

n . (1.3)

We are now ready to define the disentangling map Tμ1,...,μn which will carryus from our commutative framework to the noncommutative setting of L(X).

For j = 1, . . . , n and all s ∈ [0, 1], we take A j(s) = A j (recall that each A j isindependent of s) and, for i = 1, . . . , m, we define

Ci(s) :=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

A1(s) if i ∈ {1, . . . , m1},A2(s) if i ∈ {m1 + 1, . . . , m1 + m2},

......

An(s) if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}.(1.4)

For each m = 0, 1, . . . , let Sm denote the set of all permutations of the integers{1, . . . , m}, and given π ∈ Sm, we let

�m(π) = {(s1, . . . , sm) ∈ [0, 1]m : 0 < sπ(1) < · · · < sπ(m) < 1}.Finally, we remark that we will use the notation μk to denote μ × · · · × μ︸︷︷︸

k times

.


Definition 1.2 Tμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

)

:=∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))(μm11 × · · · × μmn

n )(ds1, . . . , dsm).

(1.5)

Then, for f (A1, . . . , An) ∈ D(A1, . . . , An) given by

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mn Am11 · · · Amn

n , (1.6)

we set Tμ1,...,μn

(f (A1, . . . , An)

)equal to

∞∑

m1,...,mn=0

cm1,...,mnTμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

). (1.7)

Remark 1.3 Even though the A j’s are independent of s, the order of theoperator products in each term of (1.5) depends on the s’s and on the measuresμ1, . . . , μn. (If the A j’s do depend on s, we obtain exactly the same expressionas seen in (1.5) and then we have a nontrivial integrand. But this situation willconcern us only marginally in this paper. For details of the time dependentsetting see, for example, the papers [8, 15, 16].)

It is worth noting that the disentangling map as defined above is a linearoperator of norm one from D(A1, . . . , An) to L(X). (See [4].) In the commu-tative setting, the right-hand side of (1.5) gives us Am1

1 · · · Amnn , the expected

result of the commutative functional calculus [4, Proposition 2.2]. (Of course,commutativity allows us to write the m operators in any desired order.) Asis usual, we shall write the operator Tμ1,...,μn f in place of Tμ1,...,μn( f ) for anelement f of D(A1, . . . , An).

We shall sometimes write the bounded linear operator

Tμ1,...,μn

(f (A1, . . . , An)

)

as fμ1,...,μn(A1, . . . , An), fμ1,...,μn(A) with A denoting the n-tuple (A1, . . . , An)

of operators, or fμ(A) with μ denoting the n–tuple (μ1, . . . , μn) of measures.In particular,

Pm1...mnμ1,...,μn

(A) = Tμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

). (1.8)

We find it convenient to use i as an index, so i denotes√−1. The real part of a

complex number z is written as �z and the imaginary part as �z. For a complexvector ζ = (ζ1, . . . , ζn) ∈ C

n, we set

�ζ = (�ζ1, . . . , �ζn), �ζ = (�ζ1, . . . , �ζn), |ζ | =√

|ζ1|2 + · · · + |ζn|2.

Remark 1.4 A family of Trotter product type formulas suitable for Feynman’soperational calculi (and mentioned in the abstract) will be established in


Theorem 2.4. Here the bounded linear operators A1, . . . , An need not be self-adjoint and X can be a Banach space. However, we will require the probabilitymeasures μ1, . . . , μn to be absolutely continuous with respect to Lebesguemeasure λ.

2 The Main Theorem

We give a detailed proof that for any n-tuple A = (A1, . . . , An) of self-adjointoperators, there exists r > 0 such that A is of ‘Paley–Wiener type’ (0, r, μ) forany μ = (μ1, . . . , μn). We will follow the statement of what this means with abrief discussion of its consequences for the enlargement of the domain of theassociated operational calculi.

Our main interest in this paper is in the Hilbert space setting. However, westate the definition of ‘Paley–Wiener type’ in the more general Banach spacesetting.

Definition 2.1 Let A1, . . . , An be bounded linear operators acting on a Banachspace X. Let μ = (μ1, . . . , μn) be an n-tuple of continuous probability mea-sures on B([0, 1]) and let

Tμ1,...,μn : D(A1, . . . , An) → L(X) (2.1)

be the disentangling map defined in Definition 1.2. If there exists C, r, s ≥ 0such that

‖Tμ1,...,μn

(ei(ζ,A)

)‖L(X) ≤ C(1 + |ζ |)ser|�ζ |, for all ζ ∈ C

n, (2.2)

then the n-tuple A = (A1, . . . , An) of operators is said to be of Paley–Wienertype (s, r, μ). (Note: Given A and ζ ∈ C

n, (ζ, A) = ζ1 A1 + · · · + ζn An.)

If the estimate (2.2) holds, then there exists a unique L(X)-valued distribu-tion Fμ,A ∈ L(C∞(Rn),L(X)) such that

Fμ,A( f ) = (2π)−n∫

RnTμ1,...,μn

(ei(ξ,A)

)f (ξ) dξ, (2.3)

for every rapidly decreasing function f ∈ S(Rn). Here

f (ξ) =∫

Rne−i(x,ξ) f (x) dx

denotes the Fourier transform of f . Moreover,

Fμ,A(Pm1,...,mn) = Pm1,...,mnμ1,...,μn

(A1, . . . , An), (2.4)

for all nonnegative integers m1, . . . , mn. Hence we have a rich extension ofthe functional calculus f �−→ fμ(A) from analytic functions with a uniformlyconvergent power series in a polydisk, to functions C∞ in a neighbourhood of


the support γμ(A) of Fμ,A. In fact, all of the distributions just mentioned arecompactly supported and so of finite order k. In the setting of Theorem 2.2,k will be the smallest integer strictly greater than n/2. (For example, k = 2,if n = 2 or 3.) Now let K = ∏n

j=1[−‖A j‖, ‖A j‖]. The functional calculus thenextends to all functions f that are k times continuously differentiable on someopen set containing K.

The support of γμ(A) of the distribution Fμ,A is defined as the μ–jointspectrum of the n–tuple A = (A1, . . . , An). The distribution Fμ,A is calledFeynman’s μ–functional calculus for A. The number rμ(A) = sup{|x| : x ∈γμ(A)} is called the μ–joint spectral radius of A. It is shown in [7] via Cliffordanalysis that the nonempty compact subset γμ(A) of R

n may be interpreted asthe set of singularities of a multidimensional analogue of the resolvent familyof a single operator. For more detail on or related to the last two paragraphs,see pages 186–192 and especially Theorem 3.1 and Proposition 3.2 of [6].

If we have more information about the particular operators and measuresthat are involved, we can sometimes further enlarge the functional calculus.Example 2.1, p.176–178 in [6] is an extreme case. Here A1 and A2 are the 2by 2, self-adjoint Pauli matrices σ1 and σ3. The measures μ1 and μ2 are anycontinuous probability measures with the support of μ1 entirely to the left ofthe support of μ2. In this case, fμ1,μ2(A1, A2) makes sense for any f which isdefined on the 4 point set {−1, 1} × {−1, 1}. This last set is the product of theordinary spectrums of σ1 and σ3.

Theorem 2.2 An n-tuple A = (A1, . . . , An) of bounded self-adjoint operatorsacting on a Hilbert space H is of Paley–Wiener type (0, r, μ) with r = (‖A1‖2 +· · · + ‖An‖2)1/2, for any n-tuple μ = (μ1, . . . , μn) of continuous probabilitymeasures on B([0, 1]).

Proof One of the keys to the proof is a use of the Martingale ConvergenceTheorem. We can apply it to the Radon–Nikodyn derivative of probabilitymeasures on [0, 1] which are absolutely continuous with respect to Lebesguemeasure λ. Such Radon–Nikodyn derivatives are nonnegative functions withL1(λ)-norm 1. However, we are only assuming that μ1, . . . , μn are continuousand so, given μi, we will begin by finding a sequence of absolutely continuousprobability measures which converge weakly to μi. (We will, but do not needto, do this even for the μi’s that are absolutely continuous with respect to λ.)

Let μ be a continuous Borel probability measure on [0, 1]. Let ρ : [0, 1] →R be a nonnegative continuous function with compact support in [0, 1) and‖ρ‖1 = 1. It will be convenient to let ρ(x) = 0 for x < 0. For every ε ∈ (0, 1],we set ρε(x) = ε−1ρ(x/ε), 0 ≤ x ≤ 1. It is easy to check that ‖ρε‖1 = 1 for 0 <

ε ≤ 1. Now we let

(ρε ∗ μ)(x) :=∫ x

0ρε(x − y)μ(dy), 0 ≤ x ≤ 1. (2.5)


We assert that (ρε ∗ μ)λ ⇀ μ as ε → 0+. Indeed, using the definition ofweak convergence, our assertion follows once we know that, for every con-tinuous function φ : [0, 1] → R,

∫ 1

0(ρε ∗ μ)(x)φ(x)dx →

∫ 1

0φ(x)μ(dx) as ε → 0+. (2.6)

We omit the proof of the limit (2.6) as it can be carried out using standard tech-niques. The basic idea of the proof is much like arguments using approximateidentities although some of the particular details follow Exercise 10, p. 194 of[17] more closely.

Now consider the partition of the interval [0, 1) into nk disjoint intervalsIk, = [( − 1)n−k, n−k), = 1, . . . , nk each of length n−k. (Below, n will bethe number of operator-measure pairs in our problem.) The collection Ak offinite disjoint unions of these intervals is an algebra – in fact a σ -algebra. Notethat Ak ⊂ Ak+1 for k = 1, 2, . . . . Let Pk : L1(λ) → L1(λ) be the conditionalexpectation operator [1, p.265] with respect to Ak; that is,

Pk f = nknk∑

=1

χIk,

∫

Ik,

f dλ. (2.7)

Note that the sequence {Pk f } is adapted with respect to the sequence {Ak}of σ -algebras [1, p. 280]. Then for each f ∈ L1(λ), by the Martingale Con-vergence Theorem [1, p. 285–286], Pk f → f in L1(λ) (and λ-a.e.) as k → ∞.

Further, by Theorem 10.1.3 from [1], we have∫ 1

0Pk f dλ =

∫ 1

0f dλ, k = 1, 2, . . . . (2.8)

(Remark The usual notation for Pk f in the probability literature is E( f |Ak).)Now set fi, j,k := Pk(ρ1/j ∗ μi) for each i = 1, . . . , n and j, k = 1, 2, . . . . Each

such step function fi, j,k is constant on each interval Ik, , = 1, . . . , nk and

fi, j,k := nknk∑

=1

χIk,

∫

Ik,

ρ1/j ∗ μi dλ. (2.9)

The function fi, j,k that we are about to define is a key to the proof:

fi, j,k := nknk−1−1∑

m=0

χIk,mn+i

∫

Ik−1,m+1

ρ1/j ∗ μi dλ. (2.10)

Note that this function has support in the finite union

Ji,k :=nk−1−1⋃

m=0

Ik,mn+i

of disjoint intervals, for each i = 1, . . . , n and Ji,k ∩ J ,k = ∅ for i �= . Theintegral

∫Ik−1,m+1

ρ1/j ∗ μi dλ in the sum (2.10) is a weight factor compensating


for the omission of terms from the sum (2.9), so that the following equalitieshold true:

∫ 1

0fi, j,k dλ =

∫ 1

0ρ1/j ∗ μi dλ =

∫ 1

0fi, j,k dλ. (2.11)

The first equality follows from (2.7), (2.8) and the definition of fi, j,k above.We turn now to the second equality in (2.11):

∫ 1

0fi, j,k dλ = nk

nk−1−1∑

m=0

1

nk

∫

Ik−1,m+1

ρ1/j ∗ μi dλ

=∫

Ik−1,1

ρ1/j ∗ μi dλ +∫

Ik−1,2

ρ1/j ∗ μi dλ + . . .

+∫

Ik−1,nk−1

ρ1/j ∗ μk dλ

=∫ 1

0ρ1/j ∗ μi dλ. (2.12)

Thus (2.11) is established.Now let φ be a continuous function on [0, 1]. Given ε > 0, use the uniform

continuity of φ to choose k so large that |φ(x) − φ(y)| < ε whenever |x − y| <

n−k+1. We wish to compare the integrals∫ 1

0 fi, j,kφ dλ and∫ 1

0 fi, j,kφ dλ. Webegin with the first of these. The RHS of the 2nd equality in (2.13) below isjust another way of writing the sum of the nk terms that appear on the LHS.

∫ 1

0fi, j,kφ dλ =

∫ 1

0

⎧⎨

⎩nknk∑

p=1

χIk,pφ

∫

Ik,p

ρ1/j ∗ μi dλ

⎫⎬

⎭dλ

=∫ 1

0nk

⎧⎨

⎩

nk−1−1∑

m=0

n∑

=1

χIk,mn+ φ

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)⎫⎬

⎭ dλ

= nknk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)(∫

Ik,mn+

φ dλ

). (2.13)

On the other hand, starting with (2.10) we have

∫ 1

0fi, j,kφ dλ =

∫ 1

0nk

nk−1−1∑

m=0

χIk,mn+iφ

(∫

Ik−1,m+1

ρ1/j ∗ μi dλ

)dλ

= nknk−1−1∑

m=0

(∫

Ik−1,m+1

ρ1/j ∗ μi dλ

)(∫

Ik,mn+i

φdλ

)

= nknk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)(∫

Ik,mn+i

φ dλ

). (2.14)


First using (2.13) and (2.14) and then the Mean Value Theorem for Inte-grals, there exist ξk, ∈ Ik, , = 1, . . . , nk such that

∣∣∣∣∫ 1

0

(fi, j,k − fi, j,k

)φ dλ

∣∣∣∣

≤nk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

) ∣∣∣∣nk∫

Ik,mn+

φ dλ − nk∫

Ik,mn+i

φ dλ

∣∣∣∣

=nk−1−1∑

m=0

n∑

=1

∫

Ik,mn+

ρ1/j ∗ μi dλ∣∣φ(ξk,mn+

)− φ(ξk,mn+i

)∣∣ < ε. (2.15)

The final inequality follows from the fact that |ξk,mn+ − ξk,mn+i| < n−k+1 foreach i ∈ {1, . . . , n} and each j ∈ {1, 2, 3, . . . }. Thus for each i ∈ {1, . . . , n} andeach j ∈ {1, . . . , m},

fi, j,kλ − fi, j,kλ ⇀ 0 as k → ∞. (2.16)

Let μi, j,k := fi, j,kλ, i = 1, . . . , n and j, k = 1, 2, . . . .

Next we summarize what we have proved so far about the weak limits andthen draw some conclusions. For each i = 1, . . . , n, the following limits obtain.

A(ρ1/j ∗ μi

)λ ⇀ μi as j → ∞.

B fi, j,kλ = Pk(ρ1/j ∗ μi

)λ ⇀

(ρ1/j ∗ μi

)λ as k → ∞.

In fact, the probability densities in B converge in L1-norm and so themeasures converge in total variation norm and so certainly converge weakly.

C fi, j,kλ − fi, j,kλ ⇀ 0 as k → ∞.

From B and C we see that

D fi, j,kλ = Pk(ρ1/j ∗ μi

)λ ⇀

(ρ1/j ∗ μi

)λ as k → ∞.

Now from D and A we have the iterated weak limits,

E lim j→∞[limk→∞

(fi, j,k

)λ]

= lim j→∞ ρ1/j ∗ μi = μi.

Since [0, 1] is a separable metric space, it follows from E that there exists asequence ( fi, j,k( j ))λ such that for each i = 1, . . . , n,

F(

fi, j,k( j )

)λ ⇀ μi as j → ∞.

(See [1, p. 309–310] and especially the paragraph preceding Theorem 11.3.3.)Note that the increasing sequence k( j ), j = 1, 2, . . . , may be chosen indepen-dently of i = 1, . . . , n by observing that the space M([0, 1], R

n) of Rn-valued

Borel measures on [0, 1] is in duality with the space C([0, 1], Rn) of R

n-valuedcontinuous functions and that closed balls in M([0, 1], R

n) are compact andmetrizable for the associated weak*-topology.


Note As we continue we will just write j, k but will understand that k = k( j )as in F above.

The measure μi, j,k is given by

μi, j,k = fi, j,kλ = nknk−1−1∑

m=0

(χIk,mn+iλ

) ∫

Ik−1,m+1

ρ1/j ∗ μi dλ. (2.17)

Before beginning the calculation below we make some simple commentsand introduce some notation.

(a) Since exponential functions are entire, the exponential functions involvedbelow are certainly in the domain of the disentangling map.

(b) We will write (ζ, A) = ζ1 A1 + · · · + ζn An where ζ = (ζ1, . . . , ζn) is an n-tuple of complex numbers and A = (A1, . . . , An). Similar notation willbe used in connection with A.

(c) Since the disentangling algebra is commutative, we have

e(ζ,A) = eζ1 A1+···+ζn An = eζ1 A1 . . . eζn An .

Because of how the function fi, j,k is supported (see (2.10)) and since μi, j,k isdefined using fi, j,k, an extension of Proposition 2.2 from [6] allows us to do thefollowing calculation (much as was done in Example 2.2 of that paper):

Tμ1, j,k,...,μn, j,k(ei(ζ,A))

=(

exp

{iζn

(∫

Ik−1,nk−1

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ

)A1

}). . .

(exp

{iζn

(∫

Ik−1,2

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,2

ρ1/j ∗ μ1dλ

)A1

}).

(exp

{iζn

(∫

Ik−1,1

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,1

ρ1/j ∗ μ1dλ

)A1

}). (2.18)

Breaking ζ1 = ξ1 + iη1 into real and imaginary parts, we have iζ1 = −η1+iξ1. Further, |eiζ1 | = |eiξ1 ||e−η1 | ≤ e|η1| = e|�ζ1|. Similarly, |eiζ2 | ≤ e|�ζ2|, . . . ,|eiζn | ≤ e|�ζn|.


Now using (2.18), the self-adjointness of Ai, i = 1, . . . , n, the standardmultiplicative Banach algebra inequality and the simple computations justabove, we can write

‖Tμ1, j,k,...,μn, j,k(ei(ζ,A))‖

≤∥∥∥∥∥exp

{iζn

(∫

Ik−1,nk−1

ρ1/j ∗ μndλ

)An

}∥∥∥∥∥ . . .

∥∥∥∥∥exp

{iζ1

(∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥∥ . . .

∥∥∥∥exp

{iζn

(∫

Ik−1,2

ρ1/j ∗ μndλ

)An

}∥∥∥∥ . . .

∥∥∥∥exp

{iζ1

(∫

Ik−1,2

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥ .

∥∥∥∥exp

{iζn

(∫

Ik−1,1

ρ1/j ∗ μndλ

)An

}∥∥∥∥ . . .

∥∥∥∥exp

{iζ1

(∫

Ik−1,1

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥ (2.19)

≤ e|�ζn|

(∫Ik−1,nk−1

ρ1/j∗μndλ

)‖An‖

. . . e|�ζ1|

(∫Ik−1,nk−1

ρ1/j∗μ1dλ

)‖A1‖

. . .

e|�ζn|

(∫Ik−1,2

ρ1/j∗μndλ)‖An‖

. . . e|�ζ1|

(∫Ik−1,2

ρ1/j∗μ1dλ)‖A1‖

.

e|�ζn|

(∫Ik−1,1

ρ1/j∗μndλ)‖An‖

. . . e|�ζ1|

(∫Ik−1,1

ρ1/j∗μ1dλ)‖A1‖

.

Now we are dealing with numbers and so the noncommutativity of theoperators is not an issue. Since

∫

Ik−1,1

ρ1/j ∗ μ1dλ +∫

Ik−1,2

ρ1/j ∗ μ1dλ + . . .

+∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ =∫ 1

0ρ1/j ∗ μ1dλ = 1,

the product of the terms in the column furthest to the right is e|�ζ1|‖A1‖.A similar thing happens in the other columns and so we obtain

‖Tμ1, j,k,...,μn, j,k(ei(ζ,A))‖ ≤ e(|�ζ1|‖A1‖+|�ζ2‖A2‖+···+|�ζn|‖An‖)

≤ e[|�ζ1|2+···+|�ζn|2]1/2[‖A1‖2+···+‖An‖2]1/2 = er|�ζ | (2.20)


where r = [‖A1‖2 + · · · + ‖An‖2]1/2 and |�ζ | is the Euclidean norm in Rn of

the vector (�ζ1, . . . , �ζn).

Theorem 1.1 and the inequality (2.20) show that the bounded self-adjoint operators A1, . . . , An are of Paley–Wiener type (0, r, μ) where μ =(μ1, . . . , μn). ��

Remark 2.3 Following a suggestion of Jefferies, Johnson and Nielsen usedthe main theorem from [11], that is, Theorem 1.1 of this paper, to prove thespecial case of Theorem 2.2 where n = 2 and the operational calculus is theWeyl calculus. In the closing remark of [11], Johnson and Nielsen asserted thatan argument similar to the one in that paper would take care of the case ofthe Weyl calculus for any n. In fact, the argument in [11] fails at a critical pointfor n ≥ 3.

The following result is a simple consequence of Theorem 1.1 and the proofof Theorem 2.2.

Corollary 2.1 Let A1, . . . , An be bounded, self-adjoint operators on the Hilbertspace H and let μ1, . . . , μn be absolutely continuous probability measures onB([0, 1]). Finally let ξ = (ξ1, . . . ξn) be an n-tuple of real numbers. Then

‖Tμ1,...,μn(ei(ξ,A))‖ = ‖Tμ1,...μn(e

iξ1 A1 · · · eiξn An)‖ = 1. (2.21)

Proof Let ζ from Theorem 2.2 (see also Definition 2.1) equal ξ as above. Thenwe see that the operator on the RHS of (2.18) is unitary and so has norm 1.But, by Theorem 1.1 and the fact that μi, j,k ⇀ μi for each i = 1, . . . , n, wehave the operator norm convergence of the sequence of operators in (2.18)to Tμ1,...,μn(e

i(ξ,A)). The equality in (2.21) follows. ��

The reader may have noticed that we did not need the self-adjointness of theoperators nor even the Hilbert space setting until late in the proof of Theorem2.2. In fact, we can use some of the early parts of the proof to establish a classof “Trotter product formulas” suitable for Feynman’s operational calculi in thegeneral Banach space setting.

Theorem 2.4 Let X be a Banach space over C and let ζ = (ζ1, . . . , ζn) bean n-tuple of complex numbers. Further, let μ1, . . . , μn be probability mea-sures on B([0, 1]) each of which is absolutely continuous with respect to λ;hence, μ1 = g1λ, . . . , μn = gnλ where g1, . . . , gn are nonnegative functions inL1([0, 1],B([0, 1]), λ). Finally, let A = (A1, . . . , An) be an n-tuple of boundedlinear operators on X. Then

‖Tμ1,...,μn(ei(ζ,A)) − Tμ1,k,...,μn,k(e

i(ζ,A))‖L(X) → 0 (2.22)


as k → ∞ where

Tμ1,k,...,μn,k(ei(ζ,A)) = Tμ1,k,...,μn,k(e

iζ1 A1 . . . eiζn An)

= exp

{iζn

(∫

Ik−1,nk−1

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,nk−1g1dλ

)A1

}. . .

exp

{iζn

(∫

Ik−1,2

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,2

g1dλ

)A1

}

exp

{iζn

(∫

Ik−1,1

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,1

g1dλ

)A1

}. (2.23)

Proof We will just comment on which parts of the proof of Theorem 2.2 areneeded here and which are not. We will also note the point at which Theorem1.1 is applied.

The first part of the proof of Theorem 2.2 is needed only for measures whichare continuous but not absolutely continuous. We have no such measures here.The second part of the earlier proof involves the Martingale ConvergenceTheorem. We need this and we obtain fi,k = Pkgi, i = 1, . . . , n with (2.9)suitably adjusted. (Note that we do not need the index j as our measures areall absolutely continuous.) The function fi,k is then defined with the integrandsin (2.10) changed to gi. The proof now moves along with no j’s involved andwith ρ1/j ∗ μi replaced by gi, i = 1, . . . , n.

When we reach the summary A-F, A is not needed. Replace B with fi,kλ =(Pkgi)λ ⇀ giλ as k → ∞ for i = 1, . . . , n. Change C to fi,kλ − fi,kλ ⇀ 0 ask → ∞ for i = 1, . . . , n. Based on the revised form of B and C, change D tofi,kλ ⇀ giλ as k → ∞; that is

μi,k ⇀ μi

as k → ∞ for i = 1, . . . , n. (Note that in (2.17) μi, j,k, fi, j,k and ρ1/j ∗ μi arereplaced by μi,k, fi,k and gi, respectively.)

The calculation in (2.18) is done in the same way but the subscript j on theLHS is missing and ρ1/j ∗ μi is replaced by gi, i = 1, . . . , n, on the RHS.

We can now apply Theorem 1.1 to finish the proof of this theorem. ��

Remark 2.5

(a) The inequalities which concerned us toward the end of the proof ofTheorem 2.2 did not concern us in the proof of Theorem 2.4 sincewe were not trying to show that (A1, . . . , An) is of Paley–Wiener type(0, r, (μ1, . . . , μn)).

(b) The Trotter products on the RHS of (2.23) look more like the usualTrotter products when special choices are made for ζ . Some examples:(1) Each ζ j equals i (or ti). (2) Each ζ j equals −1 (or −t).

(c) We hope to investigate variations and consequences of Theorem 2.4 inlater work.


Remark 2.6 In view of Corollary 3.1 of the paper [14], we may write (2.22) as

‖Tλ,...,λ(ei(ζ,[g·A])) − Tμ1,k,...,μn,k(ei(ζ,A))‖L(X) → 0 (2.24)

where [g · A] :=(

˜[g1 · A1], . . . , ˜[gn · An])

, that is the time independent opera-tors A1, . . . , An are replaced by the time dependent operators g1 · A1, . . . , gn ·An where g1 = dμ1

dλ, . . . , gn = dμn

dλ.

We now present two simple examples illustrating Theorem 2.4.

Example 2.7 For the first example we assume that, in Theorem 2.4, g1 = · · · =gn = 1; i.e. Lebesgue measure is associated to each operator. It follows from[5, Lemma 5.4] that

Tμ1,...,μn(ei(ζ,A)) = Tλ,...,λ(ei(ζ,A)) = ei(ζ,A) (2.25)

and so Theorem 2.4 tells us that

ei(ζ,A) = limk→∞

{exp

{ζn

nk−1An

}. . . exp

{ζ1

nk−1A1

}}nk−1

(2.26)

Example 2.8 In the second example, we will consider two operators, A1 andA2 and we will take μ1 = 2t dλ and μ2 = 3t2 dλ. For any nonnegative integer l,we have

∫

Ik−1,l

g1 dλ = 2l − 1

22k−2, (2.27)

and∫

Ik−1,l

g2 dλ = 3l2 − 3l + 1

23k−3. (2.28)

Theorem 2.4 tells us that

limk→∞

Tμ1,k,μ2,k ei(ζ1 A1+ζ2 A2)

= limk→∞

{exp

(iζ2

[3 · 22k−2 − 3 · 2k−1 + 1

23k−3

]A2

)·

exp

(iζ1

[2 · 2k−1 − 1

22k−2

]A1

)· · ·

exp

(iζ2

[7

23k−3

]A2

)exp

(iζ1

[3

22k−2

]A1

)·

exp

(iζ2

[1

23k−3

]A2

)exp

(iζ1

[1

22k−2

]A1

)}

= Tμ1,μ2 ei(ζ1 A1+ζ2 A2). (2.29)


Using Corollary 3.1 of [14], the last expression immediately above can be

written as Tλ,λei(ζ1 ˜[g1·A1]+ζ2 ˜[g2·A2]) where the time dependence is carried by thefunctions g1 and g2.

It is clear from Example 2.8 that we can produce an infinite number ofdistinct Trotter product formulas by varying the probability densities g1 and g2.

References

1. Dudley, R.M.: Real Analysis and Probability. Chapman and Hall Mathematics Series,New York (1989)

2. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys.Rev. 84, 108–128 (1951)

3. Jefferies, B.: Spectral properties of noncommuting operators. In: Lecture Notes inMathematics, vol. 1843. Springer, Berlin Heidelberg New York (2004)

4. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators:definitions and elementary properties. Russian J. Math. Phys. 8, 153–171 (2001)

5. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting systems ofoperators: tensors, ordered supports and disentangling an exponential factor. Math. Notes 70,815–838 (2001)

6. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting systems ofoperators: spectral theory. Infin. Dimens. Anal. Quantum Prob. 5, 171–199 (2002)

7. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: themonogenic calculus. Adv. Appl. Clifford Algebras 11, 233–265 (2002)

8. Jefferies, B., Johnson, G.W., Nielsen, L.: Feynman’s operational calculi for time-dependentnoncommuting operators. J. Korean Math. Soc. 38(2), (March 2001)

9. Johnson, G.W., Kim, B.S.: Extracting linear and bilinear factors in Feynman’s operationalcalculi. Math. Phys. Anal. Geom. 6, 181–200 (2003)

10. Johnson, G.W., Lapidus, M.L.: The Feynman integral and Feynman’s operational calculus.In: Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

11. Johnson, G.W., Nielsen, L.: A stability theorem for Feynman’s operational calculus.In: Conference Proc. Canadian Math. Soc., Conference in Honor of Sergio Albeverio’s 60thbirthday, vol. 29, pp. 351–365 (2000)

12. Maslov, V.P.: Operational Methods. Mir, Moscow (1976)13. Nazaikinskii, V.S., Shatalov, V.E., Sternin, B.Yu.: Methods of noncommutative analysis.

In: Studies in Mathematics, vol. 22. Walter de Gruyter, Berlin, Germany (1996)14. Nielsen, L.: Effects of absolute continuity in Feynman’s operational calculus. Proc. Amer.

Math. Soc. 131, 781–791 ( 2002)15. Nielsen, L.: Time dependent stability for Feynman’s operational calculus. Rocky Mountain J.

Math. 35, 1347–1368 (2005)16. Nielsen, L.: Stability properties of Feynman’s operational calculus for exponential functions

of noncommuting operators. Acta Appl. Math. 74, 265–292 (2002)17. Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill, New York (1987)18. Taylor, M.E.: Functions of several self-adjoint operators. Proc. Amer. Math. Soc. 19, 91–98

(1968)


On the Spectral Behaviour of a Non-self-adjointOperator with Complex Potential

Carmen Martínez Adame

Received: 27 September 2006 / Accepted: 26 May 2007 /Published online: 4 July 2007© Springer Science + Business Media B.V. 2007

Abstract We consider the non-self-adjoint Anderson operator with a complexpotential as a pseudo-ergodic operator in one spatial dimension and use secondorder numerical ranges to obtain tight bounds on the spectrum of the operator.We also find estimates for the size of possible holes contained in the spectrumof such an operator.

Keywords Complex potential · Non-self-adjoint Anderson model ·Numerical range · Spectrum

Mathematics Subject Classifications (2000) 47B80 · 47A12 · 60H25 · 65F15

1 Introduction and General Context

The Anderson model was developed in the early 1960’s as a theoretical modelto describe the effect of the presence of an impurity atom in a metal and thenon-self-adjoint version of this model originated in [9] motivated by the studyof superconductivity and has been studied in greater detail in [10, 11]. Sincethen however, its purely mathematical treatment has grown extensively. It hasbeen the subject of a series of papers by Davies, [1, 2], and by Goldsheid andKhoruzhenko, [5–7]; and in [13] we looked at the non-self-adjoint Andersonoperator with real potential and were able to obtain very tight bounds on thespectrum and in many cases to determine it completely.

C. Martínez Adame (B)Departamento de Matemáticas, Facultad de Ciencias, UNAM,Ciudad Universitaria, 04510, México D.F., Mexicoe-mail: [email protected]

82 C. Martínez Adame

Our interest in this paper is to study the non-self-adjoint Anderson operatorH with complex potential as a pseudo-ergodic operator defined on l2 (Z) andas such to obtain inner and outer bounds on the spectrum of said operator. InTheorem 5 we give an estimate of the size of the hole in the spectrum when 0 �∈Spec(H) and Theorems 6 and 10–12 provide precise bounds for the spectrumof the operator, which is always a bounded set of C. Theorem 13 is the mainresult of this paper, it delimits the spectrum of H and allows us to appreciatethe strength of the methods developed to study this operator as we come veryclose to producing a complete determination of the spectrum.

1.1 Pseudo-ergodic Operators

Pseudo-ergodic operators have been treated in recent papers and they arethe right context in which to work in this case as they allow us to eliminateall probabilistic aspects of the problem at hand. An account of their spectraltheory can be found in [4], for instance, and thus, we limit ourselves below toincluding the definition of these operators which is all that the reader needs tounderstand the present paper.

Definition 1 Let M1, M2, M3 be compact subsets of C and let H be an operatordefined on l2 (Z) such that

H (x, y) = 0 if |x − y| > 1,

H (x, x) ∈ M1,

H (x, x + 1) ∈ M2,

H (x, x − 1) ∈ M3.

We say that H is (Z, M1, M2, M3) pseudo-ergodic if for every ε > 0, every finitesubset F ⊂ Z and every Wr : F −→ Mr, where r = 1, 2, 3, there exists γ ∈ Z

such that

|H (γ + x, γ + x) − W1 (x)| < ε,

|H (γ + x, γ + x + 1) − W2 (x)| < ε,

|H (γ + x, γ + x − 1) − W3 (x)| < ε,

for all x ∈ F. If M2 and M3 consist solely of one point we will say that H is(Z, M1) pseudo-ergodic.

1.2 Higher Order Numerical Ranges

Many of the bounds we find for the spectrum of the operator that interests usdepend on results obtained from studying the second order numerical range.This concept was introduced in detail and in greater generality in [3, 12, 13]and it should be noted that it differs entirely from other generalisations ofthe numerical range as those that appear in [8]. In this paper we restrict thepresentation of this subject to the following brief account.

Spectral behaviour of a NSA operator 83

Let H be a bounded linear operator defined on a Hilbert space and let p (z) :C → C be a polynomial. We define

Num (p, H) ={

z : p (z) ∈ Num (p (H))}

, (1)

Numn (H) =⋂

deg(p)�n

Num (p, H) , (2)

and we call the set defined by (2) the n-th numerical range of H. It then followsthat Spec (H) ⊆ Num (p, T) and it is clear that for any n ∈ N

Spec (H) ⊆ Numn (H) . (3)

2 The Non-self-adjoint Anderson Operator with Complex Potential

We consider the (Z, M) pseudo-ergodic operator given by

H fn = ±β fn−1 + vn fn + α fn+1 (4)

acting on l2 (Z), where α, β ∈ R and M is a compact subset of C.It is easy to see that the case α, β ∈ C can be reduced to this case. To do this

we need the following lemma whose proof we omit:

Lemma 2 Given any two points z, w ∈ C there exist θ and ϕ in R such thatzeiθ−iϕ and we−iθ−iϕ are real.

Now, let α and β be any two complex numbers; we rewrite them as |α| eiθα and|β| eiθβ respectively. Let θ = 1

2

(π − θα + θβ

)and let us also define the unitary

operator Uθ on l2 (Z) given by Uθ fn = e−niθ fn. It follows that Uθ HUθ∗ has the

same spectrum as H and in fact, Uθ HUθ∗ is given by

Uθ HUθ∗ fn = βe− i

2 (π−θα+θβ) fn−1 + vn fn + αei2 (π−θα+θβ) fn+1.

If we now consider the operator defined by K = e− i2 (θα+θβ+π)Uθ HUθ

∗, itfollows that K fn = − |β| fn−1 + e− i

2 (θα+θβ+π)vn fn + |α| fn+1 and hence thedesired result follows.

Thus, let H fn = β fn−1 + vn fn + α fn+1 with α, β ∈ R and let M be a compactsubset of C. The particular case with which we are interested here is thecase M = {γ + iδ, γ − iδ, −γ + iδ, −γ − iδ} with γ, δ > 0. We will consider thecases when the coefficient of fn−1 is either β or −β separately, however wenote that it is only when we combine the results obtained in both cases that weget a more complete description of the spectrum of H.

Let us concentrate first on the case when

H fn = β fn−1 + vn fn + α fn+1.

We will assume, without loss of generality, that H has been suitably scaled sothat α − β = 1 and thus

Spec (H) ⊂ {z ∈ C : |y| � δ + 1} . (5)


If we split vn into its real and imaginary parts we may rewrite H in thefollowing forms:

H = A + iB = C + Vγ + i (B + Vδ) = H0 + V (6)

where B, C, Vγ , Vδ are self-adjoint; ‖B‖ = 1,∥∥Vγ

∥∥ = γ , ‖Vδ‖ = δ; V, Vγ , Vδ

are diagonal and H0 is normal with spectrum

{(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} ={(x, y) ∈ R

2 : x2

(2a)2 + y2 = 1

}.

These operators are defined by

Afn = afn−1 + vn fn + afn+1, V fn = vn fn,

Bfn = b fn−1 − b fn+1, Vγ fn = Re (vn) fn,

Cfn = afn−1 + afn+1, Vδ fn = Im (vn) fn.

where a = 1

2(α + β) and b = i

2.

With regard to these operators we have the following theorem which is acorollary of a more general result that Davies proved in [2]. We state it here asit will be of great use throughout this paper:

Theorem 3 The spectrum of H satisfies the following

Spec (H) ⊇ {(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} + M,

Spec (H) ⊆ conv {(α + β) cos θ + i sin θ : θ ∈ [0, 2π ]} + conv (M) ,

Spec (H) ⊆ Spec (H0) + B (0, |γ + iδ|) ,

Spec (H) ⊆⋃

m∈M

B (m, α + β) .

where B (0, |γ + iδ|) = {z ∈ C : |z| < |γ + iδ|}.

As a corollary to Theorem 3 we have the following result with respect toholes in Spec (H).

Corollary 4 If |γ + iδ| > α + β, then 0 �∈ Spec (H).

Proof This follows from the last inclusion in Theorem 3. �

We can also prove the following theorem which provides a more explicitresult.

Theorem 5 If |γ + iδ| < 1, then Spec (H) does not intersect the interior of thecurve given parametrically by

(2a cos t − |γ + iδ| cos t√

4a2 sin2 t + cos2 t, sin t − 2a |γ + iδ| sin t√

4a2 sin2 t + cos2 t

)


with t ∈ [0, 2π ], or the exterior of the curve(

2a cos t + |γ + iδ| cos t√4a2 sin2 t + cos2 t

, sin t + 2a |γ + iδ| sin t√4a2 sin2 t + cos2 t

).

Proof Theorem 3 implies that Spec (H) is contained in the set {2a cos θ +i sin θ : 0 � θ � 2π} + B (0, |γ + iδ|), or in other words, Spec (H) is containedin the union of balls of radius |γ + iδ| with centre at each point of the ellipsethat defines Spec (H0). Since |γ + iδ| < 1 there is a hole in the spectrum ofH and the origin is not contained in these balls. Thus, the envelope of thisfamily of circles is the curve that gives an estimate of the size of this hole.By construction, the envelope of this family of circles is a curve which isparallel to the ellipse at a distance |γ + iδ|. That is, a curve which is displacedfrom the ellipse by |γ + iδ| in the direction of the curve’s normal and giventhis characterization it is straightforward to see that if the ellipse is givenby ( f (t) , g (t)) = (2a cos t, sin t) then the envelope is given parametrically by(F (t) , G (t)) where

F (t) = 2a cos t − |γ + iδ| cos t√4a2 sin2 t + cos2t

G (t) = sin t − 2a|γ + iδ| sin t√4a2 sin2 t + cos2t

. �

Let us now prove the following theorem that provides outer bounds forSpec (H).

Theorem 6 The spectrum of H is contained in the set{

z ∈ C : x2 − y2 � 1

2a2 + 1γ 2 − 2δ2 − 2

}. (7)

Proof It is straightforward to see that

H2 = (C + Vγ

)2 − (B + Vδ)2 + i[(C + Vγ )(B + Vδ) + (B + Vδ)(C + Vγ )]

and hence,

Re(H2) = (

C + Vγ

)2 − (B + Vδ)2 ,

= C2 + V2γ + Vγ C + CVγ − B2 − V2

δ − Vδ B − BVδ.

Now, for any ϕ and ψ in R \ {0} we can write this equality as follows.

Re(H2) =

(ϕC + 1

ϕVγ

)2

+(

ψ B − 1

ψVδ

)2

+ (1 − ϕ2)C2 +

+(

1 − 1

ϕ2

)V2

γ − (1 + ψ2)

B2 −(

1 + 1

ψ2

)V2

δ


and since B2 + 14a2 C2 = I, it follows that

Re(H2) =

(ϕC + 1

ϕVγ

)2

+(

ψ B − 1

ψVδ

)2

+ (1 − ϕ2)

C2 +

+(

1 − 1

ϕ2

)V2

γ − (1 + ψ2) (

I − 1

4a2C2

)−(

1 + 1

ψ2

)V2

δ

�(1 − ϕ2

)C2 +

(1

4a2+ ψ2

4a2

)C2 +

(1 − 1

ϕ2

)V2

γ −

−(

1 + 1

ψ2

)V2

δ − (1 + ψ2)

I

as(ϕC + 1

ϕVγ

)2and

(ψ B − 1

ψVδ

)2are non-negative.

To optimize this last expression let ϕ2 = 2a2+12a2 and ψ = 1. This implies that

1 − ϕ2 + 1+ψ2

4a2 = 0 and hence

Re(H2)

�(

1 − 1

ϕ2

)V2

γ −(

1 + 1

ψ2

)V2

δ − (1 + ψ2)

I,

=(

ϕ2 − 1

ϕ2

)V2

γ −(

ψ2 + 1

ψ2

)V2

δ − (1 + ψ2) I,

=(

1

2a2 + 1

)V2

γ − 2V2δ − 2I.

However, as∥∥Vγ

∥∥ = γ and ‖Vδ‖ = δ this last inequality becomes

Re(H2)

�(

1

2a2 + 1

)γ 2 I − 2δ2 I − 2I.

That is,

Re(H2)

�(

1

2a2 + 1γ 2 − 2δ2 − 2

)I. (8)

Hence, if p2 (z) = z2, then the spectrum of H is contained in Num (p2, H)

and given that Re(H2)

�(

12a2+1γ 2 − 2δ2 − 2

)I it follows that Num (p2, H) ⊆{

z ∈ C : Re(z2)

� 12a2+1γ 2 − 2δ2 − 2

}and the required result follows. �

The curve x2 − y2 = 12a2+1γ 2 − 2δ2 − 2 which determines the boundary of

the set defined by (7) is a hyperbola (as expected since we are dealing with thesecond order numerical range) whose shape depends upon the values of γ andδ. In fact, it depends upon the sign of η (γ, δ), where

η (γ, δ) := 1

2a2 + 1γ 2 − 2δ2 − 2.


For η (γ, δ) to be negative, γ and δ must satisfy the inequality

1

4a2 + 2γ 2 − δ2 < 1. (9)

The set of points which satisfy (9) is a hyperbolic region in the γ δ-planewith boundary 1

4a2+2γ 2 − δ2 = 1. If (γ, δ) satisfy (9) then 12a2+1γ 2 − 2δ2 − 2 is

negative and the boundary of the hyperbolic region described by (7), which isthe hyperbola given by the equation x2 − y2 = 1

2a2+1γ 2 − 2δ2 − 2, opens about

the imaginary axis and has vertices at the points ±i√

−12a2+1γ 2 + 2δ2 + 2.

If γ and δ satisfy

1

4a2 + 2γ 2 − δ2 = 1,

then η (γ, δ) = 0 and the hyperbola defined in (7) turns into two straight linesgiven by y = ±x.

Finally, if 14a2+2γ 2 − δ2 > 1 then 1

2a2+1γ 2 − 2δ2 − 2 > 0 and the hyperbolawhich determines (7) opens about the real axis and has vertices at the points

±√

12a2+1γ 2 − 2δ2 − 2.

Remark 1 It is worth noting at this point that the case when the coefficient offn−1 is negative in the definition of H will produce a second hyperbola, thatwill not only bound the spectrum of that operator but will also provide boundsfor the spectrum of the operator we are dealing with here by means of a simplerotation and rescaling.

Now, for any for any complex number u = s + it let us define a potential Vby setting vn = vn − u. We rewrite V in terms of Vγ and Vδ as follows

V = Vγ − sI + i (Vδ − tI) .

Let H = H0 + V = C + (Vγ − sI)+ i (B + (Vδ − tI)), then

z �∈ Spec(H)

if and only if z + u �∈ Spec (H) . (10)

We can thus prove that there exists a hyperbolic region that containsSpec

(H)

and hence, a hyperbolic region in which Spec (H) lies, namely aregion determined by a hyperbola centered at u = s + it.

Theorem 7 Re(H2)

�[

12a2+1 (γ − |s|)2 − 2 (δ + |t|)2 − 2

]I.

Proof The proof follows in a straightforward manner from the proof ofTheorem 6. �

Corollary 8 For any s, t ∈ R, Spec (H) is contained in the set{

z ∈ C : (x − s)2 − (y − t)2 � 1

2a2 + 1(γ − |s|)2 − 2 (δ + |t|)2 − 2

}.


Proof This result follows from the previous theorem and (10). �

These last results imply that the spectrum of H is contained inside a non-empty region determined by a two-parameter family of curves. To obtain theenvelope of such a family we will differentiate the two-parameter function withrespect to each of the parameters.

Lemma 9 Let s, t ∈ R \ {0} and let

F (x, y, s, t) = (x − s)2 − (y − t)2 − 1

2a2 + 1(γ − |s|)2 + 2 (δ + |t|)2 + 2.

If δ � 1, then the envelope of this two-parameter family of curves lies on theellipses defined by

1

4a2

(x − sign (s) γ

)2 + (y + sign (t) δ)2 = 1

where sign (x) := x|x| for any x ∈ R \ {0}.

Proof The envelope of the family of curves defined by F is obtained by solving∂ F∂s = ∂ F

∂t = F = 0. However, given that both |s| and |t| appear in the expressionof F we will do this by cases. Let us consider first the case when both s and tare positive:

∂ F∂s

= −2 (x − s) + 2

2a2 + 1(γ − s) , (11)

∂ F∂t

= 2 (y − t) + 4 (δ + t) (12)

and setting both equations equal to 0 and substituting these values into F = 0gives

[x −

(2a2 + 1

2a2x − 1

2a2γ

)]2

− [y − (−2δ − y)]2 −

− 1

2a2 + 1

[γ −

(2a2 + 1

2a2x − 1

2a2γ

)]2

+ 2[δ + (−2δ − y)

]2 + 2

= 1

2a2

[4a2δ2 + 4a2 y2 + 8a2δy − 4a2 + x2 − 2xγ + γ 2

] = 0

or equivalently

1

4a2(x − γ )2 + (y + δ)2 = 1. (13)

However, given the fact that F describes a two-parameter family of curves weneed to determine the specific relation between these parameters that yields(13) as the envelope of F. This is done by solving (11) and (12) with respectto x and y and substituting these values into (13) to obtain a relation betweens and t.


Now, x is given by 2a2

2a2+1 s + 12a2+1γ and y = −2δ − t and substituting these

values into (13) we obtain

1

4a2

(2a2

2a2 + 1s + 1

2a2 + 1γ − γ

)2

+ (−2δ − t + δ)2 = 1.

Simplifying this expression we have that

a2

(2a2 + 1

)2 (s − γ )2 + (δ + t)2 = 1.

However, given that t is positive we observe that for this equation to bedefined we need to have δ � 1.

If s > 0 and t < 0 then

s = 2a2 + 1

2a2x − 1

2a2γ,

t = −y + 2δ

and the envelope lies on the ellipse given by

1

4a2(x − γ )2 + (y − δ)2 = 1

with the same requirement that δ � 1 as s and t need to satisfy the equationa2

(2a2+1)2 (s − γ )2 + (t − δ)2 = 1.

The other two cases are analogous, and again, only valid if δ � 1. If s < 0and t > 0 then

s = 2a2 + 1

2a2x + 1

2a2γ,

t = −y − 2δ

and the envelope lies on the ellipse 14a2 (x + γ )2 + (y + δ)2 = 1. If s and t are

both negative then

s = 2a2 + 1

2a2x + 1

2a2γ,

t = −y + 2δ

and the envelope is contained in 14a2 (x + γ )2 + (y − δ)2 = 1. �

The fact that the results stated in Lemma 9 rely heavily on the sign of s andt imply that only a certain portion of the ellipses is actually obtained as theenvelope of F. We will assume that δ � 1 and study the case when both s andt are positive in detail and obtain the other three cases from the symmetry ofthe problem. In this case we know that the envelope of F lies on 1

4a2 (x − γ )2 +(y + δ)2 = 1.

We will restrict the values of s to be between 0 and γ as Theorem 3 givesus better results than what we would obtain by this method from the casesgiven by s > γ . In fact, Theorem 3 establishes that Spec (H) is contained


in the convex hull of the four ellipses given by 14a2 (x ± γ )2 + (y ± δ)2 = 1

and contains these four ellipses and hence, gives the best possible infor-mation in certain regions of the complex plane. That is, it determines theboundary of the spectrum completely in the four quarter-planes given by{z ∈ C : |x| � |γ | and |y| � |δ|}.

Theorem 10 Let δ � 1 and let

μ = max

{γ − 2a

√1 − δ2,

1

2a2 + 1γ

}

then Spec (H) does not intersect the subset of points z of C such that μ < x < γand

max

{−δ − 1,−δ −

√4a2 − (x − γ )2

}< y < −δ −

√1 − 1

4a2 (x − γ )2

}.

Furthermore, let

= B (γ − iδ, 2a) ∩ {z ∈ C : γ − 2a < x < μ & − δ − 1 < y < −δ} .

If μ = γ − 2a√

1 − δ2 then Spec (H) does not intersect

{z ∈ C : F− (x, y, s0, t0) � 0} ∩

and if μ = 12a2+1γ then Spec (H) does not intersect

{z ∈ C : F− (x, y, s1, t1) � 0} ∩

where F (x, y, s, t) is as given in Lemma 9 and F− denotes the bottom (or left)branch of the hyperbola and

s0 = γ − 2a2 + 1

a

√1 − δ2,

t0 = 0,

s1 = 0,

t1 = −δ +√

1 − a2

(2a2 + 1

)2 γ 2.

Proof Let s > 0 and t > 0 satisfy a2

(2a2+1)2 (s − γ )2 + (t + δ)2 = 1 which is the

equation obtained in the proof of the last Lemma. Taking into account theremarks made previous to the statement of this theorem this implies that we

can restrict the interval in which s will vary to(

max{

0, γ − 2a2+1a

}, γ]

and wewill take t ∈ (0, 1 − δ].

We have that x = 2a2

2a2+1 s + 12a2+1γ , and given the interval of variation of s,

this implies that x lies either in (γ −2a, γ ) or(

12a2+1γ, γ

). Similarly, y=−2δ − t


and given that t ∈ (0, 1 − δ] this implies that y varies between −δ − 1 and −2δ.However, as the equation of the envelope given by

1

4a2(x − γ )2 + (y + δ)2 = 1

provides us with a specific relation between x and y, the fact that y can at mostbe −2δ implies that we have a bound on x as well. Indeed, let us take y = −2δ,then

1

4a2(x − γ )2 + (−2δ + δ)2 = 1,

x − γ = ±2a√

1 − δ2

and hence, x cannot be less than γ − 2a√

1 − δ2. In other words the portion of1

4a2 (x − γ )2 + (y + δ)2 = 1 that we can obtain as the envelope of the familydescribed by F in Lemma 13 will be obtained as x varies in the intervalgiven by

(max

{1

2a2 + 1γ, γ − 2a

√1 − δ2

}, γ

).

Thus, in light of Theorem 3, this renders the first statement of the theorem.To prove the other two statements we note that there exists an element of thefamily described by F that intersects the ellipse 1

4a2 (x − γ )2 + (y + δ)2 = 1 at

the point(μ, −δ −

√1 − 1

4a2 (μ − γ )2)

.

Let us suppose that μ = γ − 2a√

1 − δ2. In this case, the member of thefamily described by F is obtained by letting t → 0+. This implies that s →γ − 2a2+1

a

√1 − δ2 and hence the hyperbola we are interested in is given by the

point (s0, t0) =(γ − 2a2+1

a

√1 − δ2, 0

), that is, by the equation

(x − γ + 2a2 + 1

a

√1 − δ2

)2

− y2 +(4a2 + 1

)δ2 − 1

a2= 0.

Let us now suppose that μ = 12a2+1 . We proceed in an analogous manner and

let s → 0+. This implies that t → −δ +√

1 − a2

2a2+1γ 2 and the hyperbola thatyields the last statement of the theorem is determined by the point (s1, t1) =(

0, −δ +√

1 − a2

2a2+1γ 2)

and thus is given by the equation

x2 −(

y + δ −√

1 − a2

(2a2 + 1

)2 γ 2

)2

− 4a2 + 1(2a2 + 1

)2 γ 2 + 4 = 0. �

We note that this theorem was obtained by restricting the possible values ofs and t in Corollary 8 to R

+. If we now remove this restriction we can obtainthe following theorem from the symmetry of the problem.


Theorem 11 Let δ � 1 and μ = max{γ − 2a

√1 − δ2, 1

2a2+1γ}

. Spec (H) does

not intersect the subset of points of C such that μ < |x| < γ and

max

{−δ − 1, −δ −

√4a2 − (x − γ )2

}< |y| < −δ −

√1 − 1

4a2(x − γ )2

}.

Furthermore, if

=⋃

m∈MB (m, 2a)

⋂{z ∈ C : γ − 2a < |x| < μ and − δ − 1 < |y| < −δ

}

where M = {γ + iδ, γ − iδ, −γ + iδ, −γ − iδ} as before, then, if μ = γ −2a

√1 − δ2, Spec (H) does not intersect the set

{z ∈ C : F (x, y, s0, t0) � 0} ∩

and if μ = 12a2+1γ , then Spec (H) does not intersect the set

{z ∈ C : F (x, y, s1, t1) � 0} ∩

where F (x, y, s, t) is as given in Lemma 9 and

s0 = ±(

γ − 2a2 + 1

a

√1 − δ2

),

t0 = 0,

s1 = 0,

t1 = ±(

−δ +√

1 − a2

(2a2 + 1

)2 γ 2

).

Let us now consider

H fn = −β fn−1 + vn fn + α fn+1, (14)

we will assume that H has been scaled so that α + β = 1 and rewrite H as in(6) with a = 1

2 (α − β) and b = i2 . Spec (H) is thus contained in

{z ∈ C : |y| � δ + 1} .

and it is straightforward to see that Theorem 6, Corollary 8 and Theorem 11hold in this case with no modifications to the proofs as stated previously. Werefrain from stating these results again and proceed to combining the two casesgiven by (4) to obtain better bounds on the spectrum of a given operator. Whenthe coefficient of fn−1 in the definition of an operator of the type consideredhere is positive the spectrum contains four ellipses whose major axes areparallel to the real axis, whereas if the coefficient of fn−1 is negative thespectrum still contains four ellipses but now the major axes are parallel to theimaginary axis. This is analogous to what happens in the case of real potentialsas we have seen in [13]. We will use these facts to obtain tighter bounds for thespectrum of any given operator H defined by (4). In fact, we have the followingresult which we state for the case H fn = β fn−1 + vn fn + α fn+1 with α − β = 1.We observe that an analogous result is true for H fn = −β fn−1 + vn fn + α fn+1,


however we omit the proof as it is clear that this case can be reduced to thecase we present below.

Theorem 12 Let H be given by H fn = β fn−1 + vn fn + α fn+1 with α − β = 1.(Fig. 1). Then

Spec (H) ⊆{

z ∈ C : x2 − y2 � 1

2a2 + 1γ 2 − 2δ2 − 2

},

and

Spec (H) ⊆{

z ∈ C : −x2 + y2 � 8a2

1 + 8a2δ2 − 2γ 2 − 8a2

}.

Proof The first inclusion is that of Theorem 6. To prove the second inclusionwe define an auxiliary operator by multiplying H by i

α+β, however, to simplify

our calculations here we consider the operator H− whose spectrum is clearly

equal to Spec(

iα+β

H)

and is given by

H− fn = − β

α + βfn−1 + i

α + βvn fn + α

α + βfn+1. (15)

This operator is of the type defined in (14) and satisfies the condition thatβ

α+β+ α

α+β= 1, which is the condition we have set throughout the paper for

operators of this type.Applying the analogue of Theorem 6 to H− it follows that

Spec (H−) ⊆

⎧⎪⎨⎪⎩

z ∈ C : x2 − y2 � 1

2(

12

1α+β

)2

δ2

(α + β)2 − 2γ 2

(α + β)2 − 2

⎫⎪⎬⎪⎭

,

and hence,

Spec (H−) ⊆{

z ∈ C : x2 − y2 � 2

1 + 8a2δ2 − 1

2a2γ 2 − 2

}(16)

where a is defined in terms of H as 12 (α + β). This in turn yields a result in

terms of Spec (H) when we rescale by (α+β)

i , that is, we obtain a hyperbolathat bounds Spec (H) when we rotate the hyperbola defined in (16) by −π

2 andscale by (α + β). In other words, given the hyperbola

x2 − y2 = 2

1 + 8a2δ2 − 1

2a2γ 2 − 2,

we rotate it to obtain

−x2 + y2 = 2

1 + 8a2δ2 − 1

2a2γ 2 − 2

and scale it by a factor of α + β = 2a which produces

−x2 + y2 = (2a)2

[2

1 + 8a2δ2 − 1

2a2γ 2 − 2

]


and simplifying this we obtain

Spec (H) ⊆{

z ∈ C : −x2 + y2 � 8a2

1 + 8a2δ2 − 2γ 2 − 8a2

}. �

Now, given the operator H− fn = − β

α+βfn−1 + i

α+βvn fn + α

α+βfn+1 which we

constructed in the proof of Theorem 12 we can produce a family of hyperbolaewhich bound Spec (H) where H is given by H fn = β fn−1 + vn fn + α fn+1 asbefore. This is done by applying Theorem 7 to H−.

In other words, if s, t ∈ R and we define H− = H− − (s + it) I, then

Re(

H−2)

�[

2

8a2 + 1(δ − |s|)2 − 1

2a2(γ + |t|)2 − 2

]I,

and hence Spec (H−) is contained in the set

{z ∈ C : (x − t)2 − (y − s)2 � 2

8a2 + 1(δ − |s|)2 − 1

2a2(γ + |t|)2 − 2

}. (17)

This in turn provides a bound on Spec (H) as we see in the followingtheorem.

Theorem 13 Let H be given by H fn = β fn−1 + vn fn + α fn+1 where α − β = 1and let s, t ∈ R. The spectrum of H is contained in the set{

z ∈ C : − (x + t)2 + (y + s)2 � 8a2

1 + 8a2(δ − |s|)2 − 2 (γ + |t|)2 − 8a2

}.

Proof Given (17) and the relation that exists between H and H−, it followsthat for each pair of real numbers s, t, Spec (H) is contained in the regionobtained by rotating by −π

2 and scaling by α + β the region defined by(x − t)2 − (y − s)2 � 2

8a2+1 (δ − |s|)2 − 12a2 (γ + |t|)2 − 2. �

The figure below shows how the spectrum of a particular operator His bounded by several elements of this family of hyperbolae together withhyperbolae obtained from Corollary 8.

Fig. 1 In this exampleα = 1.5, β = 0.5, γ = 1.8and δ = 0.8 and Spec (H)

is bounded by two familiesof hyperbolae which areobtained via Theorem 12and by varying s and t inCorollary 8 and Theorem 13

–6 –4 –2 0 2 4 6

–4

–3

–2

–1

0

1

2

3

4


References

1. Davies, E.B.: Spectral properties of random non-self-adjoint matrices and operators. Proc.Roy. Soc. London Ser. A 457, 191–206 (2001)

2. Davies, E.B.: Spectral theory of pseudo-ergodic operators. Comm. Math. Phys. 216, 687–704(2001)

3. Davies, E.B.: Spectral bounds using higher order numerical ranges. LMS J. Comput. Math. 8,17–45 (2005)

4. Davies, E.B., Simon, B.: L1 properties of intrinsic schrödinger operators. J. Funct. Anal. 65,126–146 (1986)

5. Goldsheid, I.Y., Khoruzhenko, B.A.: Distribution of eigenvalues in non-hermitian Andersonmodel. Phys. Rev. Lett. 80, 2897–2901 (1998)

6. Goldsheid, I.Y., Khoruzhenko, B.A.: Eigenvalue curves of asymmetric tridiagonal randommatrices. Electron. J. Probab. 5(16), 1–28 (2000)

7. Goldsheid, I.Y., Khoruzhenko, B.A.: Regular spacings of complex eigenvalues in the one-dimensional non-hermitian Anderson model. Comm. Math. Phys. 238, 505–524 (2003)

8. Gustafson, K.E. Rao, D.K.M.: Numerical Range. The field of values of linear operators andmatrices. Springer, New York (1997)

9. Hatano, N., Nelson, D.R.: Localization transitions in non-hermitian quantum mechanics. Phys.Rev. Lett. 77, 570–573 (1996)

10. Hatano, N., Nelson, D.R.: Vortex pinning and non-hermitian quantum mechanics. Phys. Rev.B 56, 8651–8673 (1997)

11. Hatano, N., Nelson, D.R.: Non-hermitian localization and Eigenfunctions. Phys. Rev. B 58,8384–8390 (1998)

12. Martínez, C.: Spectral Properties of Tridiagonal Operators, PhD Dissertation, King’s CollegeLondon, London (2005)

13. Martínez, C.: Spectral estimates for the one-dimensional non-self-adjoint Anderson model.J. Operator Theory 56(1), 59–88 (2006)


Positivity of Lyapunov Exponents for a ContinuousMatrix-Valued Anderson Model

Hakim Boumaza

Received: 28 February 2007 / Accepted: 6 June 2007 /Published online: 11 July 2007© Springer Science + Business Media B.V. 2007

Abstract We study a continuous matrix-valued Anderson-type model. Bothleading Lyapunov exponents of this model are proved to be positive anddistinct for all energies in (2, +∞) except those in a discrete set, which leadsto absence of absolutely continuous spectrum in (2, +∞). This result is animprovement of a previous result with Stolz. The methods, based upon aresult by Breuillard and Gelander on dense subgroups in semisimple Liegroups, and a criterion by Goldsheid and Margulis, allow for singular Bernoullidistributions.

Keywords Lyapunov exponents · Anderson model

Mathematics Subject Classification (2000) 34F05

1 Introduction

We will study the question of separability of Lyapunov exponents for acontinuous matrix-valued Anderson–Bernoulli model of the form:

HAB(ω) = − d2

dx2+

(0 11 0

)+

+∑n∈Z

(ω

(n)1 χ[0,1](x − n) 0

0 ω(n)2 χ[0,1](x − n)

)(1)

H. Boumaza (B)Institut de Mathématiques de Jussieu, Université Paris,7 Denis Diderot, 2 place Jussieu, 75251 Paris, Francee-mail: [email protected]

98 H. Boumaza

acting on L2(R) ⊗ C2. This question is coming from a more general problem

on Anderson–Bernoulli models. Indeed, localization for Anderson models indimension d�2 is still an open problem if one look for arbitrary disorder,especially for Bernoulli randomness. A possible approach to the local-ization for d=2 is to discretize one direction. It leads to consider one-dimensional continuous Schrödinger operators, no longer scalar-valued, butnow N × N matrix-valued. Before considering N×N matrix-valued continu-ous Schrödinger operators, we start with the model (1) corresponding to N =2.

What is already well understood is the case of dimension one scalar-valued continuous Schrödinger operators with arbitrary randomness includ-ing Bernoulli distributions (see [6]) and discrete matrix-valued Schrödingeroperators also including the Bernoulli case (see [7] and [10]). We aim atcombining existing techniques for these cases to prove that for our model (1),the Lyapunov exponents are all positive and distinct for all energies outside adiscrete set, at least for energies in (2, +∞) (see Theorem 3).

It is already proved in [3] that for model (1), the Lyapunov exponents areseparable for all energies except those in a countable set, the critical energies.Due to Kotani’s theory (see [11]) this result already implies the absence ofabsolutely continuous spectrum in the interval (2, +∞). But the techniquesused in [3] didn’t allow us to avoid the case of an everywhere dense countableset of critical energies and we have to keep in mind that we want to be ableto use our result to prove Anderson localization and not only the absence ofabsolutely continuous spectrum. The separability of Lyapunov exponents canbe seen as a first step in order to follow a multiscale analysis scheme. The nextstep would be to prove some regularity on the integrated density of states, likelocal Hölder-continuity and then to prove a Wegner estimate and an initiallength scale estimate to start the multiscale analysis (see [13]). To prove thelocal Hölder-continuity of the integrated density of states, we need to have theseparability of the Lyapunov exponents on intervals (see [5] or [6]). But, if likein [3] we can get an everywhere dense countable set of critical energies, we willnot be able to prove local Hölder-continuity of the integrated density of states.This is the main reason of doing the present improvement of the result of [3].

Our approach of the separability of Lyapunov exponents is based uponan abstract criterion in terms of the group generated by the random transfermatrices. This criterion has been provided by Gol’dsheid and Margulis in [7]and was used to prove Anderson localization for discrete strips (see [10]). It isalso interesting because it allows for singularly distributed random parameters,including Bernoulli distributions.

We had the same approach in [3]; what changes here is the way to apply thecriterion of Gol’dsheid and Margulis. We have to prove that a certain groupis Zariski-dense in the symplectic group Sp2(R). In [3] we were constructingexplicitly a family of ten matrices linearly independent in the Lie algebrasp2(R) of Sp2(R). This construction was only possible by considering energiesin (2, +∞) and an everywhere dense countable set of critical energies. By usinga result of group theory by Breuillard and Gelander (see [4]), we are here able

Continuous matrix-valued Anderson model 99

to prove that the group involved in Gol’dsheid–Margulis’ criterion is dense inSp2(R) for all energies in (2, +∞), except those in a discrete subset.

We start in Section 2 with a presentation of the necessary background onproducts of i.i.d. symplectic matrices and with a statement of the criterionof Gol’dsheid and Margulis. We also present the result of Breuillard andGelander in this section. Then, in Section 3 we specify the assumptions madeon the model (1) and we make explicit the transfer matrices associated tothis model. In Section 4 we give the proof of our main result, Theorem 3 byfollowing the steps given by the assumptions of Theorem 2 by Breuillard andGelander.

We finish by mentioning that different methods have been used in [9] toprove localization properties for random operators on strips. They are basedupon the use of spectral averaging techniques which did not allow to handlewith singular distributions of the random parameters. So even if the methodsused in [9] (which only considers discrete strips) have potential to be applicableto continuous models, one difference between these methods and the onesused here is that, like in [3], we handle singular distributions, in particularBernoulli distributions.

2 Criterion of Separability of Lyapunov Exponents

We start with a review of some results about Lyapunov exponents and howto prove their separability. These results hold for general sequences of i.i.d.random symplectic matrices. Even if we will only use them for symplecticmatrices in M4(R), we will write these results for symplectic matrices inM2N(R) for arbitrary N.

Let N be a positive integer. Let SpN(R) denote the group of 2N × 2N realsymplectic matrices, i.e.:

SpN(R) = {M ∈ GL2N(R) | t MJM = J}

where

J =(

0 −II 0

).

Here, I = IN is the N × N identity matrix.

Definition 1 (Lyapunov exponents) Let (Aωn )n∈N be a sequence of i.i.d. ran-

dom matrices in SpN(R) with

E(log+ ||Aω1 ||) < ∞.

100 H. Boumaza

The Lyapunov exponents γ1, . . . , γ2N associated with (Aωn )n∈N are defined

inductively by

p∑i=1

γi = limn→∞

1

nE(log || ∧p (Aω

n . . . Aω1 )||).

Here, ∧p(Aωn . . . Aω

1 ) denotes the p-th exterior power of the matrix(Aω

n . . . Aω1 ), acting on the p-th exterior power of R

2N . For more details aboutthese p-th exterior powers, see [2].

One has γ1 � . . . � γ2N . Moreover, the random matrices (An)n∈N beingsymplectic, we have the symmetry property γ2N−i+1 = −γi, for i = 1, . . . , N(see [2], Proposition 3.2).

We say that the Lyapunov exponents of a sequence (Aωn )n∈N of i.i.d. random

matrices are separable when they are all distinct:

γ1 > γ2 > . . . > γ2N.

We now give a criterion of separability of the Lyapunov exponents. Forthe definitions of Lp-strong irreducibility and p-contractivity we refer to [2],Definitions A.IV.3.3 and A.IV.1.1, respectively.

Let μ be a probability measure on SpN(R). We denote by Gμ the smallestclosed subgroup of SpN(R) which contains the topological support of μ, supp μ.

Now we can set forth the main result on separability of Lyapunov exponents,which is a generalization of Furstenberg’s theorem to the case N > 1.

Proposition 1 Let (Aωn )n∈N be a sequence of i.i.d. random symplectic matrices

of order 2N and p be an integer, 1 � p � N. Let μ be the common distributionof the Aω

n . If

(a) Gμ is p-contracting and Lp-strongly irreducible,(b) E(log ‖Aω

1 ‖) < ∞,

then the following holds:

1. γp > γp+1

2. For any non zero x in Lp:

limn→∞

1

nE(log ‖ ∧p Aω

n . . . Aω1 x‖) =

p∑i=1

γi .

Proof See [2], Proposition 3.4. �

Corollary 1 If

(a) Gμ is p-contracting and Lp-strongly irreducible for p = 1, . . . , N,(b) E(log ‖Aω

1 ‖) < ∞,

then γ1 > γ2 > . . . > γN > 0.


Proof Use Proposition 1 and the symmetry property of Lyapunov exponents.�

For explicit models like (1), it can be quite difficult to check the p-contractivity and the Lp-strong irreducibility for all p. To avoid this difficulty,we will use the Gol’dsheid–Margulis theory presented in [7] which gives analgebraic criterion to check these assumptions. The idea is the following: if thegroup Gμ is large enough in an algebraic sense then Gμ is p-contractive andLp-strongly irreducible for all p.

We first recall the definition of the Zariski topology on M2N(R). We identifyM2N(R) to R

(2N)2by identifying a matrix to the list of its entries. Then for

S ⊂ R[X1, . . . , X(2N)2 ], we set:

V(S) = {x ∈ R(2N)2 | ∀P ∈ S, P(x) = 0}

So, V(S) is the set of common zeros of the polynomials in S. These sets V(S)

are the closed sets of the Zariski topology on R(2N)2

. Then, on any subset ofM2N(R) we can define the Zariski topology as the topology induced by theZariski topology on M2N(R). In particular we define in this way the Zariskitopology on SpN(R).

The Zariski closure of a subset G of SpN(R) is the smallest Zariski closedsubset that contains G. We denote it by ClZ(G). In other words, if G is asubset of SpN(R), its Zariski closure ClZ(G) is the set of zeros of polynomialsvanishing on G. A subset G′ ⊂ G is said to be Zariski-dense in G if ClZ(G′) =ClZ(G), i.e., each polynomial vanishing on G′ vanishes on G.

Being Zariski-dense is the meaning of being large enough for a subgroupof SpN(R) to be p-contractive and Lp-strongly irreducible for all p. Moreprecisely, from the results of Gol’dsheid and Margulis one gets:

Theorem 1 (Gol’dsheid–Margulis criterion, [7]) If Gμ is Zariski dense inSpN(R), then for all p, Gμ is p-contractive and Lp-strongly irreducible.

Proof It is explained in [3] how to get that criterion from the results ofGol’dhseid and Margulis stated in [7]. �

As we can see in [3], it is not easy to check directly that the group GμE

introduced there is Zariski-dense. In [3] we were reconstructing explicitly theZariski closure of GμE . But this construction was possible only for energiesnot in a dense countable subset of R. We will now give a way to prove moresystematically the Zariski-density of a subgroup of SpN(R). It is based on thefollowing result of Breuillard and Gelander:

Theorem 2 (Breuillard, Gelander [4]) Let G be a real, connected, semisimpleLie group, whose Lie algebra is g.

Then there is a neighborhood O of 1 in G, on which log = exp−1 is a welldefined diffeomorphism, such that g1, . . . , gm ∈ O generate a dense subgroupwhenever log g1, . . . , log gm generate g.

102 H. Boumaza

We will use this theorem in the sequel to prove that the subgroup generatedby the transfer matrices associated to our operator is dense, hence Zariski-dense, in SpN(R).

In the next section we will specify the assumptions on model (1) and givethe statement of our main result.

3 A Matrix-Valued Continuous Anderson Model

Let

HAB(ω) = − d2

dx2+ V0 +

∑n∈Z

(ω

(n)1 χ[0,1](x − n) 0

0 ω(n)2 χ[0,1](x − n)

)(2)

be a random Schrödinger operator acting in L2(R) ⊗ C2. Here

• χ[0,1] denotes the characteristic function of the interval [0, 1],• V0 is the constant-coefficient multiplication operator by

(0 11 0

),

• (ω(n)1 )n∈Z, (ω

(n)2 )n∈Z are two independent sequences of i.i.d. random vari-

ables with common distribution ν such that {0, 1} ⊂ supp ν.

This operator is a bounded perturbation of − d2

dx2 ⊕ − d2

dx2 . Thus it is self-adjointon the Sobolev space H2(R) ⊗ C

2.For the operator HAB(ω) defined by (2) we have the following result:

Theorem 3 Let γ1(E) and γ2(E) be the positive Lyapunov exponents associatedto HAB(ω).

There exists a discrete subset SB ⊂ R such that γ1(E) > γ2(E) > 0 for allE > 2, E /∈ SB.

Corollary 2 HAB(ω) has no absolutely continuous spectrum in the interval(2, +∞).

We will first specify some notations. We consider the differential system:

HABu = Eu, E ∈ R. (3)

For a solution u = (u1, u2) of this system we define the transfer matrix Aω(n)

n (E),n ∈ Z from n to n + 1 by the relation

⎛⎜⎜⎝

u1(n + 1)

u2(n + 1)

u′1(n + 1)

u′2(n + 1)

⎞⎟⎟⎠ = Aω(n)

n (E)

⎛⎜⎜⎝

u1(n)

u2(n)

u′1(n)

u′2(n)

⎞⎟⎟⎠ .


The sequence {Aω(n)

n (E)}n∈Z is a sequence of i.i.d. random matrices in the sym-plectic group Sp2(R). This sequence will determine the Lyapunov exponents atenergy E. In order to use Proposition 1, it is necessary to define a measure onSp2(R) adapted to the sequence {Aω(n)

n (E)}n∈Z. The distribution μE is given by:

μE(�) = ν({

ω(0) = (ω

(0)1 , ω

(0)2

) ∈ (supp ν)2 | Aω(0)

0 (E) ∈ �})

for any Borel subset � ⊂ Sp2(R). The distribution μE is defined by Aω(0)

0 (E)

alone because the random matrices Aω(n)

n (E) are i.i.d.

Definition 2 We denote by GμE the smallest closed subgroup of Sp2(R) gener-ated by the support of μE.

Since {0, 1} ⊂ supp ν, A(0,0)0 (E), A(1,0)

0 (E), A(0,1)0 (E), A(1,1)

0 (E) ∈ GμE . Weneed to work with explicit forms of these four transfer matrices. First, we set:

Mω(0) =(

ω(0)1 11 ω

(0)2

). (4)

We start by writing Aω(0)

0 (E) as an exponential. We associate to the secondorder differential system (3) the following first order differential system:

Y ′ =(

0 I2

Mω(0) − E 0

)Y (5)

with Y ∈ M4(R). If Y is the solution with initial condition Y(0) = I4, thenAω(0)

0 (E) = Y(1). Solving (5), we get:

Aω(0)

0 (E) = exp

(0 I2

Mω(0) − E 0

). (6)

To compute this exponential, we have to compute the successive powersof Mω(0) . To do this, we diagonalize the real symmetric matrix Mω(0) by anorthogonal matrix Sω(0) :

Mω(0) =(

ω(0)1 11 ω

(0)2

)= Sω(0)

(λω(0)

1 00 λω(0)

2

)S−1

ω(0) ,

104 H. Boumaza

the eigenvalues λω(0)

2 � λω(0)

1 of Mω(0) being real. We can compute these eigen-values and the corresponding matrices Sω(0) for the different values of ω(0) ∈{0, 1}2. We get:

S(0,0) = 1√2

(1 11 −1

), λ

(0,0)1 = 1, λ

(0,0)2 = −1,

S(1,1) = S(0,0), λ(1,1)1 = 2, λ

(1,1)2 = 0,

S(1,0) =⎛⎝

2√10−2

√5

2√10+2

√5

−1+√5√

10−2√

5

−1−√5√

10+2√

5

⎞⎠ , λ

(1,0)1 = 1 + √

5

2, λ

(1,0)2 = 1 − √

5

2,

S(0,1) =⎛⎝

2√10−2

√5

2√10+2

√5

1−√5√

10−2√

5

1+√5√

10+2√

5

⎞⎠ , λ

(0,1)1 = 1 + √

5

2, λ

(0,1)2 = 1 − √

5

2.

We also define the block matrices:

Rω(0) =(

Sω(0) 00 Sω(0)

).

Let E > 2 be larger than all eigenvalues of all Mω(0) . With the abbreviation

rl = rl(E, ω(0)) :=√

E − λω(0)

l , l = 1, 2,

the transfer matrices become

Aω(0)

0 (E) = Rω(0)

⎛⎜⎜⎜⎝

cos r1 0 sin r1r1

0

0 cos r2 0 sin r2r2

−r1 sin r1 0 cos r1 0

0 −r2 sin r2 0 cos r2

⎞⎟⎟⎟⎠ R−1

ω(0) . (7)

4 Proof of Theorem 3

We will show in the last part of this section that Theorem 3 can be easilydeduced from the following proposition:

Proposition 2 There exists a discrete subset SB ⊂ R such that GμE = Sp2(R) forall E > 2, E /∈ SB.

Sections 4.1–4.3 are devoted to the proof of Proposition 2. To prove thisproposition, we will follow Theorem 2 for G = Sp2(R). Let O be a neighbor-hood of the identity in G = Sp2(R) as in Theorem 2.

4.1 Elements of GμE in O

To apply Theorem 2 we need to work with elements in the neighborhood O ofthe identity. We will work with the four matrices A(0,0)

0 (E), A(1,0)0 (E), A(0,1)

0 (E)


and A(1,1)0 (E) which are in GμE . We will prove that by taking a suitable power

of each of these matrices we find four matrices in GμE which lies in an arbitrarysmall neighborhood of the identity and thus in O. For this purpose we will usea simultaneous diophantine approximation result.

Theorem 4 (Dirichlet [12]) Let α1, . . . , αN be real numbers and let M > 1 bean integer. There are integers y, x1, . . . , xN in Z such that 1 � y � M and

|αi y − xi| < M− 1N

for i = 1, . . . , N.

From this theorem we deduce the proposition:

Proposition 3 Let E ∈ (2, +∞). For all ω(0) ∈ {0, 1}2, there is an integermω(E) � 1 such that:

Aω(0)

0 (E)mω(E) ∈ O.

Proof We fix ω(0) ∈ {0, 1}2. Let M > 1 be an integer. Theorem 4 with α1 = r12π

and α2 = r22π

leads to the existence of y, x1, x2 ∈ Z, such that 1 � y � M and∣∣∣ r1

2πy − x1

∣∣∣ < M− 12 ,

∣∣∣ r2

2πy − x2

∣∣∣ < M− 12 ,

which be can be written as:

|r1 y − 2x1π | < 2π M− 12 , |r2 y − 2x2π | < 2π M− 1

2 . (8)

Let θi = yri − 2πxi, i = 1, 2. Then we have:

Aω(0)

0 (E)y = Rω(0)

⎛⎜⎜⎝

cos yr1 0 sin yr1

r10

0 cos yr2 0 sin yr2

r2−r1 sin yr1 0 cos yr1 00 −r2 sin yr2 0 cos yr2

⎞⎟⎟⎠ R−1

ω(0)

= Rω(0)

⎛⎜⎜⎝

cos θ1 0 sin θ1r1

00 cos θ2 0 sin θ2

r2−r1 sin θ1 0 cos θ1 00 −r2 sin θ2 0 cos θ2

⎞⎟⎟⎠ R−1

ω(0)

by 2π -periodicity of sinus and cosinus. Let ε > 0. If we choose M large enough,M− 1

2 will be small enough to get:∥∥∥∥∥∥∥∥

⎛⎜⎜⎝

cos θ1 0 sin θ1r1

00 cos θ2 0 sin θ2

r2−r1 sin θ1 0 cos θ1 00 −r2 sin θ2 0 cos θ2

⎞⎟⎟⎠ − I4

∥∥∥∥∥∥∥∥< ε.

106 H. Boumaza

The matrices Sω(0) being orthogonal, so are also the matrices Rω(0) . Thenconjugating by Rω(0) does not change the norm and:

‖Aω(0)

0 (E)y − I4‖ < ε.

As O depends only on the semisimple group Sp2(R), we can choose ε such thatB(I4, ε) ⊂ O. So if we set y = mω(E), we have 1 � mω(E) � M and:

Aω(0)

0 (E)mω(E) ∈ O. �

Remark It is important to note that the neighborhood O does not dependneither on E nor on ω(0). So the integer M > 1 also does not depend neitheron E nor on ω(0). It will be crucial in some step of the proof to say that evenif the integer mω(E) depends on E and ω(0), it belongs always to an interval ofintegers {1, . . . , M} independent of E and ω(0).

To apply Theorem 2, we need to show that the logarithms of the matricesAω(0)

0 (E)mω(E) generate the Lie algebra sp2(R) of Sp2(R). A first difficulty is tocompute the logarithm of Aω(0)

0 (E)mω(E) which belongs to logO.

4.2 Computation of the Logarithm of Aω0 (E)mω(E)

We fix ω(0) ∈ {0, 1}2. We assume E > 2. Let ϑi = mω(E)ri, i = 1, 2. To computethe logarithm of Aω

0 (E)mω(E), we start from its expression:

Aω(0)

0 (E)mω(E) = Rω(0)

⎛⎜⎜⎜⎝

cos ϑ1 0 sin ϑ1r1

0

0 cos ϑ2 0 sin ϑ2r2

−r1 sin ϑ1 0 cos ϑ1 0

0 −r2 sin ϑ2 0 cos ϑ2

⎞⎟⎟⎟⎠ R−1

ω(0) .

We can always permute the vectors of the orthonormal basis defined by thecolumns of Rω(0) . So there exists a permutation matrix Pω(0) (thus orthogonal)such that:

Aω(0)

0 (E)mω(E)

= Rω(0) Pω(0)

⎛⎜⎜⎜⎝

cos ϑ1sin ϑ1

r10 0

−r1 sin ϑ1 cos ϑ1 0 0

0 0 cos ϑ2sin ϑ2

r2

0 0 −r2 sin ϑ2 cos ϑ2

⎞⎟⎟⎟⎠ P−1

ω(0) R−1ω(0) .

Recall that we can choose mω(E) such that Aω0 (E)mω(E) is arbitrarily close to

the identity in Sp2(R). Particularly we can assume that:∥∥∥Aω(0)

0 (E)mω(E) − I4

∥∥∥ < 1 .


So we can use the power series of the logarithm:

log Aω(0)

0 (E)mω(E) =∑k�1

(−1)k+1

k

(Aω(0)

0 (E)mω(E) − I4)k

. (9)

To simplify our computations we will also use the complex forms of sinus andcosinus. We set:

Qω(0) =

⎛⎜⎜⎜⎝

− ir1

ir1

0 0

1 1 0 0

0 0 − ir2

ir2

0 0 1 1

⎞⎟⎟⎟⎠ .

Hence:

Q−1ω(0) = 1

2

⎛⎜⎜⎝

ir1 1 0 0−ir1 1 0 0

0 0 ir2 10 0 −ir2 1

⎞⎟⎟⎠

Let

κ±l = e±imω(E)rl , l = 1, 2. (10)

Then we have:

Aω(0)

0 (E)mω(E) − I4

= Rω(0) Pω(0) Qω(0)

⎛⎜⎜⎝

κ+1 − 1 0 0 0

0 κ−1 − 1 0 0

0 0 κ+2 − 1 0

0 0 0 κ−2 − 1

⎞⎟⎟⎠ Q−1

ω(0) P−1ω(0) R−1

ω(0) .

So by using (9) we only have to compute:

+∞∑k=1

(−1)k+1

k(κ±

l − 1)k.

Let Ln be the main determination of the complex logarithm defined on C \ R−.We want to write, for l = 1, 2:

+∞∑k=1

(−1)k+1

k(κ±

l − 1)k = Ln κ±l .

To write this, we have to assume that rl =√

E − λω(0)

l /∈ π + 2πZ. So weintroduce the discrete set

S1 = {E > 2 | E = −λω(0)

l + π + 2 jπ for j ∈ Z, l = 1, 2, ω(0) ∈ {0, 1}2}.

108 H. Boumaza

If we choose E > 2, E /∈ S1 we can write:

log Aω(0)

0 (E)mω(E)

= Rω(0) Pω(0) Qω(0)

⎛⎜⎜⎝

Ln κ+1 0 0 0

0 Ln κ−1 0 0

0 0 Ln κ+2 0

0 0 0 Ln κ−2

⎞⎟⎟⎠ Q−1

ω(0) P−1ω(0) R−1

ω(0) .

Therefore, we are left with computing Ln κ±l . We do this for l = 1, the

computation will be the same for l = 2. We have:

Ln κ+1 = i Arg κ+

1 = i Arcsin sin ϑ1

= i(

mω(E)r1 − π

⌊mω(E)r1

π+ 1

2

⌋)(−1)

⌊mω(E)r1

π+ 1

2

⌋(11)

where � · � in (11) denotes the integer part. We recall that by (8), mω(E)r1 can

be chosen arbitrarily close to 2πZ., i.e. we can assume that mω(E)r1π

is arbitrarilyclose to an even integer. It suffices to choose M such that 2M− 1

2 < 12 to have⌊

mω(E)r1π

+ 12

⌋even and more precisely equal to 2x1. Thus (11) becomes:

Ln κ+1 = i

(mω(E)r1 − π

⌊mω(E)r1

π+ 1

2

⌋). (12)

We have the corresponding equation for the conjugate logarithm:

Ln κ−1 = i

(−mω(E)r1 − π

⌊−mω(E)r1

π+ 1

2

⌋)(−1)

⌊− mω(E)r1

π+ 1

2

⌋. (13)

Then: (− ir1

ir1

1 1

)×

(Ln κ+

1 00 Ln κ−

1

)× 1

2

(ir1 1

−ir1 1

)

= 1

2

(Ln κ+

1 + Ln κ−1 − i

r1

(Ln κ+

1 − Ln κ−1

)ir1

(Ln κ+

1 − Ln κ−1

)Ln κ+

1 + Ln κ−1

)(14)

By (12) and (13) we have:

Ln κ+1 + Ln κ−

1 = −iπ(⌊

mω(E)r1

π+ 1

2

⌋+

⌊−mω(E)r1

π+ 1

2

⌋)

and, for all x ∈ R: ⌊x + 1

2

⌋+

⌊1

2− x

⌋=

{1 if x ∈ 1

2 + Z,

0 otherwise.

We can assume that mω(E)rlπ

is arbitrarily close to an even number, hence wecan assume that for l = 1, 2, mω(E)rl

πdoes not belong to 1

2 + Z. So we have:

Ln κ+1 + Ln κ−

1 = 0 (15)


and:

Ln κ+1 − Ln κ−

1

= 2imω(E)r1 − iπ(⌊

mω(E)r1

π− 1

2

⌋−

⌊−mω(E)r1

π+ 1

2

⌋)

= 2imω(E)r1 − 2iπ⌊

mω(E)r1

π− 1

2

⌋. (16)

Let, for l = 1, 2:

xl = xl(E, ω) := 1

2

⌊mω(E)rl

π− 1

2

⌋.

αl = −mω(E)r2l + 2πrlxl,

βl = mω(E) − 2πxl

rl. (17)

Putting (15) and (16) into (14), and doing the same for the block correspondingto r2, we get:

log Aω(0)

0 (E)mω(E) = Rω(0) Pω(0)

⎛⎜⎜⎝

0 β1 0 0α1 0 0 00 0 0 β2

0 0 α2 0

⎞⎟⎟⎠ P−1

ω(0) R−1ω(0)

= Rω(0)

⎛⎜⎜⎝

0 0 β1 00 0 0 β2

α1 0 0 00 α2 0 0

⎞⎟⎟⎠ R−1

ω(0)

We set:

LAω(0) := log Aω(0)

0 (E)mω(E).

We can summarize the computations we have done in this section. For allE > 2, E /∈ S1:

LAω(0) = Rω(0)

⎛⎜⎜⎝

0 0 β1 00 0 0 β2

α1 0 0 00 α2 0 0

⎞⎟⎟⎠ R−1

ω(0) . (18)

We have now to prove that the four matrices LAω(0) , for ω(0) ∈ {0, 1}2, generatethe whole Lie algebra sp2(R).

4.3 The Lie Algebra la2(E)

For E ∈ (2, +∞) \ S1, we denote by la2(E) the Lie subalgebra of sp2(R)

generated by the LAω(0) for ω(0) ∈ {0, 1}2. We will use the expressions of λω(0)

iand Sω computed in Section 3.

110 H. Boumaza

4.3.1 Notations

We set:

a1 = x1(E, (0, 0)) =⌊

m(0,0)(E)√

E − 1

π+ 1

2

⌋

a2 = x2(E, (0, 0)) =⌊

m(0,0)(E)√

E + 1

π+ 1

2

⌋

b1 = x1(E, (1, 0)) =⎢⎢⎢⎣m(1,0)(E)

√E − 1+√

52

π+ 1

2

⎥⎥⎥⎦

b2 = x2(E, (1, 0)) =⎢⎢⎢⎣m(1,0)(E)

√E − 1−√

52

π+ 1

2

⎥⎥⎥⎦

and

c1 = x1(E, (0, 1)) =⎢⎢⎢⎣m(0,1)(E)

√E − 1+√

52

π+ 1

2

⎥⎥⎥⎦

c2 = x2(E, (0, 1)) =⎢⎢⎢⎣m(0,1)(E)

√E − 1−√

52

π+ 1

2

⎥⎥⎥⎦

d1 = x1(E, (1, 1)) =⌊

m(1,1)(E)√

Eπ

+ 1

2

⌋

d2 = x2(E, (1, 1)) =⌊

m(1,1)(E)√

E − 2

π+ 1

2

⌋.

We denote by M[i, j ] the (i, j ) entry of a matrix M. We also set:

r001 = √

E − 1, r002 = √

E + 1,

r111 = √

E − 2, r112 = √

E,

r101 = r01

1 =√

E − 1 + √5

2, r10

2 = r012 :=

√E − 1 − √

5

2,

and finally we set:

D1(E) = √E − 1

√E + 1

√E − 1 + √

5

2

√E − 1 − √

5

2,

D2(E) = √E

√E − 2

√E − 1 + √

5

2

√E − 1 − √

5

2.


To prove that la2(E) = sp2(R), we will find a family of 10 matrices linearlyindependent in la2(E). First we will consider the subspace generated by theLie brackets [LAω(0) , LAω(0)].

4.3.2 The Subspace V1 Generated by the [LAω(0) , LAω(0)]

A direct computation shows that each Lie bracket [LAω(0) , LAω(0)] is ofthe form

(A 00 −t A

)(19)

for some A ∈ M2(R). Let V1 be the 4-dimensional subspace of sp2(R) ofmatrices of the form (19). We will show that outside a discrete set of energiesE, the four Lie brackets

ϒ1 = [LA(1,0), LA(0,0)], ϒ2 = [LA(1,0), LA(1,1)],ϒ3 = [LA(0,1), LA(0,0)], ϒ4 = [LA(0,1), LA(0,0)]

generate V1.

Expression of ϒ1 = [LA(1,0), LA(0,0)]. We give the expressions of the entries.By (19) it suffices to give the entries corresponding to the first diagonal 2 × 2block.

ϒ1[1, 1] = − 1

4√

5D1(E)×

× [( − π(a1r00

2 + a2r001

) + 2m00r001 r00

2

)(πb1(1 + √

5)r102 −

−πb2(1 − √5)r10

1 − 2√

5m10r101 r10

2

)]

ϒ1[1, 2] = π2 E

2√

5D1(E)

[(b1r10

2 − b2r101

)(a1r00

2 − a2r001

)]

ϒ1[2, 1] = − π

4√

5D1(E)×

× [π(a2r00

1 − 5a1r002 + 4m00r00

1 r002

)(b1r10

2 + b2r101

) +

+ (a1r00

2 − a2r001

)(2√

5m10r101 r10

2 + 2π E(b1r10

2 − b2r101

))]

ϒ1[2, 2] = π2

2√

5D1(E)

[(b1r10

2 − b2r101

)(a1r00

2 − a2r001

)].

112 H. Boumaza

Expression of ϒ2 = [LA(0,1), LA(0,0)]. We have:

ϒ2[1, 1] = − 1

20D1(E)×

×[(10

√5πm00r00

1 r002 − √

5π2(a2r001 + 3a1r00

2

))(c1r10

2 − c2r101

) ++ 5

(π2

(a1r00

2 − 3a2r001

) + 2πm00r001 r00

2

)(c1r10

2 + c2r101

) −−10

(πm01

(a1r00

2 − 3a2r001

) + 2m00m01r001 r00

2

)r10

1 r102

]

ϒ2[1, 2] = − 1

2√

5D1(E)×

×[(π2

(a1r00

2 − 3a2r001

) + π2 E(a1r00

2 − a2r001

) ++(2 + 2

√5)πm00r00

1 r002

)(c1r10

2 − c2r101

) −−√

5π2(a1r00

2 + a2r001

)(c1r10

2 + c2r101

) ++2

√5(πm01

(a1r00

2 + a2r001

) − 2m00m01r001 r00

2

)r10

1 r102

]

ϒ2[2, 1] = − 1

20D1(E)×

×[(5π2

(a1r00

2 + 3a2r001

) − 20πm00r001 r00

2

)(c1r10

2 + c2r101

) ++√

5π2(2E − 5)(a1r00

2 − a2r001

)(c1r10

2 − c2r101

) −− 10

(πm01

(a1r00

2 + 3a2r001

) − 4m00m01r001 r00

2

)r10

1 r102

]ϒ2[2, 2] = − π

10D1(E)×

×[5π

(a1r00

2 − a2r001

)(c1r10

2 + c2r101

) + 2√

5(π(a1r00

2 + a2r001

) ++ 2m00r00

1 r002

)(c1r10

2 − c2r101

) ++ 10m01

(a1r00

2 + a2r001

)r10

1 r102

].


ϒ3[1, 1] = − π

10D2(E)×

×[2√

5(2m11r11

1 r112 − π

(d1r11

2 + d2r111

))(b1r10

2 − b2r101

) ++ 5π

(d2r11

1 − d1r112

)(b1r10

2 + b2r101

) + 10m10(d1r11

2 − d2r111

)r10

1 r102

]

ϒ3[1, 2] = 1

20D2(E)×

×[√5π2

(d1r11

2 − d2r111

)(2E − 3)

(b1r10

2 − b2r101

)+


+ (5π2

(d1r11

2 + 3d2r111

) − 20πm11r111 r11

2

)(b1r10

2 + b2r101

) ++ (

40m11m10r111 r11

2 − 10πm10(d1r11

2 + 3d2r111

))r10

1 r102

]

ϒ3[2, 1] = − 1

10D2(E)×

× [(2π2

√5(2d2r11

1 − d1r112

) + π2√

5E(d1r11

2 − d2r111

) −−2 π

√5m11r11

1 r112

)(b1r10

2 − b2r101

) ++ (

10πm11r111 r11

2 − 5π2(d1r11

2 + d2r111

))(b1r10

2 + b2r101

) ++ (

10πm10(d1r11

2 + d2r111

) − 20m11m00r111 r11

2

)r10

1 r102

]

ϒ3[2, 2] = − 1

20D2(E)×

×[(10π

√5m11r11

1 r112 − π2

√5(3d1r11

2 + 7d2r111

))(b 1r10

2 − b 2r101

) ++ (

5π2(3d2r11

1 − d1r112

) − 10πm11r111 r11

2

)(b1r10

2 + b2r101

) ++ (

10πm10(d1r11

2 − 3d2r111

) + 20m11m00r111 r11

2

)r10

1 r102

].


ϒ4[1, 1] = π2

2√

5D2(E)

[(d1r11

2 − d2r111

)(c1r10

2 − c2r101

)]

ϒ4[1, 2] = π

4√

5D2(E)×

×[(π(d1r11

2 + 3d2r111

) + 2π E(d1r11

2 − d2r111

) − 4m11r111 r11

2

) ×× (

c1r102 − c2r10

1

) + √5π

(d2r11

1 − d1r112

)(c1r10

2 + c2r101

)+ 2

√5m01

(d1r11

2 − d2r111

)r10

1 r102

]

ϒ4[2, 1] = − π2

2√

5D2(E)

[(E − 1)

(d1r11

2 − d2r111

)(c1r10

2 − c2r101

)]

ϒ4[2, 2] = − 1

4√

5D2(E)×

× [(2m11r11

1 r112 − π

(d1r11

2 + d2r111

))(2√

5m01r101 r10

2 ++ π

(c1r10

2 − c2r101

) − √5π

(c1r10

2 + c2r101

))].

To prove that la2(E) = sp2(R), we will build a family of 10 matrices linearlyindependent in la2(E). First we will consider the subspace generated by theLie brackets [LAω(0) , LAω(0)].

114 H. Boumaza

Then we consider the determinant of these entries:

det

⎛⎜⎜⎝

ϒ1[1, 1] ϒ2[1, 1] ϒ3[1, 1] ϒ4[1, 1]ϒ1[1, 2] ϒ2[1, 2] ϒ3[1, 2] ϒ4[1, 2]ϒ1[2, 1] ϒ2[2, 1] ϒ3[2, 1] ϒ4[2, 1]ϒ1[2, 2] ϒ2[2, 2] ϒ3[2, 2] ϒ4[2, 2]

⎞⎟⎟⎠

= f1(E) = f1(a1, a2, b1, b2, c1, c2, d1, d2, m00, m01, m10, m11, E) (20)

where f1(X1, . . . , X12, Y) is a polynomial function in X1, . . . , X12, analytic inY. Indeed, the determinant (20) is a rational function in the r jk

i which areanalytic functions in E not vanishing on the interval (2, +∞).

Note that all coefficients a1, . . . , d2, m00, . . . , m11 depend also on E and arenot analytic in E. Hence f1 is a priori not analytic in E. We will now explainhow to avoid this difficulty.

We recall that for all E and ω, 1 � mω(E) � M with M independent of Eand ω. Thus mω(E) only take a finite number of values in the set {1, . . . , M}.

Then we consider the sequence of intervals I2 = (2, 3], I3 = [3, 4], and forall k � 3, Ik = [k, k + 1]. These intervals cover (2, +∞). We fix k � 2 and weassume that E ∈ Ik. Then the integers

xωi (E) =

⌊mω(E)

√E − λω

i

π+ 1

2

⌋

are bounded by a constant depending only on M and Ik. Indeed, the eigenval-ues λω

i are all in the fixed interval [−2, 2], mω(E) take its values in {1, . . . , M}and E ∈ Ik. So the integers xω

i (E) take only a finite number of values in a set{0, . . . , Nk}.

To study the zeros of the function f1 on Ik, we have only to study the zerosof a finite number of analytic functions:

f1,p,l : E �→ f1(p1, . . . , p8, l1, . . . , l4, E)

for pi ∈ {0, . . . , Nk} and l j ∈ {1, . . . , M}. We have to show that the functionsf1,p,l do not vanish identically on Ik. In fact, the only bad case is when all thexω

i are zero. Indeed, f1(0, . . . , 0, X9, . . . , X12, Y) is identically zero. But if welook at the values of xω

i for E > 2 and mω(E) � 1, we get that a2 � 1. We cancompute the term of the determinant (20) involving only a2. We get:

m210m2

01m211π

2a22

E + 1� π2

E + 1> 0.


By observing all entries of (20), this term is the only one involving E onlyby this power of E + 1 = (r00

2 )2 and no other power of the rklj . So this term

cannot be cancelled uniformly in E by another term of the development ofthe determinant (20), whatever values taken by the integers a1, b1, . . . , d2 andm00, . . . , m11. So the only case where f1,p,l could identically vanish does nothappen. We set:

J1 = {(a1, b1, . . . , d2, m00, . . . , m11) | 0 � a1, c1, . . . , d2 � Nk,

1 � b1 � Nk, 1 � mij � M}.

Then, as (a1, . . . , m11) ∈ J1 the set of zeros of f1 in Ik is included in thefollowing finite union of discrete sets:

{E ∈ Ik | f1(E) = 0} ⊂⋃

(p,l )∈J1

{E ∈ Ik | f1,p,l(E) = 0}

Thus this set is also discrete in Ik. We finally get that:

{E ∈ (2, +∞) | f1(E) = 0} =⋃k�2

{E ∈ Ik | f1(E) = 0}

is discrete in (2, +∞). We set:

S2 = {E > 2 | f1(E) = 0}.

Let E > 2, E /∈ S1 ∪ S2. As the determinant (20) is not zero, it follows thatthe four matrices ϒ1, . . . , ϒ4 are linearly independent in the subspace V1 ⊂sp2(R) of dimension 4. Thus, they generate V1. We deduce that:

for all E ∈ (2, +∞) \ (S1 ∪ S2), V1 ⊂ la2(E) (21)

We now have to find another family of six matrices linearly independent ina complement of V1 in sp2(R).

4.3.3 The Orthogonal V2 of V1 in sp2(R)

We begin by giving the expressions of the three matrices

LA(1,0) − LA(0,0), LA(1,0) − LA(1,1), LA(0,1) − LA(0,0). (22)

116 H. Boumaza

Looking at the form of LAω(0) given by (18) we already know that all thesedifferences are of the form:

⎛⎜⎜⎝

0 0 e g0 0 g fa c 0 0c b 0 0

⎞⎟⎟⎠ (23)

for (a, b , c, e, f, g) ∈ R6. Let V2 ⊂ sp2(R) be the 6-dimensional subspace of

matrices of the form (23). We have the direct sum decomposition sp2(R) =V1 ⊕ V2. By (23) it suffices to compute the [3, 1], [3, 2], [4, 2], [1, 3],[1, 4] and[2, 4] entries of the three matrices (22).

Expression of �1 = LA(1,0) − LA(0,0). We have:

�1[3, 1] = m10(1 − E) + m00 E − π

2

(a1r00

2 + a2r001

) +

+ π

2√

5

(b1r10

2 − b2r101

) + π

2

(b1r10

2 + b2r101

)

�1[3, 2] = m10 − m00 + π

2

(a2r00

1 − a1r002

) + π√5

(b1r10

2 − b2r101

)

�1[4, 2] = (m00 − m10)E − π

2

(a1r00

2 + a2r001

) −

− π

2√

5

(b1r10

2 − b2r101

) + π

2

(b1r10

2 + b2r101

)

�1[1, 3] = m10 − m00 + π

2

(a1

r002

− a2

r001

)+

+ π

2√

5

(b2

r101

− b1

r102

)− π

2

(b1

r102

+ b2

r101

)

�1[1, 4] = π

2

(a1

r002

− a2

r001

)+ π√

5

(b2

r101

− b1

r102

)

�1[2, 4] = m10 − m00 + π

2

(a1

r002

+ a2

r001

)+

+ π

2√

5

(b1

r102

− b2

r101

)− π

2

(b1

r102

+ b2

r101

)



�2[3, 1] = m10 + (m11 − m10)E − π

2

(d1r11

2 + d2r111

) +

+ π

2√

5

(b1r10

2 − b2r101

) + π

2

(b1r10

2 + b2r101

)

�2[3, 2] = m10 + m11 + π

2

(d2r11

1 − d1r112

) + π√5

(b1r10

2 − b2r101

)

�2[4, 2] = (m11 − m10)E − π

2

(d1r11

2 + d2r111

) −

− π

2√

5

(b1r10

2 − b2r101

) + π

2

(b1r10

2 + b2r101

)

�2[1, 3] = m10 − m11 + π

2

(d1

r002

+ d2

r001

)−

− π

2√

5

(b1

r102

− b2

r101

)− π

2

(b1

r102

+ b2

r101

)

�2[1, 4] = π

2

(d1

r002

− d2

r001

)− π√

5

(b1

r102

− b2

r101

)

�2[2, 4] = m10 − m11 + π

2

(d1

r002

+ d2

r001

)+

+ π

2√

5

(b1

r102

− b2

r101

)− π

2

(b1

r102

+ b2

r101

)


�3[3, 1] = m01 + (m00 − m01)E − π

2

(a1r00

2 + a2r001

) +

+ π

2√

5

(c1r10

2 − c2r101

) + π

2

(c1r10

2 + c2r101

)

�3[3, 2] = − (m00 + m01) + π

2

(a2r00

1 − a1r002

) +

+ π√5

(c2r10

1 − c1r102

)

�3[4, 2] = (m00 − m01)E − π

2

(a1r00

2 + a2r001

) +

+ π

2√

5

(c1r10

2 − c2r101

) − π

2

(c1r10

2 + c2r101

)

118 H. Boumaza

�3[1, 3] = m01 − m00 + π

2

(a1

r002

+ a2

r001

)+

+ π

2√

5

(c2

r101

− c1

r102

)− π

2

(c1

r102

+ c2

r101

)

�3[1, 4] = π

2

(a1

r002

− a2

r001

)− π√

5

(c2

r101

− c1

r102

)

�3[2, 4] = m01 − m00 + π

2

(a1

r002

+ a2

r001

)−

− π

2√

5

(c2

r101

− c1

r102

)− π

2

(c1

r102

+ c2

r101

)

Now we assume that E ∈ (2, +∞) \ (S1 ∪ S2). Then V1 ⊂ la2(E) and inparticular the following matrices are in la2(E):

Z1 =

⎛⎜⎜⎝

1 0 0 00 0 0 00 0 −1 00 0 0 0

⎞⎟⎟⎠ , Z2 =

⎛⎜⎜⎝

0 0 0 00 1 0 00 0 0 00 0 0 −1

⎞⎟⎟⎠ , Z3 =

⎛⎜⎜⎝

0 1 0 00 0 0 00 0 0 00 0 −1 0

⎞⎟⎟⎠ .

So we can consider the three matrices of la2(E):

[LA(1,0) − LA(0,0), Z1], [LA(1,0) − LA(1,1), Z2], [LA(0,1) − LA(0,0), Z3].We can check that in general the Lie bracket of an element of V1 and anelement of V2 is still in V2. So, to write this three matrices we will only have togive explicitly six of their entries.

Expression of �4 = [LA(1,0) − LA(0,0), Z1]. We have:

�4[3, 1] = 2m10 + 2(m00 − m10)E − π(a1r00

2 + a2r001

) ++ π

(b1r10

2 + b2r101

) + π√5

(b1r10

2 − b2r101

)

�4[3, 2] = m10 − m00 + π(a2r00

1 − a1r002

) + π√5

(b1r10

2 − b2r101

)

�4[4, 2] = 0

�4[1, 3] = 2(m00 − m10) − π

(a1

r002

+ a2

r001

)+

+ π

(b1

r102

+ b2

r101

)+ π√

5

(b1

r102

− b2

r101

)

�4[1, 4] = π

2

(a2

r001

− a1

r002

)+ π√

5

(b1

r102

− b2

r101

)

�4[2, 4] = 0.



�5[3, 1] = 0

�5[3, 2] = m10 + m11 + π

2

(d2r11

1 − d1r112

) + π√5

(b1r10

2 − b2r101

)

�5[4, 2] = 2(m11 − m10)E − 2m11 − π(d1r11

2 + d2r111

) ++ π

(b1r10

2 + b2r101

) − π√5

(b1r10

2 − b2r101

)

�5[1, 3] = 0

�5[1, 4] = π

2

(d2

r111

− d1

r112

)+ π√

5

(b1

r102

− b2

r101

)

�5[2, 4] = 2(m11 − m10) − π

(d1

r112

+ d2

r111

)+

+ π

(b1

r102

+ b2

r101

)− π√

5

(b1

r102

− b2

r101

)


�6[3, 1] = 0

�6[3, 2] = m01 + (m00 − m01)E − π

2

(a1r00

2 + a2r001

) +

+ π

2

(c1r10

2 + c2r101

) + π

2√

5

(c1r10

2 − c2r101

)

�6[4, 2] = −2(m00 + m01) + π(a2r00

1 − a1r002

) − 2π√5

(c1r10

2 − c2r101

)

�6[1, 3] = π

(a2

r001

− a1

r002

)− 2π√

5

(c1

r102

− c2

r101

)

�6[1, 4] = m00 − m01 − π

2

(a1

r002

+ a2

r001

)+

+ π

2

(c1

r102

+ c2

r101

)− π

2√

5

(c1

r102

− c2

r101

)

�6[2, 4] = 0.

It remains to check that these six matrices are linearly independent, atleast for all E > 2 except those in a discrete set. We denote by f2(E) thedeterminant of the 6 × 6 matrix whose columns are representing the 6 matriceswe just compute. Each column is made of the 6 entries we compute for eachmatrix. We also set:

f2(E) = f2(a1, a2, b1, b2, c1, c2, d1, d2, m00, m01, m10, m11, E) (24)

120 H. Boumaza

where f2(X1, . . . , X12, Y) is polynomial in the coefficients X1, . . . , X12 andanalytic in Y.

We define the functions f2,p,l as we defined the functions f1,p,l. We can showthat the f2,p,l do not vanish identically on Ik. More precisely we can look at theterm in the development of the determinant (24) involving only a2:

m10(m11 − m10)π2a2

2

4(E + 1)3×

× [πa2m11(10

√E + 1E3 − 8(E + 1)3/2 E3

− 9(E + 1)7/2 + (E + 1)5/2 + 28√

E + 1E2 + 14(E + 1)3/2 E −− 2(E + 1)3/2 E2 − 11(E + 1)5/2 E + 8(E + 1)7/2 E + 26

√E + 1E +

+ 8(E + 1)3/2 + 8√

E + 1) + πa2m10(10(E + 1)5/2 + 2(E + 1)7/2 ++ 8(E + 1)3/2 E3 + 14(E + 1)5/2 E − 8(E + 1)7/2 E − 29

√E + 1E2 −

− (E + 1)3/2 E + 10(E + 1)3/2 E2 − 28√

E + 1E − 3(E + 1)3/2 −− 9

√E + 1−10

√E + 1E3)+m10m11(16E4+32E3−16E2−64E − 32)

].

This term is different from 0 for a2 � 1, m10 � 1, m11 � 1 and m10 �= m11. Butwe can always assume that these two integers are distinct. Indeed, in the proofof Proposition 3, we can replace m10 by 2m10 and multiply by 2 the integers x10

1and x10

1 . And of course m10 and 2m10 cannot be both equal to m11.The term we just computed is the only one in the development of the deter-

minant (24) involving exactly those powers of E and E + 1 in the numeratorand in the denominator. So this term cannot be cancelled uniformly in Eby another term of the development of the determinant (24). As before, thefunctions f2,p,l do not vanish identically on Ik whenever (p, l ) ∈ J2 with:

J2 = {(p1, . . . , p8, l1, . . . , l4) | 0 � p1, p3, . . . , p8 � Nk, 1 � p2 � Nk,

1 � l j � M, l3 �= l4}.As we have justified that (a1, . . . , m11) ∈ J2, we have:

{E ∈ Ik | f2(E) = 0} ⊂⋃

(p,l)∈J2

{E ∈ Ik | f2,p,l(E) = 0}.

So the set of zeros of f2 is a discrete subset in (2, +∞). If we set:

S3 = {E > 2 | f2(E) = 0},S3 is discrete, and for E > 2, E /∈ S1 ∪ S2 ∪ S3, then f2(E) �= 0. For theseenergies, the matrices

LA(1,0) − LA(0,0), LA(1,0) − LA(1,1), LA(0,1) − LA(0,0),

[LA(1,0) − LA(0,0), Z1], [LA(1,0) − LA(1,1), Z2], [LA(0,1) − LA(0,0), Z3]


are linearly independent in the 6-dimensional subspace V2. Thus,

for all E > 2, E /∈ S1 ∪ S2 ∪ S3, V2 ⊂ la2(E).

Finally, we set

SB = S1 ∪ S2 ∪ S3.

We fix E > 2, E /∈ SB. We have V1 ⊂ la2(E) and V2 ⊂ la2(E). As V1 ⊕ V2 =sp2(R), we get:

for all E > 2, E /∈ SB, sp2(R) ⊂ la2(E)

We have proven:

for all E > 2, E /∈ SB, sp2(R) = la2(E).

This ends our study of the Lie algebra la2(E). We actually have proven thatfor E > 2, E /∈ SB, we can apply Theorem 2 to the four matrices

A(0,0)0 (E)m00(E), A(1,0)

0 (E)m10(E), A(0,1)0 (E)m01(E), A(1,1)

0 (E)m11(E).

Indeed, they are all in O and their logarithms generate the whole Lie algebrasp2(R). So this achieves the proof of Proposition 2.

4.4 End of the Proof of Theorem 3

It remains to explain how we deduce Theorem 3 from Proposition 2. Let E > 2,E /∈ SB be fixed. By Proposition 2, GμE is dense, therefore Zariski-dense, inSp2(R). So, applying Theorem 1, we get that GμE is p-contractive and Lp-strongly irreducible for all p. Then applying Corollary 1 we get the separabilityof the Lyapunov exponents of the operator HAB(ω) and the positivity of thetwo leading exponents. Thus we obtain Theorem 3: for all E > 2, E /∈ SB,we have

γ1(E) > γ2(E) > 0.

4.5 Proof of Corollary 2

Corollary 2 says that HAB(ω) has no absolutely continuous spectrum in(2, +∞). For this we refer to Kotani’s theory in [11]. Note that [11] considersR-ergodic systems, while our model is Z-ergodic. But we can use the suspensionmethod provided in [8] to extend the Kotani’s theory to Z-ergodic operators.So, non-vanishing of all Lyapunov exponents for all energies except those in adiscrete set allows to show the absence of absolutely continuous spectrum viaTheorem 7.2 of [11].

Acknowledgements The author would like to thank Anne Boutet de Monvel and Günter Stolzfor numerous helpful suggestions and remarks, and also for their constant encouragements duringthis work.

122 H. Boumaza

References

1. Benoist, Y.: Sous-groupes discrets des groupes de Lie. European Summer School in GroupTheory (1997)

2. Bougerol, P., Lacroix, J.: Products of random matrices with applications to Schrödinger oper-ators. Prog. Probab. Stat., vol. 8. Birkhäuser, Boston (1985)

3. Boumaza, H., Stolz, G.: Positivity of Lyapunov exponents for Anderson-type models on twocoupled strings. Electron. J. Differential Equations 2007(47), 1–18 (electronic)

4. Breuillard, E., Gelander, T.: On dense free subgroups of Lie groups. J. Algebra 261(2),448–467 (2003)

5. Carmona, R., Lacroix, J.: Spectral Theory of Random Schrödinger Operators, Probability andIts Applications. Birkhäuser, Boston (1990)

6. Damanik, D., Sims, R., Stolz, G.: Localization for one-dimensional, continuum, Bernoulli-Anderson models. Duke Math. J. 114, 59–99 (2002)

7. Gol’dsheid, I.Ya., Margulis, G.A.: Lyapunov indices of a product of random matrices. RussianMath. Surveys 44(5), 11–71 (1989)

8. Kirsch, W.: On a class of random Schrödinger operators. Adv. in Appl. Math. 6, 177–187 (1985)9. Kirsch, W., Molchanov, S., Pastur, L., Vainberg, B.: Quasi 1D localization: deterministic and

random potentials. Markov Process. Related Fields 9, 687–708 (2003)10. Klein, A., Lacroix, J., Speis, A.: Localization for the Anderson model on a strip with singular

potentials. J. Funct. Anal. 94, 135–155 (1990)11. Kotani, S., Simon, B.: Stochastic Schrödinger operators and Jacobi matrices on the strip.

Comm. Math. Phys. 119(3), 403–429 (1988)12. Schmidt, W.: Diophantine approximation. Lecture Notes in Mathematics, vol. 785. Springer,

Berlin (1980)13. Stollmann, P.: Caught by disorder – bound states in random media. Progress in Mathematical

Physics, vol. 20. Birkhäuser (2001)14. Stolz, G.: Strategies in localization proofs for one-dimensional random Schrödinger operators.

Proc. Indian Acad. Sci. 112, 229–243 (2002)


Super D-Differentiation for R∞-Supermanifolds

D. Hurley · M. Vandyck

Received: 27 April 2007 / Accepted: 13 July 2007 /Published online: 31 July 2007© Springer Science + Business Media B.V. 2007

Abstract The concept of ‘D-Differentiation’, which, in the context of smoothmanifolds, generalises Lie and covariant differentiation, is extended toR∞-supermanifolds under the name of ‘Super D-Differentiation’. This is doneby defining new (non-linear) mappings, called ‘μ-mappings’ and by relatingtheir non-linearity to the Leibniz rule that a derivation must satisfy when itacts on a tensor product. The resulting axiomatics, which is basis-independentand coordinate-free, is then expressed in a general basis (not necessarily holo-nomic). Super Lie and Super covariant differentiation are, amongst others,special cases of Super D-Differentiation. In particular, the transformationrules for the connection coefficients and the commutation coefficients ofnon-holonomic bases are obtained. These special cases are found to be inagreement with the DeWitt Super covariant and Super Lie derivatives.

Keywords Graded manifolds · R∞-supermanifolds · D-differentiation

Mathematics Subject Classifications (2000) 58C20 · 58A50 · 58C50

1 Introduction

Since the introduction of anticommuting variables in Physics, much effort hasbeen devoted to the construction of a fully comprehensive framework for‘supergeometry’, be it under the name of a ‘superspace’, a ‘supermanifold’ or a

D. Hurley (B)Mathematics Department, National University of Ireland, Cork, Irelande-mail: [email protected]

M. VandyckPhysics Department, National University of Ireland, Cork, Ireland

124 D. Hurley, M. Vandyck

‘graded manifold’. This is not the place for a detailed description of the variousmethods successively employed to reach this goal (See, for instance, [1–3] fora comprehensive list of references, and a careful analysis).

Let us only recall that the first satisfactory attempt [1], from the mathe-matical point of view, was that of Berezin, Leites and Kostant [4, 9], andwas based on algebraic geometry. Unfortunately, in the context of physicalapplications, such ‘graded manifolds’ (also called ‘supermanifolds’ in theRussian literature), led to real-valued spinor fields, which is incompatible withthe requirements of supersymmetry.

A different approach, proposed by DeWitt [5] and Rogers [10, 11], defineda ‘supermanifold’ by analogy with an ordinary (C∞) manifold, using an atlas.It turned out, however, that this created difficulties, because of the delicatenature of the concept of differentiability of the transition functions betweencoordinate patches [1]. In particular, the choice of the ‘ground algebra’ � of‘supernumbers’ plays a role in the differentiability issue.

Rothstein [1, 12] introduced then a set of axioms to single out ‘well-behaved’ supermanifolds (called here ‘R-supermanifolds’ for brevity). It wasthought that R-supermanifolds reduced to the Berezin–Leites–Kostant gradedmanifolds when � is commutative, and to certain extensions of the RogersG∞-supermanifolds when � is a finite-dimensional exterior algebra. However,the situation was re-investigated later [2], and both statements were found tobe false.

In the course of the analysis, Bartocci, Bruzzo, Hernández-Ruipérez andPestov defined a new structure [2], called an ‘R∞-supermanifold’, which re-quires � to be a ‘Banach algebra of Grassmann origin’ (BGO-algebra). Ex-amples of BGO-algebras are the finite-dimensional Grassmann algebras, theinfinite-dimensional algebra B∞ employed by Rogers, and (trivially) the real(or complex) numbers, but DeWitt’s original algebra �∞ is not of that kind.

R∞-supermanifolds based on R or C coincide with Berezin–Leites–Kostantgraded manifolds, whereas they generalise Rogers’s G∞-supermanifolds (ina well-defined sense [2]) when the ground algebra is B∞. As a consequence,R∞-supermanifolds resolve the problems encountered with Rothstein’s R-supermanifolds.

Owing to the desirability to include DeWitt’s �∞ in the set of accept-able ground-algebras, R∞-supermanifolds were finally developed [3] over‘Arens–Michael algebras of Grassmann origin’ (AMGO-algebras), which aregeneralisations of BGO-algebras that do contain �∞. In spite of the fact thatAMGO-algebras are exceedingly general (so much so that all algebras everused for supernumbers belong [3] to this category), R∞-supermanifolds stillconstitute ‘workable’ structures, in the sense that they allow for a substantialamount of theory to be developed within their framework. For instance, theyshed light [3] on DeWitt supermanifolds.

Let us now return to ordinary differential geometry. In this context, oneoften introduces Lie differentiation L and covariant differentiation ∇ as twofundamentally important derivations of the space of tensors. However, oneseldom avails of the possibility [7] of defining them as special cases of the much

Super D-differentiation for R∞-supermanifolds 125

larger family of derivations called ‘D-differentiation’. Not only is this alterna-tive method more economical, but it also provides greater conceptual insightinto L and ∇, through generalisations of the notions of torsion, curvature,metricity, geodesics, etc. In addition, certain operators of D-differentiation,which are neither L nor ∇, sometimes prove beneficial in the study of someareas of Physics, such as the dynamics of rigid bodies, the semi-classical motionof electrons in crystals, or optics in General Relativity [6–8].

It would therefore seem reasonable to try to construct an operator of ‘superD-differentiation’ that would, in an R∞-supermanifold, play the same roleas D-differentiation does in an ordinary C∞ manifold. If this programme issuccessful, it will automatically provide super Lie differentiation and super co-variant differentiation (amongst other operators), and it will illustrate furtherthat R∞-supermanifolds are rich enough to support a considerable amount ofstructure.

Such will be our aim in the present article. However, to avoid cumber-some technicalities, we shall restrict attention (loosely speaking) to superD-differentiation of vectors. The extension of the formalism to tensors willnot be presented here.

In the same spirit, we shall confine ourselves to the local construction ofD-differentiation. It is well known [1] that covariant differentiation does notalways exist globally, so that the same must be true of D-differentiation, whichcontains it as a special case.

To fix the notation, we shall begin by a brief summary of R∞-supermanifoldsin Section 2. Then, we shall define, in Section 3, non-linear mappings (called‘μ-mappings’), which will be closely related to super D-differentiation. Theoperator D itself will be defined in Section 4, and some of its properties willbe studied. Examples will be provided in Section 5, which will concentrateon Lie and covariant differentiation. Finally, future developments will becontemplated in the conclusion.

2 Summary of R∞-Supermanifolds

We do not wish to present here a detailed introduction to R∞-supermanifoldsbased on an AMGO-algebra; the reader is referred to [2, 3], in which allthe details are available. We shall rather summarise the fundamental axiomsin a manner that will emphasise exclusively the aspects required for ourconstruction.

Let � be a graded-commutative algebra (and technically an AMGO-algebra) over the real numbers. (It would be possible to have � over C.)A graded module over � is said to be ‘free of rank m|n’ if it is isomor-phic [3] to a module of the form �m|n ≡ � ⊗ Rm|n, where Rm|n is the standardgraded real vector space of dimension m|n. The ‘linear superspace’ �m,n ofdimension (m, n) is then the even part of �m|n. It can be endowed with twofundamentally different topologies: the fine (or Rogers) one and the coarse(or DeWitt) one (see [3]).


A superspace (not necessarily linear) over � is a triple (X,A, ev), where Xis a paracompact topological space, A denotes a sheaf of graded-commutativealgebras over �, and ev is a morphism of sheaves between A and the sheaf CX

of �-valued continuous functions on X.Let p be a point of X, and let U be an open neighbourhood containing p.

A graded derivation V of A|U is an endomorphism of sheaves V : A|U → A|Usatisfying the graded Leibniz rule:

V( f · g) = V( f ) · g + (−1)| f ||V| f · V(g), (2.1)

where vertical bars around an object denote the parity (even or odd) ofthis object, assumed homogeneous. Derivations form a presheaf, which, afterappropriate completion, yields the sheaf DerA of graded modules over A. Theelements of DerA(U) will informally be called ‘vectors’.

The sheaf DerA possesses a dual, denoted by Der∗A, which is also a sheaf ofgraded modules over A. The elements of Der∗A(U) will informally be called‘forms’. For all homogeneous elements f of A(U), it is possible to define theform df by

(df )[V] ≡ (−1)| f ||V| V( f ), (2.2)

and d constitutes a morphism of sheaves from A to Der∗A.An R∞-supermanifold of dimension (m, n) over � is now introduced as a

superspace (X,A, ev) over �, satisfying the following four axioms:

Axiom 1 Der∗A is a locally-free graded module of rank (m, n). For everypoint p of X there exist an open neighbourhood U of p and sections xi, 1 � i �m + n of A, such that xμ, 1 � μ � m (respectively xA, m + 1 � A � m + n),belongs to the even part A(U)0 of A(U) (respectively the odd part A(U)1

of A(U)), with the property that the set {dxi, 1 � i � m + n} ≡ {dxμ (1 � μ �m), dxA (m + 1 � A � m + n)} forms a graded basis of Der∗A(U) over A(U).

As one can see, Greek (respectively capital Latin) indices lie in therange {1, · · · , m} (respectively {m + 1, · · · , m + n}), whereas lower-case Latinones take values in {1, · · · , m + n}. By definition, the parity of a Greek indexis zero, that of a capital Latin index is one, and that of a lower-case Latinindex is either zero or one, according as it belongs to the set {1, · · · , m} or{m + 1, · · · , m + n}.

We shall also systematically adopt Einstein’s summation convention, withthe amendment that two repeated indices, of which one is surrounded byvertical bars (indicating parity), are not summed over, unless an additionalrepeated index is present, which is not surrounded by vertical bars. Forinstance, there is no sum over a in (−1)|a| αa, whereas the index a does undergosummation in (−1)|a| αa Xa.

Moreover, given a basis {dxi, 1 � i � m + n}, an arbitrary set of linearcombinations of {dxi, 1 � i � m + n} that forms a basis will be denoted by{e(i ), 1 � i � m + n}. Of course, it will be assumed that the basic elements e(i )


are numbered in such a manner that the first m vectors are even, and the last nones are odd.

In addition, it follows from the local freeness of Der∗A that DerA is alsolocally free of rank (m, n). The basis of DerA(U) to which {e(i ), 1 � i � m + n}is dual will be denoted by {(i )e, 1 � i � m + n}. In the special case where e(i ) =dxi, the corresponding (i )e will be denoted by ∂/∂xi.

Axiom 2 Given a so-called ‘coordinate chart’, namely a collection (U,

(dx1, · · · , dxm+n)) as in Axiom 1, the assignment

p �→ ((ev(x1))(p), · · · , (ev(xm+n))(p)) (2.3)

is a homeomorphism of U onto an open subset of �m,n.

Axiom 3 Let Ip denote the graded ideal of the stalk Ap of A at p defined as

Ip ≡ {φ ∈ Ap : (ev(φ ))(p) = 0}. (2.4)

Then, Ip is finitely generated for all p in X.

It is shown in [2] that this axiom is closely related to the existence of aTaylor-like expansion for the germs φ in Ap.

Axiom 4 For every open subset U of X, the algebra A(U) is complete andHausdorff.

This axiom requires, obviously, that a topology be specified in A(U). For ourpurposes, It suffices to recall that the ground-algebra � possesses a topology,which is determined by continuous submultiplicative prenorms π .

If f is an element of A(U), the action L( f ) of a differential operator L on falso belongs to A(U), because such an operator is an element of the gradedA-module D(A) generated multiplicatively by DerA over A. By applyingthe evaluation morphism ev to L( f ), one obtains a �-valued function, sothat (ev(L( f )))(p) belongs to �, for all p in X. The topology on A(U) is thendescribed in terms of π((ev(L( f )))(p)). Details may be found in [3].

At this stage, we are ready to construct super D-differentiation. Looselyspeaking, one would like DV W to be a vector when V and W are vectors. Thus,one is led to consider vector-valued mappings μ taking two vectors, V and W,as arguments. Such mappings, having suitable properties, will be obtained inthe following section. Later, in Section 4, we shall re-interpret some of them interms of an operator of super D-differentiation.

Remarks

1. When, after Axiom 1, we described the bases {e(i ), 1 � i � m + n} and{(i )e, 1 � i � m + n} as being dual, the graded dual was understood, inthe sense

e(a)[(b)e] = (−1)|a|bδa, (2.5)


where bδa stands for the usual Kronecker symbol (with its indices stag-gered, in accordance with the conventions of [5]). In the graded situationthat occupies us, it is convenient to introduce ‘double’ duality, whichconsists in allowing a vector X to act on a form α as

X[α] ≡ (−1)|X||α|α[X]. (2.6)

From this new point of view, the duality-relation (2.5) of the bases may bereformulated as

(b)e[e(a)] = b δa, (2.7)

which is simpler than (2.5).Furthermore, with the same notation, the defining-relation (2.2) of theexterior differential df becomes

V[df ] = (−1)| f ||V| (df )[V] = V( f ). (2.8)

On the left-hand side, the square brackets emphasise that V is acting ona form (by double duality), whereas, on the right-hand side, V acts as aderivation on the element f of the algebra A.

2. There is no difficulty in establishing that, by virtue of the definition (2.2),the expression of the exterior differential df reads, in a pair of dual bases,

df = (df )a e(a) = (−1)|a|(| f |+1)(a)e( f ) e(a), (2.9)

which will be useful later.

3 Construction of μ-Mappings

As in the previous section, let p be a point of X, and let U denote an openset containing p. We now wish to consider mappings μ defined on DerA(U) ×DerA(U), with values in DerA(U), satisfying the axioms

μ(V + W, Z ) = μ(V, Z ) + μ(W, Z ) (3.1)

μ(Z , V + W) = μ(Z , V) + μ(Z , W) (3.2)

μ(V, W · f ) = μ(V, W) · f + (−1)| f |(|V|+|W|) A(df, V, W) (3.3)

μ( f · V, W) = f · μ(V, W) − (−1)|V||W| B(df, W, V), (3.4)

where A and B are even vector-valued (graded) multilinear mappings. In thecontext of vector-valued mappings, linearity means (by definition) that the�-valued mappings A′ and B′ constructed from A and B as

A′ : (α, V, W, β) �→ A′(α, V, W, β) ≡ (A(α, V, W))[β] (3.5)

B′ : (α, V, W, β) �→ B′(α, V, W, β) ≡ (B(α, V, W))[β] (3.6)

are (graded)-linear in the entries α, V, W and β, for all V and W in DerA(U),and all α and β in Der∗A(U). Note that, in (3.5) and (3.6), the vectorsA(α, V, W) and B(α, V, W) act on β in accordance with (2.6).


The presence of A and B in (3.3) and (3.4) prevents μ from being linear,in general. However, the ‘linearity-breaking’ objects A and B are, themselves,linear mappings. In this sense, one may consider μ as being next in simplicityafter linear mappings.

One readily establishes, from (3.1–3.4), the following propositions, whichwe state without proof:

Proposition 1 The parity of μ is given by

|μ(V, W)| = |V| + |W|, (3.7)

for all homogeneous V and W.

Proposition 2 For all vectors V and W, and for all elements k of �, one has

μ(V, W · k · 1) = μ(V, W) · k · 1, μ(k · 1 · V, W) = k · 1 · μ(V, W), (3.8)

where 1 denotes the unit of the algebra A(U).

The comparison of (3.8) with (3.3) and (3.4) expresses in what sense μ is‘�-linear’, but not ‘A(U)-linear’.

Proposition 3 Let the commutator [V, W] be defined by

[V, W] ≡ V ◦ W − (−1)|V||W|W ◦ V. (3.9)

Then, [V, W] is a mapping of the type (3.1–3.4), with A and B given by

A(α, V, W) = B(α, V, W) = α[V] · W ≡ LA(α, V, W). (3.10)

In this proposition, we have introduced the definition of the mapping LA,which will be convenient later.

At this stage, we have focussed attention on the family (3.1–3.4) of map-pings, which we shall call ‘arbitrary μ-mappings’. This family is, however,too large for our purposes (although it can be studied in full generality).Therefore, in the next section, we shall restrict ourselves to the subfamily ofthe ‘generalisable μ-mappings’.

4 Generalisable μ-Mappings

Let us now specialise μ by requiring that, in (3.3), the mapping A be givenby LA, which was defined in (3.10). It is then obvious that (3.3) is replaced by

μ(V, W · f ) = μ(V, W) · f + (−1)|V||W|W · V( f ). (4.1)

This relationship, which is investigated in a broader context in the appendix,enables one to consider μ(V, W) as the ‘rate of change’ DV W of W alongV, also called the ‘D-derivative of W along V’. Indeed, in terms of thesymbol D, (4.1) reads

DV (W · f ) = (DV W) · f + (−1)|V||W|W · V( f ). (4.2)


The right-hand side of (4.2) corresponds to that produced by an operator DV

(for V fixed) satisfying the (graded) Leibniz rule for the product W · f of Wby the element f of A(U).

With the same notation, the axiom (3.4) becomes

D f ·V W = f · DV W − (−1)|V||W| B(df, W, V), (4.3)

which specifies the behaviour of the operator D with respect to its ‘differenti-ating slot’. This behaviour depends on the properties of the mapping B, so thateach operator D is characterised by its own B. (Examples will be provided inthe next section.)

The axioms that we have adopted imply that the operator D is entirelydetermined. Its local expression in a basis is given by the following theorem.

Theorem 1 Let {(i )e, 1 � i � m + n} be a basis of DerA(U). For all V and W,the D-derivative DV W reads

DV W = {V(Wi ) + (−1)|V||W|W j

j�i(V)

}(i )e (4.4)

j�i(V) ≡ (−1)(|V|+|b |)| j| Vb

jbλi−(−1)(|V|+|b |)(|a|+| j|)+|a|(a)e(Vb )

ajb Bi, (4.5)

in which the elements jb λi and ajb Bi of A(U) uniquely satisfy

(−1)| j||b |jbλi

(i )e = D(b)e ( j )e (4.6)

−(df )a (−1)| j||b | ajb Bi

(i )e = D{ f ·(b)e} ( j )e − f · D(b)e ( j )e. (4.7)

In other words, the operator D is determined by its action D(i )e ( j )e on a basis,

which is enciphered in the coefficients ijλk, and by the relationship between

D{ f ·(i )e} ( j )e and f · D(i )e ( j )e, which is represented by the components a

ij Bb .

Under a change of basis, the quantities aij Bb transform in an obvious

fashion, because they are the components of the linear mapping B of (4.3).On the other hand, the coefficients ijλ

k transform as follows:

Theorem 2 Let {(i )e′, 1 � i � m + n} be a basis of DerA(U), related to thebasis {( j )e, 1 � j � m + n} by

(i )e = i M j( j )e′, (i )e′ = i N j

( j )e, MN = NM = 1. (4.8)

Then, the coefficients λ′ are given in terms of λ by

ijλ′k = (−1)|a|(|b |+| j|)

i NajNb

abλcc Mk + (−1)|i|| j|

jNa(a)e(i Nb ) b Mk −

−(−1)(|a|+|c|)(| j|+|d|)+|a|i Nc

(a)e( jNd) acd Bb

b Mk. (4.9)

It goes without saying that Theorem 1 and Theorem 2 reduce to theanalogous results for ordinary D-differentiation in C∞ manifolfds when all theobjects involved are purely even. The reader is referred to [7] or [8] for details.

After these considerations valid for an arbitrary operator D, we are nowgoing to investigate two special cases: super Lie differentiation L and super


covariant differentiation ∇. This will enable us to establish a closer contactwith the supermanifold literature.

5 Examples

Let us begin with the case of the commutator [V, W] ≡ LV W. By virtueof (4.6), we find successively

(−1)| j||b |jbλi

(i )e = [(b)e, ( j )e] (5.1)

≡ b jCi(i )e (5.2)

= −(−1)| j||b |jb Ci

(i )e, (5.3)

where we have introduced the ‘commutation coefficients’ ijCk of the basis,and we have exploited their property [5] of graded antisymmetry. As aconsequence of (5.3), the value of jbλi pertaining to the commutator reads

jb λi = − jb Ci, (5.4)

from which the transformation rule for the commutation coefficients jb Ci isalso obtainable, through (4.9).

Moreover, the expression (2.9) of the exterior differential in components,combined with the definition (4.7) of the quantities a

jb Bi and the properties ofthe commutator, yields

− (−1)|a|(| f |+1)(a)e( f ) a

jb Bi(i )e = −(df )a

ajb Bi

(i )e (5.5)

= (−1)| j||b |{[ f · (b)e, ( j )e]− f · [(b)e, ( j )e]}

(5.6)

= −(df )[( j )e] · (b)e (5.7)

= −(−1)| f || j|( j )e( f ) (b)e (5.8)

= −(−1)| f ||a|(a)e( f ) aδ j b δi

(i )e, (5.9)

which implies

ajb Bi = (−1)|a| aδ j bδi. (5.10)

The result (5.10) is the translation in coordinate-language of the coordinate-free statement (3.10).

When the values (5.4) and (5.10) of jbλi and ajb Bi are inserted in the

decomposition (4.4), (4.5), the final expression for the commutator becomes

[V, W] =LV W = {

V(Wi ) − (−1)|V||W|W(Vi ) + (−1)(|W|+| j|)|k| Vk W jkjCi}

(i )e. (5.11)

In (5.11), the term involving b jCi arises from the failure of two basic vectors tocommute, in general. The special bases for which all the b jCi vanish are called‘holonomic bases’, ‘natural bases’ or ‘coordinate bases’ [5].


Apart from trivial differences in the notation, (5.11) coincides with theDeWitt Lie derivative of a vector W. (Note that, in [5], the formula for L isonly displayed in a holonomic basis.)

On the other hand, the case of covariant differentiation is fundamentallydifferent from that of Lie differentiation, in the sense that ∇ is linear in itsdifferentiating slot:

∇ f ·V = f · ∇V, (5.12)

which compels acd Bb to vanish, by virtue of (4.7). Furthermore, with the

definition

∇(i )e ( j )e ≡ ji

k(k)e, (5.13)

where the symbols are the so-called ‘connection’ coefficients, the coordinate-expressions (4.4) and (4.5) become

∇V W = {V(Wi ) + (−1)|k|(|W|+| j|) Vk W j

jki }

(i )e. (5.14)

In addition, the transformation rule (4.9) simplifies as

ij′k = (−1)| j|(|a|+|i|)

i NajNb

abcc Mk + jNa

(a)e(i Nb ) b Mk. (5.15)

The results (5.14) and (5.15) reproduce those of the DeWitt super covariantderivative [5], apart from minor differences in the notation.

6 Conclusion

In this article, we adopted the framework of R∞-supermanifolds, andwe constructed locally an operator of differentiation, called ‘(super) D-differentiation’. This operator, as presented here, acts on vectors. Its extensionto tensors will be investigated elsewhere.

We showed how (super) Lie and (super) covariant differentiation are con-tained in (super) D-differentiation as special cases. Lie and covariant differen-tiation had, separately, been considered before in the literature, for instance byDeWitt [5], within his own theory of supermanifolds, and by Bartocci, Bruzzoand Hernández-Ruipérez, for the so-called ‘G-supermanifolds’. Both kinds ofsupermanifold are included within the category of R∞-supermanifolds, so thatD-differentiation provides a unified approach of a considerable generality.

In physical applications, the DeWitt formalism is frequently employed. Thepresent construction may be considered as an illustration of how some of theoutcomes of this formalism (such as the transformation rule for connectioncoefficients) may be obtained from the point of view of R∞-supermanifolds.Therefore, not only do R∞-supermanifolds provide the benefit of unifyingvarious supermanifold theories, but they are also a natural context for someof the DeWitt formalism, which is familiar to, and employed by, Physicists.


Appendix

In order to interpret the two kinds of μ-mappings from a deeper point of view,let us assume that we have at our disposal an operator of D-differentiation thatacts on tensors. For a vector V fixed, DV must satisfy the Leibniz rule for thetensor product ⊗, namely

DV(T1 ⊗ T2) = (DV T1) ⊗ T2 + (−1)|V||T1| T1 ⊗ DV T2. (7.1)

Furthermore, DV must be ‘compatible’ with V, in the sense

DV f = V( f ), (7.2)

for all elements f of A.Let us now consider f as a tensor of rank (0, 0). Then, with T2 = f , the

Leibniz rule (7.1) reduces to

DV(T · f ) = DV(T ⊗ f ) = (DV T) ⊗ f + (−1)|V||T| T ⊗ DV f (7.3)

= (DV T) · f + (−1)|V||T| T · DV f . (7.4)

Even if T is restricted to being a vector W, the right-hand side of (7.4) has nomeaning in terms of μ-mappings, because such mappings cannot act on f .

On the other hand, if (7.2) is inserted in (7.4), one finds

DV(T · f ) = (DV T) · f + (−1)|V||T| T · V( f ). (7.5)

When T is a vector W, the requirement (7.5) is expressible in the language ofμ-mappings, by putting

μ(V, W · f ) = μ(V, W) · f + (−1)|V||W| W · V( f ) (7.6)

μ(V, W) ≡ DV W. (7.7)

The observation that (7.6) coincides with (4.1) implies that, for a μ-mappingto be the restriction to vectors of an operator of D-differentiation defined ontensors, it is necessary that this mapping be of the type called ‘generalisable’ inSection 4.

If, in addition, the general operator D is assumed to preserve tensor rankand to commute with tensor contractions, then D can uniquely be constructedfrom a generalisable μ-mapping, so that the above condition is also sufficient.In practice, this statement means that the components of the D-derivative ofany tensor are (locally) determined by the quantities ijλ

k and ijk Bl appearing

in (4.5). However, we shall not present these developments here.

References

1. Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D.: The Geometry of Supermanifolds. KluwerAcademic Publishers, Dordrecht (1991)

2. Bartocci, C., Bruzzo, U., Hernández-Ruipérez, D., Pestov, V.G.: Foundations of supermani-fold theory: the axiomatic approach. Differential Geom. Appl. 3, 135–155 (1993)


3. Bruzzo, U., Pestov, V.G.: On the structure of DeWitt supermanifolds. J. Geom. Phys. 30,147–168 (1999)

4. Berezin, F.A., Leites, D.A.: Supermanifolds. Soviet Math. Dokl. 16, 1218–1222 (1975)5. DeWitt, B.S.: Supermanifolds. Cambridge University Press, London (1984)6. Hurley, D., Vandyck, M.: An application of D-differentiation to solid-state Physics. J. Phys. A

33, 6981–6991 (2000)7. Hurley, D., Vandyck, M.: A unified framework for Lie and covariant differentiation. J. Math.

Phys. 42, 1869–1886 (2001)8. Hurley, D., Vandyck, M.: Topics in Differential Geometry; A New Approach using D-

differentiation. Springer-Praxis, Chichester (2002)9. Kostant B.: Graded manifolds, graded Lie theory, and prequantization. In: Differential Geo-

metric Methods in Mathematical Physics. Lecture Notes in Mathematics, vol. 570, pp. 177–306.Springer, Berlin (1977)

10. Rogers, A.: A global theory of supermanifolds. J. Math. Phys. 21, 1352–1365 (1980)11. Rogers, A.: Graded manifolds, supermanifolds and infinite-dimensional Grassmann algebras.

Comm. Math. Phys. 105, 375–384 (1986)12. Rothstein, M.J.: The axioms of supermanifolds and a new structure arising from them. Trans.

Amer. Math. Soc. 297, 159–180 (1986)


L p-Spectral Theory of Locally Symmetric Spaceswith Q-Rank One

Andreas Weber

Received: 15 March 2007 / Accepted: 26 July 2007 /Published online: 12 September 2007© Springer Science + Business Media B.V. 2007

Abstract We study the Lp-spectrum of the Laplace–Beltrami operator oncertain complete locally symmetric spaces M = �\X with finite volume andarithmetic fundamental group � whose universal covering X is a symmetricspace of non-compact type. We also show, how the obtained results for locallysymmetric spaces can be generalized to manifolds with cusps of rank one.

Keywords Arithmetic lattices · Heat semigroup on Lp-spaces ·Laplace–Beltrami operator · Locally symmetric space · Lp-spectrum ·Manifolds with cusps of rank one

Mathematics Subject Classifications (2000) Primary 58J50 · 11F72 ·Secondary 53C35 · 35P05

1 Introduction

Our main concern in this paper is to study the Lp-spectrum σ(�M,p), p ∈(1, ∞), of the Laplace–Beltrami operator on a complete non-compact locallysymmetric space M = �\X with finite volume, such that

(1) X is a symmetric space of non-compact type,(2) � ⊂ Isom0(X) is a torsion-free arithmetic subgroup with Q-rank(�) = 1.

We also treat the case of manifolds with cusps of rank one which are moregeneral than the locally symmetric spaces defined above.

A. Weber (B)Institut für Algebra und Geometrie, Universität Karlsruhe (TH),Englerstr. 2, 76128 Karlsruhe , Germanye-mail: [email protected]

136 A. Weber

Whether the Lp-spectrum of a complete Riemannian manifold M dependson p or not is related to the geometry of M. More precisely, Sturm provedin [23] that the Lp-spectrum is p-independent if the Ricci curvature of M isbounded from below and the volume of balls in M grows uniformly subexpo-nentially (with respect to their radius). This is for example true if M is compactor if M is the n-dimensional euclidean space Rn.

On the other hand, if the Ricci curvature of M is bounded from belowand the volume density of M grows exponentially in every direction (withrespect to geodesic normal coordinates around some point p ∈ M with emptycut locus) then the Lp-spectrum actually depends on p. More precisely,Sturm showed that in this case inf Re σ(�M,1) = 0 whereas inf σ(�M,2) > 0. Anexample where this happens is M = Hn, the n-dimensional hyperbolic space.

In the latter case and for more general hyperbolic manifolds of the formM = �\Hn where � denotes a geometrically finite discrete subgroup of theisometry group of Hn such that either M has finite volume or M is cusp free,the Lp-spectrum was completely determined by Davies, Simon, and Taylor in[8]. They proved that σ(�M,p) coincides with the union of a parabolic regionPp and a (possibly empty) finite subset {λ0, . . . , λm} of R�0 that consists ofeigenvalues for �M,p. Note, that we have P2 = [ (n−1)2

4 , ∞).Taylor generalized this result in [24] to symmetric spaces X of non-compact

type, i.e. he proved that the Lp-spectrum of X coincides with a certainparabolic region Pp (now defined in terms of X) that degenerates in thecase p = 2 to the interval [||ρ||2, ∞), where a definition of ρ can be found inSection 2.2. He also showed that the methods from [8] can be used in orderto prove the following:

Proposition 1.1 (cf. Proposition 3.3 in [24]) Let X denote a symmetric spaceof non-compact type and M = �\X a locally symmetric space with finitevolume. If

σ(�M,2) ⊂ {λ0, . . . , λm} ∪ [||ρ||2, ∞), (1.1)

where λ j ∈ [0, ||ρ||2) are eigenvalues of finite multiplicity, then we have forp ∈ [1, ∞):

σ(�M,p) ⊂ {λ0, . . . , λm} ∪ Pp.

However, for non-compact locally symmetric spaces �\X with finite volumethe assumption (1.1) is in general not fulfilled: If X is a symmetric space ofnon-compact type and � ⊂ Isom0(X) an arithmetic subgroup such that thequotient M = �\X is a complete, non-compact locally symmetric space, thecontinuous L2-spectrum of M contains the interval [||ρ||2, ∞) but is in generalstrictly larger.

Another upper bound for the Lp-spectrum σ(�M,p) is the sector

{z ∈ C \ {0} : | arg(z)| � arctan

|p − 2|2√

p − 1

}∪ {0}

Lp-spectral theory of locally symmetric spaces... 137

Fig. 1 The parabolic regionPp if p = 3

which is indicated in Fig. 1. This actually holds in a much more general setting,i.e. for generators of so-called sub-Markovian semigroups (cf. Section 2.1).

We are going to prove in Section 3 that a certain parabolic region (in generaldifferent from the one in Proposition 1.1) is contained in the Lp-spectrumσ(�M,p) of a locally symmetric space M = �\X with the properties mentionedin the beginning. In the case where X is a rank one symmetric space it happensthat our parabolic region and the one in Taylor’s result coincide. Therefore,we are able to determine explicitly the Lp-spectrum in the latter case.

In Section 4 we briefly explain, how the results from Section 3 can begeneralized to manifolds with cusps of rank one. For these manifolds, everycusp defines a parabolic region that is contained in the Lp-spectrum. Incontrast to the class of locally symmetric spaces however, these parabolicregions need not coincide. This is due to the fact that the volume growth indifferent cusps may be different in manifolds with cusps of rank one whereasthis can not happen for locally symmetric spaces as above. Consequently, thenumber of (different) parabolic regions in the Lp-spectrum σ(�M,p), p �= 2, ofa manifold with cusps of rank one seems to be a lower bound for the number ofcusps of M. As in the case p = 2 the Laplace–Beltrami operator is self-adjoint,we obtain in this case only the trivial lower bound one. Therefore, it seems thatmore geometric information is encoded in the Lp-spectrum for some p �= 2than in the L2-spectrum. Note however, that nothing new can be expected forcompact manifolds as the Lp-spectrum does not depend on p in this case.

For results concerning the Lp-spectrum of locally symmetric spaces withinfinite volume see [26, 27].

2 Preliminaries

2.1 Heat Semigroup on Lp-spaces

In this section M denotes an arbitrary complete Riemannian manifold. TheLaplace–Beltrami operator �M := −div(grad) with domain C∞

c (M) (the set

138 A. Weber

of differentiable functions with compact support) is essentially self-adjointand hence, its closure (also denoted by �M) is a self-adjoint operator on theHilbert space L2(M). Since �M is positive, −�M generates a bounded analyticsemigroup e−t�M on L2(M) which can be defined by the spectral theorem forunbounded self-adjoint operators. The semigroup e−t�M is a submarkoviansemigroup (i.e., e−t�M is positive and a contraction on L∞(M) for any t � 0)and we therefore have the following:

(1) The semigroup e−t�M leaves the set L1(M) ∩ L∞(M) ⊂ L2(M) invariantand hence, e−t�M |L1∩L∞ may be extended to a positive contraction semi-group Tp(t) on Lp(M) for any p ∈ [1, ∞]. These semigroups are stronglycontinuous if p ∈ [1, ∞) and consistent in the sense that Tp(t)|Lp∩Lq =Tq(t)|Lp∩Lq .

(2) Furthermore, if p ∈ (1, ∞), the semigroup Tp(t) is a bounded analyticsemigroup with angle of analyticity θp � π

2 − arctan |p−2|2√

p−1.

For a proof of (1) we refer to [7, Theorem 1.4.1]. For (2) see [19]. In general, thesemigroup T1(t) needs not be analytic. However, if M has bounded geometryT1(t) is analytic in some sector (cf. [6, 25]).

In the following, we denote by −�M,p the generator of Tp(t) (note, that�M = �M,2) and by σ(�M,p) the spectrum of �M,p. Furthermore, we will writee−t�M,p for the semigroup Tp(t). Because of (2) from above, the Lp-spectrumσ(�M,p) has to be contained in the sector

{z ∈ C \ {0} : | arg(z)| � π

2− θp

}∪ {0} ⊂

⊂{

z ∈ C \ {0} : | arg(z)| � arctan|p − 2|

2√

p − 1

}∪ {0}.

If we identify as usual the dual space of Lp(M), 1 � p < ∞, with Lp′(M),

1p + 1

p′ = 1, the dual operator of �M,p equals �M,p′ and therefore we alwayshave σ(�M,p) = σ(�M,p′).

2.2 Symmetric Spaces

Let X denote always a symmetric space of non-compact type. Then G :=Isom0(X) is a non-compact, semi-simple Lie group with trivial center that actstransitively on X and X = G/K, where K ⊂ G is a maximal compact subgroupof G. We denote the respective Lie algebras by g and k. Given a correspondingCartan involution θ : g → g we obtain the Cartan decomposition g = k ⊕ p

of g into the eigenspaces of θ . The subspace p of g can be identified withthe tangent space TeK X. We assume, that the Riemannian metric 〈·, ·〉 of Xin p ∼= TeK X coincides with the restriction of the Killing form B(Y, Z ) :=tr(adY ◦ adZ ), Y, Z ∈ g, to p.


For any maximal abelian subspace a ⊂ p we refer to = (g, a) as the setof restricted roots for the pair (g, a), i.e. contains all α ∈ a∗ \ {0} such that

hα := {Y ∈ g : ad(H)(Y) = α(H)Y for all H ∈ a} �= {0}.These subspaces hα �= {0} are called root spaces.

Once a positive Weyl chamber a+ in a is chosen, we denote by + the subsetof positive roots and by ρ := 1

2

∑α∈+(dim hα)α half the sum of the positive

roots (counted according to their multiplicity).

2.2.1 Arithmetic Groups and Q-rank

Since G = Isom0(X) is a non-compact, semi-simple Lie group with trivialcenter, we can find a connected, semi-simple algebraic group G ⊂ GL(n, C)

defined over Q such that the groups G and G(R)0 are isomorphic as Lie groups(cf. [9, Proposition 1.14.6]).

Let us denote by TK ⊂ G (K = R or K = Q) a maximal K-split algebraictorus in G. Remember that we call a closed subgroup T of G a torus if T isdiagonalizable over C, or equivalently if T is abelian and every element of T issemi-simple. Such a torus T is called R-split if T is diagonalizable over R andQ-split if T is defined over Q and diagonalizable over Q.

All maximal K-split tori in G are conjugate under G(K), and we call theircommon dimension K-rank of G. It turns out that the R-rank of G coincideswith the rank of the symmetric space X = G/K, i.e. the dimension of amaximal flat subspace in X.

Since we are only interested in non-uniform lattices � ⊂ G, we may definearithmetic lattices in the following way (cf. [29, Corollary 6.1.10] and its proof):

Definition 2.1 A non-uniform lattice � ⊂ G in a connected semi-simpleLie group G with trivial center and no compact factors is called arithmetic ifthere are

(1) a semi-simple algebraic group G ⊂ GL(n, C) defined over Q and(2) an isomorphism

ϕ : G(R)0 → G

such that ϕ(G(Z) ∩ G(R)0) and � are commensurable, i.e. ϕ(G(Z) ∩ G(R)0) ∩� has finite index in both ϕ(G(Z) ∩ G(R)0) and �.

For the general definition of arithmetic lattices see [29, Definition 6.1.1].A well-known and fundamental result due to Margulis ensures that this is

usually the only way to obtain a lattice. More precisely, every irreducible lattice� ⊂ G in a connected, semi-simple Lie group G with trivial center, no compactfactors and R-rank(G) � 2 is arithmetic ([20, 29]).

Further results due to Corlette (cf. [5]) and Gromov and Schoen (cf. [12])extended this result to all connected semi-simple Lie groups with trivial centerexcept SO(1, n) and SU(1, n). In SO(1, n) (for all n ∈ N) and in SU(1, n) (forn = 2, 3) actually non-arithmetic lattices are known to exist (see e.g. [11, 20]).

140 A. Weber

Definition 2.2 (Q-rank of an arithmetic lattice) Suppose � ⊂ G is an arith-metic lattice in a connected semi-simple Lie group G with trivial center andno compact factors. Then Q-rank(�) is by definition the Q-rank of G, where Gis an algebraic group as in Definition 2.1.

The theory of algebraic groups shows that the definition of the Q-rank of anarithmetic lattice does not depend on the choice of the algebraic group G inDefinition 2.1. A proof of this fact can be found in [28, Corollary 9.12].

We already mentioned a geometric interpretation of the R-rank: TheR-rank of G as above coincides with the rank of the corresponding symmetricspace X = G/K. For the Q-rank of an arithmetic lattice � that acts freely onX there is also a geometric interpretation in terms of the large scale geometryof the corresponding locally symmetric space �\X:

Let us fix an arbitrary point p ∈ M = �\X. The tangent cone at infinity ofM is the (pointed) Gromov–Hausdorff limit of the sequence

(M, p, 1

n dM)

ofpointed metric spaces. Heuristically speaking, this means that we are lookingat the locally symmetric space M from farther and farther away. The precisedefinition can be found in [22, Chapter 10]. We have the following geometricinterpretation of Q-rank(�). For a proof see [13, 18] or [28].

Theorem 2.3 Let X = G/K denote a symmetric space of non-compact type and� ⊂ G an arithmetic lattice that acts freely on X. Then, the tangent cone atinfinity of �\X is isometric to a Euclidean cone over a finite simplicial complexwhose dimension is Q-rank(�).

An immediate consequence of this theorem is that Q-rank(�) = 0 if andonly if the locally symmetric space �\X is compact.

2.2.2 Siegel Sets and Reduction Theory

Let us denote in this subsection by G again a connected, semi-simple algebraicgroup defined over Q with trivial center and by X = G/K the correspondingsymmetric space of non-compact type with G = G0(R). Our main referencesin this subsection are [1, 4, 16].

Langlands decomposition of rational parabolic subgroups.

Definition 2.4 A closed subgroup P ⊂ G defined over Q is called rationalparabolic subgroup if P contains a maximal, connected solvable subgroupof G. (These subgroups are called Borel subgroups of G.)

For any rational parabolic subgroup P of G we denote by NP the unipotentradical of P, i.e. the largest unipotent normal subgroup of P and by NP :=NP(R) the real points of NP. The Levi quotient LP := P/NP is reductive andboth NP and LP are defined over Q. If we denote by SP the maximal Q-splittorus in the center of LP and by AP := SP(R)0 the connected component of


SP(R) containing the identity, we obtain the decomposition of LP(R) into AP

and the real points MP of a reductive algebraic group MP defined over Q:

LP(R) = AP MP ∼= AP × MP.

After fixing a certain basepoint x0 ∈ X, we can lift the groups LP, SP and MP

into P such that their images LP,x0 , SP,x0 and MP,x0 are algebraic groups definedover Q (this is in general not true for every choice of a basepoint x0) and giverise to the rational Langlands decomposition of P := P(R):

P ∼= NP × AP,x0 × MP,x0 .

More precisely, this means that the map

P → NP × AP,x0 × MP,x0 , g �→ (n(g), a(g), m(g))

is a real analytic diffeomorphism.Denoting by XP,x0 the boundary symmetric space

XP,x0 := MP,x0/K ∩ MP,x0

we obtain, since the subgroup P acts transitively on the symmetric spaceX = G/K (we actually have G = PK), the following rational horocyclicdecomposition of X:

X ∼= NP × AP,x0 × XP,x0 .

More precisely, if we denote by τ : MP,x0 → XP,x0 the canonical projection, wehave an analytic diffeomorphism

μ : NP × AP,x0 × XP,x0 → X, (n, a, τ (m)) �→ nam · x0. (2.1)

Note, that the boundary symmetric space XP,x0 is a Riemannian product ofa symmetric space of non-compact type by a Euclidean space.

For minimal rational parabolic subgroups, i.e. Borel subgroups P, we have

dim AP,x0 = Q-rank(G).

In the following we omit the reference to the chosen basepoint x0 in thesubscripts.

Q-Roots. Let us fix some minimal rational parabolic subgroup P of G. Wedenote in the following by g, aP, and nP the Lie algebras of the (real) Lie groupsG, AP, and NP defined above. Associated with the pair (g, aP) there is –similarto Section 2.2 – a system (g, aP) of so-called Q-roots. If we define for α ∈ (g, aP) the root spaces

gα := {Z ∈ g : ad(H)(Y) = α(H)(Y) for all H ∈ aP},we have the root space decomposition

g = g0 ⊕⊕

α∈ (g,aP)

gα,

142 A. Weber

where g0 is the Lie algebra of Z (SP(R)), the center of SP(R). Furthermore,the minimal rational parabolic subgroup P defines an ordering of (g, aP)

such that

nP =⊕

α∈ +(g,aP)

gα.

The root spaces gα, gβ to distinct positive roots α, β ∈ +(g, aP) are orthogonalwith respect to the Killing form:

B(gα, gβ) = {0}.In analogy to Section 2.2 we define

ρP :=∑

α∈ +(g,aP)

(dim gα)α.

Furthermore, we denote by ++(g, aP) the set of simple positive roots. Recall,that we call a positive root α ∈ +(g, aP) simple if 1

2α is not a root.

Remark 2.5 The elements of (g, aP) are differentials of characters of themaximal Q-split torus SP. For convenience, we identify the Q-roots with char-acters. If restricted to AP we denote therefore the values of these charactersby α(a), (a ∈ AP, α ∈ (g, aP)) which is defined by

α(a) := exp α(log a).

Siegel sets. Since we will consider in the succeeding section only (non-uniform) arithmetic lattices � with Q-rank(�) = 1, we restrict ourselves fromnow on to the case

Q-rank(G) = 1.

For these groups we summarize several facts in the next lemma.

Lemma 2.6 Assume Q-rank(G) = 1. Then the Following holds:

(1) For any proper rational parabolic subgroup P of G, we have dim AP = 1.(2) All proper rational parabolic subgroups are minimal.(3) The set ++(g, aP) of simple positive Q-roots contains only a single

element:

++(g, aP) = {α}.

For any rational parabolic subgroup P of G and any t > 1, we define

AP,t := {a ∈ AP : α(a) > t},where α denotes the unique root in ++(g, aP).

If we choose a0 ∈ AP with the property α(a0) = t, the set AP,t is just a shiftof the positive Weyl chamber AP,1 by a0:

AP,t = AP,1a0.


Before we define Siegel sets, we recall the rational horocyclic decompositionof the symmetric space X = G/K:

X ∼= NP × AP × XP.

Definition 2.7 Let P denote a rational parabolic subgroup of the algebraicgroup G with Q-rank(G) = 1. For any bounded set ω ⊂ NP × XP and anyt > 1, the set

SP,ω,t := ω × AP,t ⊂ X

is called Siegel set.

Precise reduction theory. We fix an arithmetic lattice � ⊂ G = G(R) in thealgebraic group G with Q-rank(G) = 1. Recall, that by a well known resultdue to A. Borel and Harish–Chandra there are only finitely many �-conjugacyclasses of minimal parabolic subgroups (see e.g. [1]). Using the Siegel setsdefined above, we can state the precise reduction theory in the Q-rank one caseas follows:

Theorem 2.8 Let G denote a semi-simple algebraic group defined over Q withQ-rank(G) = 1 and � an arithmetic lattice in G. We further denote by P1, . . . , Pk

representatives of the �-conjugacy classes of all rational proper (i.e. minimal)parabolic subgroups of G. Then there exist a bounded set �0 ⊂ X and Siegelsets ω j × AP j,t j ( j = 1, . . . , k) such that the following holds:

(1) Under the canonical projection π : X → �\X each Siegel set ω j × AP j,t j ismapped injectively into �\X, i = 1, . . . , k.

(2) The image of ω j in (� ∩ Pj)\NP j × XP j is compact ( j = 1, . . . , k).(3) The subset

�0 ∪k∐

j=1

ω j × AP j,t j

is an open fundamental domain for �. In particular, �\X equals the closureof π(�0) ∪∐k

j=1 π(ω j × AP j,t j).

Geometrically this means that the closure of each set π(ω j × AP j,t j) cor-responds to one cusp of the locally symmetric space �\X and the numberst j are chosen large enough such that these sets do not overlap. Then theinterior of the bounded set π(�0) is just the complement of the closure of∐k

j=1 π(ω j × AP j,t j) cf. Fig. 2.Since in the case Q-rank(G) = 1 all rational proper parabolic subgroups

are minimal, these subgroups are conjugate under G(Q) (cf. [1, Theorem11.4]). Therefore, the root systems (g, aP j) with respect to the rational proper

144 A. Weber

Fig. 2 Disjoint decomposi-tion of a Q-rank-1 space

parabolic subgroups P j, j = 1 . . . k, are canonically isomorphic (cf. [1, 11.9])and moreover, we can conclude ||ρP1 || = . . . = ||ρPk ||.

2.2.3 Rational Horocyclic Coordinates

For all α ∈ +(g, aP) we define on nP =⊕α∈ +(g,aP) gα a left invariant bilinearform hα by

hα :={ 〈·, ·〉, on gα

0, else,

where 〈Y, Z 〉 := −B(Y, θ Z ) denotes the usual Ad(K)-invariant bilinear formon g induced from the Killing form B. We then have (cf. [2, Proposition 1.6] or[3, Proposition 4.3]):

Proposition 2.9

(a) For any x = (n, τ (m), a) ∈ X ∼= NP × XP × AP the tangent spaces at x tothe submanifolds {n} × XP × {a}, {n} × {τ(m)} × AP, and NP × {τ(m)} ×{a} are mutually orthogonal.

(b) The pullback μ∗g of the metric g on X to NP × XP × AP is given by

ds2(n,τ (m),a) = 1

2

∑

α∈ +(g,aP)

e−2α(log a)hα ⊕ d(τ (m))2 ⊕ da2.

If we choose orthonormal bases {N1, . . . , Nr} of nP, {Y1, . . . , Yl} of sometangent space Tτ(m) XP and H ∈ a

+P with ||H|| = 1, we obtain rational horo-

cyclic coordinates

ϕ : NP × XP × AP → Rr × Rl × R,

⎛

⎝exp

⎛

⎝r∑

j=1

x jN j

⎞

⎠ , exp

⎛

⎝l∑

j=1

x j+rYj

⎞

⎠ , exp(yH)

⎞

⎠ �→ (x1, . . . , xr+l, y).


In the following, we will abbreviate (x1, . . . , xr+l, y) as (x, y). The represen-tation of the metric ds2 with respect to these coordinates is given by the matrix

(gij)i, j(n, τ (m), a) =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

12 e−2α1(log a)

. . .12 e−2αr(log a)

0

0 hkm

1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

where the positive roots αi ∈ +(g, aP) appear according to their multiplicityand the (l × l)-submatrix (hkm)l

k,m=1 represents the metric d(τ (m))2 on theboundary symmetric space XP.

Corollary 2.10 The volume form of NP × XP × AP with respect to rationalhorocyclic coordinates is given by

√det(gij)(n, τ (m), a) dxdy =

(1

2

)r/2√det(hkm(τ (m)) e−2ρP(log a)dxdy

=(

1

2

)r/2√det(hkm(τ (m)) e−2||ρP||ydxdy,

where log a = yH.

A straightforward calculation yields.

Corollary 2.11 The Laplacian � on NP × XP × AP in rational horocycliccoordinates is

� = −2r∑

j=1

e2α j∂2

∂x2j

+ �XP − ∂2

∂y2+ 2||ρP|| ∂

∂y, (2.2)

where �XP denotes the Laplacian on the boundary symmetric space XP and e2α j

is short hand for the function (x, y) �→ e2yα j(H).

3 Lp-Spectrum

In this section X = G/K denotes again a symmetric space of non-compacttype whose metric coincides on TeK(G/K) ∼= p with the Killing form of theLie algebra g of G. Furthermore, � ⊂ G is an arithmetic (non-uniform) latticewith Q-rank(�) = 1. We also assume that � is torsion-free.

The corresponding locally symmetric space M = �\X has finitely manycusps and each cusp corresponds to a �-conjugacy class of a minimal ratio-nal parabolic subgroup P ⊂ G. Let P1, . . . , Pk denote representatives of the

146 A. Weber

�-conjugacy classes. Since these subgroups are conjugate under G(Q) and therespective root systems are isomorphic (cf. Section 2.2.2), we consider in thefollowing only the rational parabolic subgroup P := P1. We denote by ρP as inthe preceding section half the sum of the positive roots (counted according totheir multiplicity) with respect to the pair (g, aP).

We define for any p ∈ [1, ∞) the parabolic region

Pp :=

⎧⎪⎨

⎪⎩z = x + iy ∈ C : x � 4||ρP||2

p

(1 − 1

p

)+ y2

4||ρP||2(

1 − 2p

)2

⎫⎪⎬

⎪⎭

if p �= 2 and P2 := [||ρP||2, ∞).Note, that the boundary ∂ Pp of Pp is parametrized by the curve

R → C, s �→ 4||ρP||2p

(1 − 1

p

)+ s2 + 2i||ρP||s

(1 − 2

p

)

=(

2||ρP||p

+ is)(

2||ρP|| − 2||ρP||p

− is)

and that this parabolic region coincides with the one in Proposition 1.1 if andonly if ||ρP|| = ||ρ||.

Our main result in this chapter reads as follows:

Theorem 3.1 Let X = G/K denote a symmetric space of non-compact type and� ⊂ G an arithmetic lattice with Q-rank(�) = 1 that acts freely on X. If wedenote by M := �\X the corresponding locally symmetric space, the parabolicregion Pp is contained in the spectrum of �M,p, p ∈ (1, ∞):

Pp ⊂ σ(�M,p).

Lemma 3.2 Let M denote a Riemannian manifold with finite volume. For 1 �p � q < ∞, we have

e−t�M,q�M,q ⊂ �M,p e−t�M,q .

Proof Since the volume of M is finite, it follows by Hölder’s inequality

Lq(M) ↪→ Lp(M),

i.e. Lq(M) is continuously embedded in Lp(M). Therefore, we obtain theboundedness of the operators

e−t�M,q : Lq(M) → Lp(M). (3.1)

To prove the lemma, we choose an f ∈ dom(�M,q) = dom(e−t�M,q�M,q).Because of e−t�M,q f ∈ Lp(M) ∩ dom(�M,q) and the consistency of the


semigroups e−t�M,p, p ∈ [1, ∞), we have e−s�M,pe−t�M,q f = e−(t+s)�M,q f and ob-tain by using (3.1):

∥∥∥∥

1

s(e−s�M,pe−t�M,q f − e−t�M,q f ) − e−t�M,q�M,q f

∥∥∥∥

Lp

� C

∥∥∥∥

1

s(e−s�M,q f − f ) − �M,q f

∥∥∥∥

Lq

→ 0 (s → 0+).

Thus, the function e−t�M,q f is contained in the domain of �M,p and we alsohave the equality

e−t�M,q�M,q f = �M,p e−t�M,q f.

��

The following proposition follows from the preceding lemma as in [15,Proposition 3.1] or [14, Proposition 2.1]. For the sake of completeness we workout the details.

Proposition 3.3 Let M denote a Riemannian manifold with finite volume. For2 � p � q < ∞, we have the inclusion

σ(�M,p) ⊂ σ(�M,q).

Proof The statement of the proposition is obviously equivalent to the reverseinclusion for the respective resolvent sets:

ρ(�M,q) ⊂ ρ(�M,p).

We are going to show that for λ ∈ ρ(�M,q) ∩ ρ(�M,p) the resolvents coincideon Lq(M) ∩ Lp(M). From Lemma 3.2 above, we conclude for these λ

(λ − �M,p)−1 e−t�M,q = (λ − �M,p)

−1 e−t�M,q(λ − �M,q)(λ − �M,q)−1

= (λ − �M,p)−1(λ − �M,p) e−t�M,q(λ − �M,q)

−1

= e−t�M,q(λ − �M,q)−1, (3.2)

where the equality is meant between bounded operators from Lq(M) toLp(M). If t → 0, we obtain

(λ − �M,p)−1∣∣Lq∩Lp = (λ − �M,q)

−1∣∣

Lq∩Lp .

For 1q + 1

q′ = 1 (in particular, this implies q′ � p � q) and λ ∈ ρ(�M,q) =ρ(�M,q′) we have by the preceding calculation

(λ − �M,q′)−1∣∣

Lq∩Lq′ = (λ − �M,q)−1∣∣

Lq∩Lq′ .

The Riesz–Thorin interpolation theorem implies that (λ − �M,q)−1 is bounded

if considered as an operator Rλ on Lp(M).In the remainder of the proof we show that Rλ coincides with (λ − �M,p)

−1

and hence ρ(�M,q) ⊂ ρ(�M,p). Notice, that (3.2) implies

(λ − �M,p)e−t�M,q(λ − �M,q)−1 f = e−t�M,q f,

148 A. Weber

for all f ∈ Lp(M) ∩ Lq(M). Since �M,p is a closed operator, we obtain fort → 0 the limit

(λ − �M,p)Rλ f = f.

As Lq(M) ∩ Lp(M) is dense in Lp(M) and �M,p is closed, it follows (λ −�M,p)Rλ f = f for all f ∈ Lp(M). Therefore, (λ − �M,p) is onto. If we assumethat (λ − �M,p) is not one-to-one, λ would be an eigenvalue of �M,p. Assumef �= 0 is an eigenfunction of �M,p for the eigenvalue λ. Then it follows fromLemma 3.2:

λe−t�M,p f = �M,q′e−t�M,p f.

Since e−t�M,p is strongly continuous there is a t0 > 0 such that e−t0�M,p f �= 0 ande−t0�M,p f is therefore an eigenfunction of �M,q′ for the eigenvalue λ. But thiscontradicts λ ∈ ρ(�M,q) = ρ(�M,q′). We finally obtain Rλ = (λ − �M,p)

−1. ��

Proposition 3.4 For 1 � p < ∞ the boundary ∂ Pp of the parabolic region Pp

is contained in the approximate point spectrum of �M,p:

∂ Pp ⊂ σapp(�M,p).

Proof In this proof we construct for any z ∈ ∂ Pp a sequence fn of differen-tiable functions in Lp(X) with support in a fundamental domain for � suchthat

||�X,p fn − zfn||Lp

|| fn||Lp→ 0 (n → ∞).

Since such a sequence ( fn) descends to a sequence of differentiable functionsin Lp(M) this is enough to prove the proposition.

Recall that a fundamental domain for � is given by a subset of the form

�0 ∪k∐

i=1

ωi × APi,ti ⊂ X

(cf. Theorem 2.8), and each Siegel set ωi × APi,ti is mapped injectively into�\X. Furthermore, the closure of π(ωi × APi,ti) fully covers an end of �\X(for any i ∈ {1, . . . , k}).

Now, we choose some

z = z(s) =(

2||ρP||p

+ is)(

2||ρP|| − 2||ρP||p

− is)

∈ ∂ Pp.

Furthermore, we take the Siegel set ω × AP,t := ω1 × AP1,t1 where AP,t = {a ∈AP : α(a) > t}, and define a sequence fn of smooth functions with support inω × AP,t with respect to rational horocyclic coordinates by

fn(x, y) := cn(y)e(

2p ||ρP||+is

)y,

where cn ∈ C∞c

((log t||α|| , ∞

))is a so-far arbitrary sequence of differentiable

functions with support in(

log t||α|| , ∞

). Since ω is bounded, each fn is clearly


contained in Lp(X). Furthermore, the condition supp(cn) ⊂(

log t||α|| , ∞

)ensures

that the supports of the sequence fn are contained in the Siegel set ω × AP,t.Using formula (2.2) for the Laplacian in rational horocyclic coordinates, we

obtain after a straightforward calculation

�X,p fn(x, y) − zfn(x, y)

=(

−c′′n(y) +

(2||ρP|| − 2

(2

p||ρP|| + is

))c′

n(y)

)e(

2||ρP ||p +is

)y,

and therefore

||�X,p fn − zfn||pLp

=∫

ω×AP,t

|�X,p fn − zfn|pdvolX

=(

1

2

)r/2 ∫

ω×AP,t

|�X,p fn(x, y) − zfn(x, y)|p√

det(hkm(τ (m)) e−2||ρP||y dxdy

= C∫ ∞

0

∣∣∣∣−c′′

n(y) +(

2||ρP|| − 2

(2||ρP||

p+ is

))c′

n(y)

∣∣∣∣

p

dy,

where C := ( 12

)r/2 ∫ω

√det(hkm(τ (m))dx<∞ because ω ⊂ NP×XP is bounded.

This yields after an application of the triangle inequality

||�X,p fn − zfn||Lp � C1

(∫ ∞

0|c′′

n(y)|p dy)1/p

+ C2

(∫ ∞

0|c′

n(y)|p dy)1/p

.

By an analogous calculation we obtain

|| fn||Lp = C3

(∫ ∞

0|cn(y)|p dy

)1/p

.

We choose a function ψ ∈ C∞c (R), not identically zero, with supp(ψ) ⊂ (1, 2),

a sequence rn > 0 with rn → ∞ (if n → ∞), and we eventually define

cn(y) := ψ

(yrn

).

For large enough n, we have supp(cn) ⊂(

log t||α|| , ∞

). An easy calculation gives

∫ ∞

0|cn(y)|p dy = rn

∫ 2

1|ψ(u)|p du,

∫ ∞

0|c′

n(y)|p dy1 = r1−pn

∫ 2

1|ψ ′(u)|p du,

∫ ∞

0|c′′

n(y)|p dy1 = r1−2pn

∫ 2

1|ψ ′′(u)|p du.

150 A. Weber

In the end, this leads to the inequality

||�X,p fn − zfn||p

|| fn||p� C4

rn+ C5

r2n

−→ 0 (n → ∞),

where C4, C5 > 0 denote positive constants, and the proof is complete. ��

Proof of Theorem 3.1 The inclusion

Pp ⊂ σ(�M,p)

for p ∈ [2, ∞) follows immediately from Proposition 3.3 and Proposition 3.4by observing

Pp =⋃

q∈[2,p]∂ Pq.

The inclusion for all p ∈ (1, ∞) follows by duality as Pp = Pp′ if 1p + 1

p′ = 1.��

Up to now, we considered non-uniform arithmetic lattices � ⊂ G with Q-rank one. We made no assumption concerning the rank of the respectivesymmetric space X = G/K of non-compact type. However, if rank(X) = 1,we are able to sharpen the result of Theorem 3.1 considerably. In the caseQ-rank(�) = rank(X) = 1, the one dimensional abelian subgroup AP of G(with respect to some rational minimal parabolic subgroup) defines a maximalflat subspace, i.e. a geodesic, AP · x0 of X. Hence, the Q-roots coincide with theroots defined in Section 2.2 and for any rational minimal parabolic subgroup Pwe have in particular

||ρP|| = ||ρ||.

Corollary 3.5 Let X = G/K denote a symmetric space of non-compact typewith rank(X) = 1. Furthermore, � ⊂ G denotes a non-uniform arithmetic latticethat acts freely on X and M = �\X the corresponding locally symmetric space.Then, we have for all p ∈ (1, ∞) the equality

σ(�M,p) = {λ0, . . . , λm} ∪ Pp,

where 0 = λ0, . . . , λm ∈ [0, ||ρ||2) are eigenvalues of �M,2 with finitemultiplicity.

Proof Langlands’ theory of Eisenstein series implies (see e.g. [17] or thesurveys in [16] or [4])

σ(�M,2) = {λ0, . . . , λm} ∪ [||ρ||2, ∞),

where 0 = λ0, . . . , λm ∈ [0, ||ρ||2) are eigenvalues of �M,2 with finite multiplic-ity. Thus, we can apply Proposition 1.1 and obtain

σ(�M,p) ⊂ {λ0, . . . , λm} ∪ Pp.


As in the proof of [8, Lemma 6] one sees that the discrete part of theL2-spectrum {λ0, . . . , λm} is also contained in σ(�M,p) for any p ∈ (1, ∞).Together with Theorem 3.1 and the remark above this concludes the proof. ��

As remarked in [24] one can prove as in [8] that every L2-eigenfunctionof the Laplace–Beltrami operator �M,2 with respect to the eigenvalue λ j,j = 0, . . . , m, lies in Lp(M) if λ j is not contained in Pp.

Remark 3.6 Because of the description of fundamental domains for generallattices in semi-simple Lie groups with R-rank one (see [10]) it seems that thearithmeticity of � in Corollary 3.5 is not needed.

4 Manifolds with Cusps of Rank One

In this chapter we consider a class of Riemannian manifolds that is larger thanthe class of Q-rank one locally symmetric spaces. This larger class consists ofthose manifolds which are isometric – after the removal of a compact set –to a disjoint union of rank one cusps. Manifolds with cusps of rank one wereprobably first introduced and studied by W. Müller (see e.g. [21]).

4.1 Definition

Recall, that we denoted by ω × AP,t ⊂ X Siegel sets of a symmetric spaceX = G/K of non-compact type. The projection π(ω × AP,t) of certain Siegelsets to a corresponding Q-rank one locally symmetric space �\X is a cusp andevery cusp of �\X is of this form (cf. Section 2.2.2).

Definition 4.1 A Riemannian manifold is called cusp of rank one if it isisometric to a cusp π(ω × AP,t) of a Q-rank one locally symmetric space.

Definition 4.2 A complete Riemannian manifold M is called manifold withcusps of rank one if it has a decomposition

M = M0 ∪k⋃

j=1

Mj

such that the following holds:

(1) M0 is a compact manifold with boundary.(2) The subsets Mj, j ∈ {0, . . . , k}, are pairwise disjoint.(3) For each j ∈ {1, . . . , k} there exists a cusp of rank one isometric to Mj.

Such manifolds certainly have finite volume as there is only a finite numberof cusps possible and every cusp of rank one has finite volume.

152 A. Weber

From Theorem 2.8 it follows that any Q-rank one locally symmetric space isa manifold with cusps of rank one. But since we can perturb the metric on thecompact manifold M0 without leaving the class of manifolds with cusps of rankone, not every such manifold is locally symmetric. Of course, they are locallysymmetric on each cusp and we can say that they are locally symmetric nearinfinity.

4.2 Lp-Spectrum and Geometry

Precisely as in Proposition 3.4 one sees that we can find for every cusp Mj, j ∈{1, . . . , k} of a manifold M = M0 ∪⋃k

j=1 Mj with cusps of rank one a parabolic

region P( j)p such that the boundary ∂ P( j)

p is contained in the approximate pointspectrum of �M,p. Here, the parabolic regions are defined as the parabolicregion in the preceding section, where the constant ||ρP|| is replaced by ananalogous quantity, say ||ρP j||, coming from the respective cusp Mj. That is tosay, we have the following lemma:

Lemma 4.3 Let M denote a manifold with cusps of rank one. Then we have forp ∈ [1, ∞) and j = 1, . . . , k:

∂ P( j)p ⊂ σapp(�M,p).

Since the volume of a manifold with cusps of rank one is finite, we can applyProposition 3.3 in order to prove (cf. the proof of Theorem 3.1) the following:

Theorem 4.4 Let M = M0 ∪⋃kj=1 Mj denote a manifold with cusps of rank

one. Then, for p ∈ (1, ∞), every cusp Mj defines a parabolic region P( j)p that

is contained in the Lp-spectrum (cf. Fig. 3):

k⋃

j=1

P( j)p ⊂ σ(�M,p).

Fig. 3 The union of twoparabolic regions P(1)

p and

P(2)p if p �= 2


Of course, the compact submanifold M0 contributes some discrete set tothe Lp-spectrum, and 0 is always an eigenvalue as the volume of M is finite.It seems to be very likely that besides some discrete spectrum the union ofthe parabolic regions in Theorem 4.4 is already the complete spectrum. But atpresent, I do not know how to prove this result. The methods used in [8] or[24] to prove a similar result need either that the manifold is homogeneous orthat the injectivity radius is bounded from below, and it is not clear how onecould adapt the methods therein to our case.

Nevertheless, given the Lp-spectrum for some p �= 2, we have the followinggeometric consequences:

Corollary 4.5 Let M = M0 ∪⋃kj=1 Mj denote a manifold with cusps of rank one

such that

σ(�M,p) = {λ0, . . . , λr} ∪ Pp,

for some p �= 2 and some parabolic region Pp. Then every cusp Mj is of theform π(ω j × AP j,t j) with volume form

(1

2

)r j/2

e−2yc dxdy,

where c is a positive constant.

Proof Since all parabolic regions P( j)p induced by the cusps Mj coincide, the

quantities ||ρP j|| coincide. Therefore, we can take c := ||ρP1 ||. ��

This result generalizes to the case where the continuous spectrum consistsof a finite number of parabolic regions in an obvious manner.

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http://www.math.okstate.edu/~dwitte

http://www.math.okstate.edu/~dwitte


Asymptotic Behaviour of the Spectrumof a Waveguide with Distant Perturbations

Denis Borisov

Received: 23 May 2007 / Accepted: 11 August 2007 /Published online: 1 September 2007© Springer Science + Business Media B.V. 2007

Abstract We consider a quantum waveguide modelled by an infinite straighttube with arbitrary cross-section in n-dimensional space. The operator westudy is the Dirichlet Laplacian perturbed by two distant perturbations. Theperturbations are described by arbitrary abstract operators “localized” in acertain sense. We study the asymptotic behaviour of the discrete spectrumof such system as the distance between the “supports” of localized pertur-bations tends to infinity. The main results are a convergence theorem andthe asymptotics expansions for the eigenvalues. The asymptotic behaviour ofthe associated eigenfunctions is described as well. We provide a list of theoperators, which can be chosen as distant perturbations. In particular, thedistant perturbations may be a potential, a second order differential operator,a magnetic Schrödinger operator, an arbitrary geometric deformation of thestraight waveguide, a delta interaction, and an integral operator.

Keywords Distant perturbation · Waveguide · Asymptotics ·Eigenvalue · Eigenfunction

Mathematics Subject Classifications (2000) 35P05 · 35B20 · 35B40

The research was supported by Marie Curie International Fellowship within 6th EuropeanCommunity Framework Programm (MIF1-CT-2005-006254). The author is also supported bythe Russian Foundation for Basic Researches (No. 07-01-00037) and by the Czech Academyof Sciences and Ministry of Education, Youth and Sports (LC06002). The author gratefullyacknowledges the support from Deligne 2004 Balzan prize in mathematics and the grant ofRepublic Bashkortostan for young scientists and young scientific collectives.

D. Borisov (B)Nuclear Physics Institute, Academy of Sciences, Rež near Prague 25068, Czechia,Bashkir State Pedagogical University, October rev. st. 3a, 450000 Ufa, Russiae-mail: [email protected]

156 D. Borisov

1 Introduction

The multiple well problem for the Schrödinger operator attracted much atten-tion of many researches. The example of such operator with two wells is asfollows,

−�� + V1(x − a1) + V2(x − a2), (1.1)

where ai are some points, and Vi are real-valued functions satisfying certainsmoothness conditions and being either compactly supported or decayingsufficiently fast at infinity. A lot of works were devoted to the study of thediscrete spectrum in the semi-classical case, i.e., as � → 0 (see, for instance,[8, 11, 22], and references therein). The results obtained in this case are closein a certain sense to that obtained in the regime when the distances betweenthe wells tend to infinity, for instance, as � = 1 and |a1 − a2| → +∞ in (1.1).We mention the papers [14, 16, 18] as well as the book [12, Sec. 8.6] devoted tosuch problems (see also bibliography of these works). The main result of theseworks was a description of the asymptotic behaviour of the eigenvalues andthe eigenfunctions as the distance between wells tended to infinity. In [15] adouble-well problem for the Dirac operator with large distance between wellswas studied. The convergence and certain asymptotic results were established.We also mention the paper [19], where the usual potential was replaced bya delta-potential supported by a curve. The results of this paper imply theasymptotic estimate for the lowest spectral gap in the case the curve consistsof several disjoint components and the distances between components tendto infinity.

One of the possible ways to generalize the mentioned problems is toconsider them not in R

n, but in some other unbounded domains. An exampleof physical relevance is an infinite tube, since such domain arises in thewaveguide theory. An additional motivation is that the tube is infinite in onedimension only, and it simplifies the considerations. One of such problemshas already been treated in [4]. Here we considered the Dirichlet Laplacianin a two-dimensional straight infinite strip. The perturbation consisted of twosegments of the same length on the boundary, on which the Dirichlet conditionwas switched to the Neumann one. As the distance between the segmentstended to infinity, a convergence result and the asymptotic expansions for theeigenvalues and the eigenfunctions were obtained. The technique used in [4]employed essentially the symmetry of the problem.

In the present paper we consider an infinite straight tube in n-dimensionalspace and consider the Dirichlet Laplacian −�(D) in this tube. The pertur-bation consists of two arbitrary operators L± “localized” in a certain senseand satisfying certain sufficiently weak conditions. The distance between their“supports” is assumed to be a large parameter. In what follows we will callthese operators distant perturbations. The main distinction to the articlescited is that we don not specify the nature of these operators. Their precisedescription will be given in the next section; we only say here that a lot of

Asymptotic behaviour of the spectrum... 157

interesting examples are particular cases of these operators. In particular,the classical example, a compactly supported potential can be chosen as oneof the distant perturbation; the same is true for a second order differentialoperators with compactly supported coefficients. In addition, one can alsoconsider integral operators as a distant perturbation, a delta interaction or ageometric deformation of the tube can be treated as well. We refer to Section 7for detailed presentation of these examples.

Our main results are as follows. First we prove the convergence for theeigenvalues of the perturbed operator, and show that the limiting valuesfor these eigenvalues are the discrete eigenvalues of the limiting operators−�(D) + L± and the threshold of the essential spectrum. The most nontrivialresult of the article is the asymptotic expansions for the perturbed eigenvalues.Namely, we obtain a scalar equation for these eigenvalues. Based on thisequation, we construct the leading terms of their asymptotic expansions. Wealso characterize the asymptotic behaviour of the perturbed eigenfunctions.

If the distant perturbations consisted of two potential wells as in (1.1), andone of the wells can be translated into the other, it is well-known that eachlimiting eigenvalue splits into pair of two perturbed eigenvalues one of whichbeing larger than a limiting eigenvalue while the other being less. It is alsoknown that the next to leading terms of their asymptotics are exponentiallysmall w.r.t. the distance between the wells, and have the same modulus butdifferent signs. In the present article we show that the similar phenomenonoccur in the general situation, too (see Theorem 2.8).

Let us discuss the technique used in the paper. The core of the approach isa scheme which allows us to reduce the eigenvalue equation for the perturbedoperator to an equivalent operator equation in an auxiliary Hilbert space. Themain motivation of such reduction is that the original distant perturbations arereplaced by an operator which is meromorphic w.r.t. the spectral parameterand is multiplied by a small parameter. We solve this equation explicitly by themodification of the Birman–Schwinger technique suggested in [13], and in thisway we obtain the described results. We stress that our approach requires nosymmetry restriction in contrast to [4]. One of the advantages of our approachis that it is independent of the type of boundary condition, for instance, similarproblem for the Neumann Laplacian can be solved effectively, too. Moreover,our technique can applied to other problems with distant perturbations. Wecan refer to [5] where we studied the Dirichlet Laplacian in two adjacentstraight infinite strips; the perturbation consisted of two finite segments cutout in the common boundary and separated by a large distance. This problemwas a natural generalization of that treated in [4] and it can not be describedby the operators L± considered in the present paper. At the same time, weshowed in [5] that the main ideas of the present paper could be adapted tothe aforementioned problem with minor changes. We also cite [6] where weconsidered the Laplacian in a multi-dimensional space with a finite numberof distant perturbations. We followed the ideas of the present paper to treatthis problem. On the other hand, the waveguide we consider here is infinite inone dimension only, while in [6] the domain was infinite in many dimensions.

158 D. Borisov

Because of this we had to employ an additional technique; we should also saythat the results of [6] are less explicit than those in the present paper.

The article is organized as follows. In the next section we give precisestatement of the problem and formulate the main results. In the second sectionwe prove that the essential spectrum of the perturbed operator is invariantw.r.t. the perturbations and occupies a real semi-axis, while the discretespectrum contains finitely many eigenvalues. The third section is devoted tothe study of the limiting operators; we collect there some preliminary factsrequired for the proof of the main results. In the fourth section we providethe aforementioned scheme transforming the original perturbed eigenvalueequation to an equivalent operator equation. Then we solve this equationexplicitly. The fifth section is devoted to the proof of the convergence result.The asymptotic formulas for the perturbed eigenelements are established inthe sixth section. The final seventh section contains some examples of L±.

2 Statement of the Problem and Formulation of the Results

Let x = (x1, x′) and x′ = (x2, . . . , xn), be Cartesian coordinates in Rn and R

n−1,respectively, n � 2, and let ω be a bounded domain in R

n−1 having infinitelydifferentiable boundary. By � we denote an infinite tube R × ω. Given anybounded domain Q ⊂ �, by L2(�, Q) we denote the subset of the functionsfrom L2(�) whose support lies inside Q. For any domain � ⊆ � and any (n −1)-dimensional manifold S ⊂ � the symbol W j

2,0(�, S) will indicate the subset

of the functions from W j2(�) vanishing on S. If S = ∂�, we will write shortly

W j2,0(�).Let �± be a pair of bounded subdomains of � defined as �± := (−a±, a±) ×

ω, where a± ∈ R are fixed positive numbers. We let γ± := (−a±, a±) × ∂ω.By L± we denote a pair of bounded linear operators from W j

2,0(�±, γ±) into

L2(�, �±). We assume that for all u1, u2 ∈ W j2,0(�±, γ±) the identity

(L±u1, u2)L2(�±) = (u1,L±u2)L2(�±) (2.1)

holds true. We also suppose that the operators L± satisfy the estimate

(L±u, u)L2(�±) � −c0‖∇u‖2L2(�±) − c1‖u‖2

L2(�±) (2.2)

for all u ∈ W22,0(�±, γ±), where the constants c0, c1 are independent of u, and

c0 < 1. (2.3)

Since the restriction of each function u ∈ W22,0(�) on �± belongs to

W22,0(�±, γ±), we may also regard the operators L± as unbounded ones in

L2(�) with the domain W22,0(�).

By S(a) we denote a shift operator in L2(�) acting as (S(a)u)(x) := u(x1 +a, x′), and for any l > 0 we introduce the operator

Ll := S(l)L−S(−l) + S(−l)L+S(l).


Clearly, this operator depends on values its argument takes on the set {x :(x1 + l, x′) ∈ �−} ∪ {x : (x1 − l, x′) ∈ �+}. As l → +∞, this set consists of twocomponents separated by the distance 2l. This is why we can regard theoperator Ll as the distant perturbations formed by L− and L+.

We introduce the operator Hl := −�(D) + Ll in L2(�) with the domainW2

2,0(�). Here −�(D) indicates the Laplacian in L2(�) with the domainW2

2,0(�). We suppose that the operators L± are such that the operator Hl

is self-adjoint. The main aim of this paper is to study the behaviour of thespectrum of Hl as l → +∞.

In order to formulate the main results we need to introduce additionalnotations. We will employ the symbols σ(·), σess(·), σdisc(·) to indicate thespectrum, the essential spectrum and the discrete one of an operator. Wedenote H± := −�(D) + L±, and suppose that these operators with the domainW2

2,0(�) are self-adjoint in L2(�).

Remark 2.1 We note that the assumptions (2.1), (2.2), (2.3) do not imply theself-adjointness of Hl and H±, and we can employ here neither the KLMNtheorem no the Kato–Rellich theorem. On the other hand, if the operators L±satisfy stricter assumption and are −�(D)-bounded with the bound less thanone, it implies the self-adjointness of Hl and H±.

Let ν1 > 0 be the lowest eigenvalue of the Dirichlet Laplacian in ω.Our first result is

Theorem 2.1 The essential spectra of Hl , H+, H− coincide with the semi-axis[ν1, +∞). The discrete spectra of Hl , H+, H− consist of finitely many realeigenvalues.

We denote σ∗ := σdisc(H−) ∪ σdisc(H+). Let λ∗ ∈ σ∗ be a p−-multiple eigen-value of H− and p+-multiple eigenvalue of H+, where we set p± equal to zero,if λ∗ �∈ σdisc(H±). In this case we will say that λ∗ is (p− + p+)-multiple.

Theorem 2.2 Each discrete eigenvalue of Hl converges to one of the numbers inσ∗ or to ν1 as l → +∞.

Theorem 2.3 If λ∗ ∈ σ∗ is (p− + p+)-multiple, the total multiplicity of the eigen-values of Hl converging to λ∗ equals p− + p+.

In what follows we will employ the symbols (·, ·)X and ‖ · ‖X to indicate thescalar product and the norm in a Hilbert space X.

Suppose that λ∗ ∈ σ∗ is (p− + p+)-multiple, and ψ±i , i = 1, . . . , p±, are the

eigenfunctions of H± associated with λ∗ and orthonormalized in L2(�). Ifp− � 1, we denote

φi(·, l) := (0;L+S(2l)ψ−i ) ∈ L2(�−) ⊕ L2(�+),

T (i)1 f := ( f−, ψ−

i )L2(�−), i = 1, . . . , p−,

160 D. Borisov

where f := ( f−; f+) ∈ L2(�−) ⊕ L2(�+). If p+ � 1, we denote

φi+p−(·, l) := (L−S(−2l)ψ+i ; 0) ∈ L2(�−) ⊕ L2(�+),

T (i+p−)

1 f := ( f+, ψ+i )L2(�+), i = 1, . . . , p+.

In the fourth section we will show that the operator

T2(λ, l) f := (L−S(−2l)(H+ − λ)−1 f+;L+S(2l)(H− − λ)−1 f−

)(2.4)

in L2(�−) ⊕ L2(�+) satisfies the relation

T2(λ, l) = − 1

λ − λ∗

p∑

i=1

φi(·, l)T (i)1 + T3(λ, l), (2.5)

for λ close to λ∗, where p := p− + p+, and the norm of T3 tends to zero asl → +∞ uniformly in λ. We introduce the matrix

A(λ, l) :=⎛

⎜⎝

A11(λ, l) . . . A1p(λ, l)...

...

Ap1(λ, l) . . . App(λ, l)

⎞

⎟⎠ ,

where Aij(λ, l) := T (i)1 (I + T3(λ, l))−1φ j(·, l).

Theorem 2.4 Let λ∗ ∈ σ∗ be (p− + p+)-multiple, and let λi = λi(l) −−−−→l→+∞

λ∗,

i = 1, . . . , p, p := p− + p+, be the eigenvalues of Hl taken counting multiplicityand ordered as follows

0 � |λ1(l) − λ∗| � |λ2(l) − λ∗| � . . . � |λp(l) − λ∗|. (2.6)

These eigenvalues solve the equation

det((λ − λ∗)E − A(λ, l)

) = 0, (2.7)

and satisfy the asymptotic formulas

λi(l) = λ∗ + τi(l)(

1 + O(

l2p e− 4l

p

√ν1−λ∗

)), l → +∞. (2.8)

Here

τi = τi(l) = O(

e−2l√

ν1−λ∗)

, l → +∞, (2.9)

are the zeroes of the polynomial det(τE − A(λ∗, l)

)taken counting multiplicity

and ordered as follows

0 � |τ1(l)| � |τ2(l)| � . . . � |τp(l)|. (2.10)

The eigenfunctions associated with λi satisfy the asymptotic representation

ψi(x, l) =p−∑

i=1

ki, jψ−j (x1 + l, x′) +

p+∑

i=1

ki, j+p−ψ+j (x1 − l, x′) + O

(e−2l

√ν1−λ∗

),

(2.11)


as l → +∞ in W22(�)-norm. The numbers ki, j are the components of the vectors

ki = ki(l) = (ki,1(l) . . . ki,p(l)

)tsolving the system

((λ − λ∗)E − A(λ, l)

)k = 0 (2.12)

for λ = λi(l), and satisfying the condition

(ki, k j)Cp ={

1, if i = j,

O(

le−2l√

ν1−λ∗)

, if i �= j.(2.13)

According to this theorem, the leading terms of the asymptotics expansionsfor the eigenvalues λi are determined by the matrix A(λ∗, l). On the otherhand, in applications it could be quite complicated to calculate this matrixexplicitly. This is why in the following theorems we provide one more wayof calculating the asymptotic expansions.

We will say that a square matrix A(l) satisfies the condition (A), if it isdiagonalizable and the determinant of the matrix formed by the normalizedeigenvectors of A(l) is separated from zero uniformly in l large enough.

Theorem 2.5 Let the hypothesis of Theorem 2.4 hold true. Suppose that thematrix A(λ∗, l) can be represented as

A(λ∗, l) = A0(l) + A1(l), (2.14)

where the matrix A0 satisfies the condition (A), and ‖A1(l)‖ → 0 as l → +∞. Inthis case the eigenvalues λi of Hl satisfy the asymptotic formulas

λi = λ∗ + τ(0)

i

(1 + O

(l2e−4l

√ν1−λ∗

))+ O(‖A1(l)‖), l → +∞. (2.15)

Here τ(0)

i =τ(0)

i (l) are the roots of the polynomial det(τE−A0(l)

)taken counting

multiplicity and ordered as follows

0 �∣∣τ (0)

1 (l)∣∣ �

∣∣τ (0)

2 (l)∣∣ � . . . �

∣∣τ (0)

p (l)∣∣.

Each of these roots satisfies the estimate

τ(0)

i (l) = O(‖A0(l)‖), l → +∞. (2.16)

This theorem states that the leading terms of the asymptotics for theeigenvalues can be determined by that of the asymptotics for A(λ∗, l). Weobserve that the estimate for the error term in (2.15) can be worse than thatin (2.8). In the following theorem we apply Theorem 2.5 to several importantparticular cases.

Let ν2 > ν1 be the second eigenvalue of the negative Dirichlet Laplacianin ω, and φ1 = φ1(x′) be the eigenvalue associated with ν1 and normalized inL2(ω). In the fifth section we will prove

162 D. Borisov

Lemma 2.1 Let the hypothesis of Theorem 2.4 hold true, and p± � 1. Then thefunctions ψ±

i can be chosen so that

ψ±1 (x) = β±e±√

ν1−λ∗x1φ1(x′) + O(

e±√ν2−λ∗x1

),

ψ±i (x) = O

(e±√

ν2−λ∗x1

), (2.17)

as x1 → ∓∞, i = 2, . . . , p±, β± are some number, and the functions ψ±i are

orthonormalized in L2(�).

In what follows we assume that the functions ψ±i are chosen in accordance

with this lemma.

Theorem 2.6 Let the hypothesis of Theorem 2.5 hold true, and p+ = 0. Thenthe eigenvalues λi satisfy the asymptotic formulas

λi(l) = λ∗ + O(

e−2l(√

ν1−λ∗+√ν2−λ∗)

), i = 1, . . . , p − 1,

λp(l) = λ∗ − 2√

ν1 − λ∗|β−|2β−e−4l√

ν1−λ∗ ++O

(e−2l(

√ν1−λ∗+√

ν2−λ∗) + l2e−6l√

ν1−λ∗), (2.18)

where the constant β− is determined uniquely by the identity

U+(x) = β−e−√ν1−λ∗x1φ1(x′) + O

(e

√ν2−λ∗x1

), x1 → −∞,

U+ : = (H+ − λ∗)−1L+(e−√

ν1−λ∗x1φ1(x′)). (2.19)

Theorem 2.7 Let the hypothesis of Theorem 2.5 hold true, and p− = 0. Thenthe eigenvalues λi satisfy the asymptotic formulas

λi(l) = λ∗ + O(

e−2l(√

ν1−λ∗+√ν2−λ∗)

), i = 1, . . . , p − 1,

λp(l) = λ∗ − 2√

ν1 − λ∗|β+|2β+e−4l√

ν1−λ∗ ++O

(e−2l(

√ν1−λ∗+√

ν2−λ∗) + l2e−6l√

ν1−λ∗)

, (2.20)

where the constant β+ is uniquely determined by the identity

U+(x) = β+e√

ν1−λ∗x1φ1(x′) + O(

e√

ν2−λ∗x1

), x1 → +∞,

U+ := (H− − λ∗)−1L+(

e−√ν1−λ∗x1φ1(x′)

)

These two theorems treat the first possible case when the number λ∗ ∈ σ isthe eigenvalue of one of the operators H± only. The formulas (2.18), (2.20)give the asymptotic expansion for the eigenvalue λp, and the asymptoticestimates for the other eigenvalues. At the same time, given generic L± andan eigenvalue λ∗ of H±, this eigenvalue is simple. In this case p = 1, and by


Theorem 2.3 there exists the unique perturbed eigenvalue converging to λ∗,and Theorems 2.6, 2.7 provide its asymptotics.

Theorem 2.8 Let the hypothesis of Theorem 2.4 hold true, and p± � 1. Thenthe eigenvalues λi satisfy the asymptotic formulas

λi(l) = λ∗ + O(

le−4l√

ν1−λ∗)

, i = 1, . . . , p − 2,

λp−1(l) = λ∗ − 2|β−β+|√ν1 − λ∗e−2l√

ν1−λ∗ + O(

le−4l√

ν1−λ∗)

,

λp(l) = λ∗ + 2|β−β+|√ν1 − λ∗e−2l√

ν1−λ∗ + O(

le−4l√

ν1−λ∗)

, (2.21)

as l → +∞.

This theorem deals with the second possible case when the number λ∗ ∈ σ

is an eigenvalue of both operators H±. Similarly to Theorems 2.6, 2.7, theformulas (2.21) give the asymptotic expansions for λp−1 and λp, and theasymptotic estimates for the other eigenvalues. The most probable case is thatλ∗ is a simple eigenvalue of H+ and H−. In this case there exist only twoperturbed eigenvalues converging to λ∗, and Theorem 2.8 give their asymptoticexpansions.

Suppose now that under the hypothesis of Theorem 2.8 the inequalityβ−β+ �= 0 holds true. In this case the next to leading terms of the asymp-totic expansions for λp−1 and λp have the same modulus but different signs.Moreover, these eigenvalues are simple. This situation is similar to what onehas when dealing with a double-well problem with symmetric wells. We stressthat in our case we assume no additional restrictions except the belonging λ∗ ∈σdisc(H−) ∩ σdisc(H+). Hence, the latter condition is sufficient for the mentionedphenomenon to occur regardless of whether L+ and L− are equal or not. Wealso observe that the formulas (2.21) allow us to estimate the spectral gapbetween λp−1 and λp,

λ2(l) − λ1(l) = 4|β−β+|√ν1 − λ∗e−2l√

ν1−λ∗ + O(le−4l√

ν1−λ∗), l → +∞.

Finally we note that it is possible to calculate the asymptotic expansionsfor the eigenvalues λi, i � p − 1, in Theorem 2.6, 2.7, and for λi, i � p − 2,in Theorem 2.8. In order to do it, one should employ the technique of theproofs of the mentioned theorems and extract the next-to-leading term of theasymptotic for A(λ∗, l). Then these terms should be treated as the matrix A0(l)in (2.14). We refrain from presenting such calculations here in order not tooverburden the text by quite bulky and technical details.

3 Proof of Theorem 2.1

Let � be a bounded non-empty subdomain of � defined as � := (−a, a) × ω,where a ∈ R, a > 0, γ := (−a, a) × ∂ω, and let L be an arbitrary bounded

164 D. Borisov

operator from W22,0(�, γ ) into L2(�, �). Assume that for all u, u1, u2 ∈

W22,0(�, γ ) the relations

(Lu1, u2)L2(�) = (u1,Lu2)L2(�),

(Lu, u)L2(�) � −c0‖∇u‖2L2(�) − c1‖u‖2

L2(�) (3.1)

hold true, where c0, c1 are constants, and c0 obeys (2.3). As in the case of theoperators L±, we can also regard L as an unbounded operator in L2(�) withthe domain W2

2,0(�). Suppose also that the operator HL := −�(D) + L withthe domain W2

2,0(�) is self-adjoint in L2(�).

Lemma 3.1 σess(HL) = [ν1, +∞).

Proof Let λ ∈ [ν1, +∞). Since λ ∈ σess(−�(D)), it is well-known that thereexists a Weyl sequence of −�(D) at λ and this sequence of compactly supportedfunctions; without loss of generality we can also assume that the supportsof these functions do not intersect with �. Now it is easy to check thatthe mentioned sequence is also a Weyl sequence for HL at λ. Therefore,[ν1, +∞) ⊆ σess(HL). To complete the proof, it is sufficient to show thatinf σess(HL) = ν1.

The threshold of the essential spectrum of HL is given by Agmon–Perssonformula

inf σess(HL) = limR→+∞

infu∈C∞

0 (�−R∪�+

R)

(HLu, u)L2(�)

‖u‖2L2(�)

, (3.2)

where �±R := � ∩ {x :±x1 >±R}. The proof of this formula for the Schrödinger

operator was given in [21]; the case of more general elliptic operator wastreated in [1, Ch. 3, Th. 3.2] and also in [23, Ch. 14, Sec. 14.3, Prop. 14.8].In our case the proof of this formula reproduces word by word the proof givenin [23]; this is why we do not give it here. Since (HLu, u)L2(�) = ‖∇u‖2

L2(�),if u ∈ C∞

0 (�−R ∪ �+

R) and R > a, it follows from (3.2) that inf σess(HL) =inf σess(−�(D)) = ν1. ��

Lemma 3.2 The discrete spectrum of HL consists of finitely many eigenvalues.

Proof Let χ = χ(t) ∈ C∞(R) be a cut-off function which takes values in [0, 1],is identically equal to one for t < 0, and vanishes for t > 1. Due to (3.1)we have

(u, HLu)L2(�) � ‖∇u‖2L2(�) − c0‖∇u‖2

L2(�) − c1‖u‖2L2(�) �

�(∇u, (1 − c0χ(|x1| − a))∇u

)L2(�)

− (u, c1χ(|x1| − a)u

)L2(�)

. (3.3)

By H±L we denote the Laplacian in L2(�

±a+1) whose domain consists of the

functions in W22,0(�

±a+1, ∂�±

a+1 \ (±a ± 1) × ω) satisfying Neumann boundarycondition on (±a ± 1) × ω. The symbol H0

L denotes the operator

− div(1 − c0χ(|x1| − a)

)∇ − c1χ(|x1| − a)


in L2(�0), �0 := (−a − 1, a + 1) × ω, with the domain formed by the functions

in W22,0(�

0, (−a − 1, a + 1) × ∂ω) satisfying Neumann condition on {−a − 1} ×ω and {a + 1} × ω. The inequality (3.3) implies that

HL � HL := H−L ⊕ H0

L ⊕ H+L. (3.4)

It is easy to see that H(±)

L are self-adjoint operators and σ(H±L) = σess(H±

L) =[ν1, +∞). The self-adjoint operator H0

L is lower semibounded due to (2.3),its spectrum is purely discrete and this is why it has finitely many eigenval-ues in (−∞, ν1]. Hence, the discrete spectrum of HL contains finitely manyeigenvalues. Due to (3.4) and the minimax principle we can claim that the k-theigenvalue of HL is estimated from above by the k-th eigenvalue of HL. Theformer having finitely many discrete eigenvalues, the same is true for HL. ��

The statement of Theorem 2.1 follows from Lemmas 3.1, 3.2, if one choosesL = L+, � = �+; L = L−, � = �−; L = Ll, � = ω × (−l − a−, l + a+).

4 Analysis of H±

In this section we establish certain properties of the operators H± which willbe employed in the proof of Theorems 2.2–2.7.

By Sδ we indicate the set of all complex numbers separated from the half-line [ν1, +∞) by a distance greater than δ. We choose δ so that σdisc(H±) ⊂ Sδ .

Lemma 4.1 The operator (H±−λ)−1 : L2(�)→W22,0(�) is bounded and mero-

morphic w.r.t. λ ∈ Sδ . The poles of this operator are the eigenvalues of H±. Forany λ close to p-multiple eigenvalue λ∗ of H± the representation

(H± − λ)−1 = −p∑

j=1

ψ±j (·, ψ±

j )L2(�)

λ − λ∗+ T ±

4 (λ) (4.1)

holds true. Here ψ±j are the eigenfunctions associated with λ± and orthonormal-

ized in L2(�), while T ±4 (λ) : L2(�) → W2

2,0(�) is a bounded operator beingholomorphic w.r.t. λ in a small neighbourhood of λ±. The relations

(T ±

4 (λ) f, ψ±j

)

L2(�)= 0, j = 1, . . . , p, (4.2)

are valid.

Proof According to [17, Ch. V, Sec. 3.5], the resolvent (H± − λ)−1 consideredas an operator L2(�) is bounded and meromorphic w.r.t. λ ∈ Sδ and its polesare the eigenvalues of H±. The same is true, if we consider the resolvent as theoperator from L2(�) into W2

2,0(�); this fact follows from the obvious identity

(H± − λ − η)−1 − (H± − λ)−1 = η(H± − λ)−1(I − η(H± − λ)−1)−1(H± − λ)−1.

166 D. Borisov

Consider λ ranging in a small neighbourhood of λ∗. The formula (3.21) in [17,Ch. V, Sec. 3.5] gives rise to (4.1), where the operator T ±

4 (λ) : L2(�) → L2(�)

is bounded and holomorphic w.r.t. λ. The self-adjointness of H± and (4.1) yieldthat for any f ∈ L2(�) the identities (4.2) and

T ±4 (λ) f = (H± − λ)−1 f , f = f −

p∑

j=1

ψ±j ( f, ψ±

j )L2(�) (4.3)

hold true. Let �⊥ be a subspace of the functions in L2(�) which are orthogonalto ψ±

j , j = m, . . . , m + p − 1. The identities (4.3) mean that

T ±4 (λ)

∣∣�⊥ = (H± − λ)−1

∣∣�⊥ . (4.4)

The operator (H± − λ)−1∣∣�⊥ is holomorphic w.r.t. λ as an operator from �⊥

into W22,0(�) ∩ �⊥. This fact is due to holomorphy in λ of the operator (H± −

λ) : W22,0(�) ∩ �⊥ → �⊥ and the invertibility of this operator (see [17, Ch. VII,

Sec. 1.1]). Therefore, the restriction of T ±4 (λ) on �⊥ is holomorphic w.r.t. λ as

an operator into W22,0(�). Taking into account (4.3), we conclude that for any

f ∈ L2(�) the function T ±4 (λ) f is holomorphic w.r.t. λ. The holomorphy in a

weak sense implies the holomorphy in the norm sense [17, Ch. VII, Sec. 1.1],and we arrive at the statement of the lemma. ��

Let 0 < ν1 < ν2 � . . . � ν j � . . . be the eigenvalues of the negative DirichletLaplacian in ω taken in a non-decreasing order counting multiplicity, and φi =φi(x′) be the associated eigenfunctions orthonormalized in L2(ω). We denotes j(λ) := √

ν j − λ, where the branch of the roof is specified by the requirement√1 = 1. We remind that �±

a := � ∩ {x : ±x1 > ±a}.

Lemma 4.2 Suppose that u ∈ W12(�

±a ) is a solution to the boundary value

problem

(� + λ)u = 0, x ∈ �±a , u = 0, x ∈ ∂� ∩ �

±a ,

where λ ∈ Sδ . Then the function u can be represented as

u(x) =∞∑

j=1

α je−s j(λ)(±x1−a)φ j(x′), α j =∫

ω

u(a, x′)φ j(x′) dx′. (4.5)

The series (4.5) converges in W p2 (�±

b ) for any �±b ⊂ �±

a , p � 0. The coefficientsα j satisfy the identity

∞∑

j=1

|α j|2 = ‖u(·, a)‖2L2(ω). (4.6)


Proof In view of the obvious change of variables it is sufficient to prove thelemma for �+

0 . It is clear that v ∈ C∞(�+0 \ ω0), where ω0 := {0} × ω. Since

v ∈ W12(�

+0 ) and v(x, λ) = 0 as x ∈ ∂�+

0 \ ω0, the representation

u(x, λ) =∞∑

j=1

φ j(x′)u j(x1, λ), u j(x1, λ) :=∫

ω

u(x1, t, λ)φ j(t) dt,

holds true for any x1 � 0 in L2(ω). We have the identity

∞∑

j=1

|u j(x1, λ)|2 = ‖u(x1, ·, λ)‖2L2(ω) (4.7)

for any x1 � 0. Employing the equation for u we obtain

d2u j

dx21

= −∫

ω

φ j(x′) (�x′ + λ) u(x, λ) dx′

= −∫

ω

u(x, λ) (�x′ + λ) φ j(x′) dx′ = s2ju j

as x1 > 0. Since v → W12(�

+0 ), the identity obtained implies that

u j(x1, λ) = α je−√

ν j−λx1 . (4.8)

We have employed here that the functions u j are continuous at 0 since

∣∣u j(t, λ) − u j(0, λ)

∣∣ �

∣∣∣∣

∫

ω

t∫

0

∂u∂x1

φ j dx

∣∣∣∣ �

√|ω||t|∥∥∥∥

∂u∂x1

∥∥∥∥

L2(�+0 )

.

The identities (4.7), (4.8) yield (4.6). For x1 � b > 0 the coefficients of theseries in (4.5) decays exponentially as j → +∞, that implies the convergenceof this series in W p

2 (�+b ), p � 0. ��

For l � a− + a+ we introduce the operators T ±6 (λ, l) : L2(�, �∓) →

L2(�, �±),

T ±6 (λ, l) := L±S(±2l)(H∓ − λ)−1. (4.9)

Lemma 4.3 The operator T ±6 is bounded and meromorphic w.r.t. λ ∈ Sδ . For

any compact set K ⊂ Sδ separated from σdisc(H∓) by a positive distance theestimates

∥∥∥∥∂ iT ±

6

∂λi

∥∥∥∥ � Clie−2l Re s1(λ), i = 0, 1, λ ∈ K, (4.10)

168 D. Borisov

hold true, where the constant C is independent of λ ∈ K and l � a− + a+. Forany λ close to a p-multiple eigenvalue λ∗ of H∓ the representation

T ±6 (λ, l) = −

p∑

j=1

ϕ∓j (·, ψ∓

j )L2(�∓)

λ − λ∗+ T ±

7 (λ, l), ϕ∓j := L±S(±2l)ψ∓

j , (4.11)

is valid. Here ψ∓j are the eigenfunctions associated with λ∗ and orthonormalized

in L2(�), while the operator T ±7 (λ, l) : L2(�, �∓) → L2(�, �±) is bounded

and holomorphic w.r.t. λ close to λ∗ and satisfies the estimates∥∥∥∥∂ iT ±

7

∂λi

∥∥∥∥ � Cli+1e−2l Re s1(λ), i = 0, 1, (4.12)

where the constant C is independent of λ close to λ∗ and l � a− + a+. Theidentities

T ±7 (λ∗, l) = L±S(±2l)T ∓

4 (λ∗) (4.13)

hold true.

Proof We prove the lemma for T +6 only; the proof for T −

6 is similar. Let f ∈L2(�, �−), and denote u := (H− − λ)−1 f . The function f having a compactsupport, by Lemma 4.2 the function u can be represented as the series (4.5) forx1 � a−. Hence,

(S(2l)u

)(x) =

∞∑

j=1

α je−2s j(λ)le−s j(λ)(±x1−a)φ j(x′).

Employing this representation and (4.6), we obtain∥∥∥∥

∞∑

j=1

α je−2s j(λ)le−s j(λ)(x1−a−)φ j(x′)∥∥∥∥

W22 (�+)

� C∞∑

j=1

|α j|e−2l Re s j(λ)‖e−s j(λ)(·−a−)‖W22 (−a+,a+)×

× (‖�x′φ j‖L2(ω) + ‖φ j‖L2(ω)

)(4.14)

� C∞∑

j=1

|α j||s j(λ)|3/2ν je−(2l−a−−a+) Re s j(λ)

� C

⎛

⎝∞∑

j=1

|α j|2⎞

⎠

1/2 ⎛

⎝∞∑

j=1

(ν

7/2j + |λ|7/2

)e−2(2l−a−−a+) Re s j(λ)

⎞

⎠

1/2

� C(1 + |λ|7/4

)e−(2l−a−−a+) Re s1(λ)‖u(a, ·)‖L2(ω), (4.15)

where the constant C is independent of λ ∈ Sδ and l � a− + a+. Here we havealso applied the well-known estimate [20, Ch. IV, Sec. 1.5, Theorem 5]

cj2

n−1 � ν j � Cj2

n−1 . (4.16)


By direct calculations we check that

(∂S(2l)u

∂λ

)(x, λ, l) =

∞∑

j=1

α j

2s j(λ)(x1 − a− + 2l)e−2s j(λ)le−s j(λ)(x1−a−)φ j(x′).

Proceeding in the same way as in (4.15) we obtain∥∥∥∥∂S(2l)u

∂λ

∥∥∥∥

W22 (�+)

� Cl(1 + |λ|5/4

)e−(2l−a−−a+) Re s1(λ)‖u(a, ·)‖L2(ω), (4.17)

where the constant C is independent of λ ∈ Sδ and l � a− + a+. In the sameway we check that

∥∥∥∥∂2S(2l)u

∂λ2

∥∥∥∥

W22 (�+)

� Cl(1 + |λ|3/4) e−(2l−a−−a+) Re s1(λ)‖u(a, ·)‖L2(ω), (4.18)

where the constant C is independent of λ ∈ Sδ and l � a− + a+. Lemma 4.1implies

u(a−, ·) = −p∑

j=1

ψ−j (a−, ·)( f, ψ−

j )L2(�)

λ − λ∗+ (

T −4 (λ) f

)(a−, ·).

This representation and (4.6), (4.15), (4.17), (4.18) lead us to the statement ofthe lemma. ��

5 Reduction of the Eigenvalue Equation for Hl

In this section we reduce the eigenvalue equation

Hlψ = λψ (5.1)

to an operator equation in L2(�−) ⊕ L2(�+). The reduction will be one of thekey ingredients in the proofs of Theorems 2.2–2.7. Hereafter we assume thatl � a− + a+.

Let f± ∈ L2(�, �±) be a pair of arbitrary functions, and let functions u±satisfy the equations

(H± − λ)u± = f±, (5.2)

where λ ∈ Sδ . We choose δ so that σdisc(H±) ⊂ Sδ . We construct a solution to(5.1) as

ψ = S(l)u− + S(−l)u+. (5.3)

Suppose that the function ψ defined in this way satisfies (5.1). We substitute(5.3) into (5.1) to obtain

0 = (Hl − λ)ψ = (Hl − λ)S(l)u− + (Hl − λ)S(−l)u+.

170 D. Borisov

By direct calculations we check that

(Hl − λ)S(l)u− =S(l)(−�(D) + L− − λ)S(−l)S(l)u− ++ S(−l)L+S(2l)u− = S(l) f− + S(−l)L+S(2l)u−,

(Hl − λ)S(−l)u+ =S(−l) f+ + S(l)L−S(−2l)u+.

Hence,

S(l)(

f− + L−S(−2l)u+) + S(−l)

(f+ + L+S(2l)u−

) = 0.

Since the functions f± + L±S(±2l)u∓ are compactly supported, it follows thatthe functions S(∓l)

(f± + L±S(±2l)u∓

)are compactly supported, too, and

their supports are disjoint. Thus, the last equation obtained is equivalent tothe pair of the equations

f− + L−S(−2l)u+ = 0, f+ + L+S(2l)u− = 0. (5.4)

These equations are equivalent to (5.1). The proof is the subject of

Lemma 5.1 To any solution f := ( f−, f+) ∈ L2(�−) ⊕ L2(�+) of (5.4) andfunctions u± solving (5.2) there exists the unique solution of (5.1) given by (5.2),(5.3). For any solution ψ of (5.1) there exists the unique f ∈ L2(�−) ⊕ L2(�+)

solving (5.4) and the unique functions u± satisfying (5.2) such that ψ is given by(5.2), (5.3). The equivalence holds for any λ ∈ Sδ .

Proof It was shown above that if f ∈ L2(�−) ⊕ L2(�+) solves (5.4), andthe functions u± are the solutions to (5.2), the function ψ defined by (5.3)solves (5.1).

Suppose that ψ is a solution of (5.1). This functions satisfies the equation

(−� − λ)ψ = 0, −l + a− < x1 < l − a+, x′ ∈ ω, (5.5)

and vanishes as −l + a− < x1 < l − a+, x′ ∈ ∂ω. Due to standard smoothnessimproving theorems (see, for instance, [20, Ch. IV, Sec. 2.2]) it implies thatψ ∈ C∞({x : −l + a− < x1 < l − a+, x′ ∈ ω}). Hence, the numbers

ρ±j = ρ±

j (l, λ) := (1/2)

∫

ω

ϒ±(x′, l, λ)φ j(x′) dx′, ϒ± :=(

ψ ± 1

s j

∂ψ

∂x1

) ∣∣∣∣x1=0

,

are well-defined. Employing (5.5), the identity ψ = 0 as x ∈ ∂�, and thesmoothness of ψ we integrate by parts,

ρ±j = − 1

2ν j

∫

ω

ϒ±�x′φ j dx′ = − 1

2ν j

∫

ω

φ j�x′ϒ± dx′

= 1

2ν j

∫

ω

φ j

(∂2

∂x21

+ λ

)ϒ± dx′ = 1

2νpj

∫

ω

φ j

(∂2

∂x21

+ λ

)p

ϒ± dx′


for any p ∈ N. In view of (4.16) it yields that∞∑

j=1jp|ρ±

j | < ∞ for any p ∈ N.

Now we introduce the functions u±,

u±(x1 ∓ l, x′, l) :=∞∑

j=1

ρ±j (l)e±s j(λ)x1φ j(x′), x ∈ �∓

0 ,

u±(x1 ∓ l, x′, l) := ψ(x, l) − u∓(x1 ± l, x′, l), x ∈ �±0 ,

and conclude that they satisfy (5.3), and

u−(x1 + l, x′, l), u+(x1 − l, x′, l) ∈ C∞(�±

0

)∩ W2

2,0

(�±

0 , ∂� ∩ ∂�±0

).

The smoothness of ψ gives rise to the representations

ψ∣∣x1=0 =

∞∑

j=1

(ρ+

j + ρ−j

)φ j,

∂ψ

∂x1

∣∣∣∣x1=0

=∞∑

j=1

(ρ+

j − ρ−j

)s jφ j.

These relations and the aforementioned smoothness of u± imply that u± ∈W2

2,0(�). We define a vector f =( f−; f+)∈ L2(�−) ⊕ L2(�+) by f− := −L−u+(x1 − 2l, x′, l), f+ := −L+u−(x1 + 2l, x′, l). Let us check that the functions u±satisfy (5.2); it will imply that f solves (5.6).

The definition of u± yields

(� + λ)u±(x1 ∓ l, x′, l) = 0, x ∈ �∓0 .

Thus,

(� + λ)u− = 0, x ∈ �+l , (� + λ)u+ = 0, x ∈ �

−−l.

These equations, the equation for ψ , and the definitions of u−(x1 + l, x′, l) forx ∈ �−

0 , follow that for x ∈ �−l

(−� − λ + L−)u−(x, l)

= (−� − λ + L−)(ψ(x1 − l, x′, l) − u+(x1 − 2l, x′, l)

) == −(−� − λ + L−)u+(x1 − 2l, x′, l) = −L−u+(x1 − 2l, x′, l) = f−(x).

Therefore, (H− − λ)u− = f−. The relation (H+ − λ)u+ = f+ can be establishedin the same way. ��

Suppose that λ ∈ Sδ \ σ∗. In this case u± = (H± − λ)−1 f±. This fact togetherwith the definition (4.9) of T ±

6 implies that L±S(±2l)u∓ = T ±6 (λ, l) f±. Substi-

tuting the identity obtained into (5.4), we arrive at the equation

f + T2(λ, l) f = 0, (5.6)

where f := ( f−; f+) ∈ L2(�−) ⊕ L2(�+), where the operator T2 : L2(�−) ⊕L2(�+) → L2(�−) ⊕ L2(�+) is defined in (2.4).

Proof of Theorem 2.2 The inequality (3.3) yields that the operator Hl is lowersemibounded with lower bound −c1. Together with Theorem 2.1 it implies that

172 D. Borisov

the discrete eigenvalues of the operator are located in [−c1, ν1). The set Kδ :=[−c1, ν1 − δ) \ ⋃

λ∈σ∗(λ − δ, λ + δ) satisfies the hypothesis of Lemma 4.3, and the

estimate (4.10) implies

‖T2(λ, l)‖ � C(δ)e−2l Re s1(λ) < 1,

if l is large enough and λ ∈ Kδ . Therefore, (5.6) has no nontrivial solutions,if l is large enough and λ ∈ Kδ . Since Kδ ∩ σ∗ = ∅, the identity f = 0 impliesthat u± = (H± − λ)−1 f± = 0, i.e., ψ = 0. Thus, (5.1) has no nontrivial solutionfor λ ∈ Kδ , i.e., Kδ ∩ σdisc(Hl) = ∅, if l is large enough. The number δ beingarbitrary, the last identity completes the proof. ��

Assume that λ∗ ∈ σ∗ is (p− + p+)-multiple. Lemma 4.3 implies that for λ

close to λ∗ the representation (2.5) holds true, where

T3(λ, l) f := (T −7 (λ, l) f+; T +

6 (λ, l) f−), if p− = 0,

T3(λ, l) f := (T −6 (λ, l) f+; T +

7 (λ, l) f−), if p+ = 0,

T3(λ, l) f := (T −7 (λ, l) f+; T +

7 (λ, l) f−), if p± �= 0.

Lemma 4.3 yields also that the operator T3(λ, l) is bounded and holomorphicw.r.t. λ in a small neighbourhood of λ∗, and satisfies the estimate

∥∥∥∥∂ iT3

∂λi

∥∥∥∥ � Cli+1e−2l Re s1(λ), i = 0, 1, (5.7)

where the constant C is independent of l � a− + a+ and λ close to λ∗.Suppose that λ �= λ∗ is an eigenvalue of Hl converging to λ∗. In this case the

identity f = 0 leads us to the relations u± = (H± − λ)−1 f± = 0, ψ = 0. Thus,the corresponding equation (5.6) has a nontrivial solution. Let us solve thisequation.

We substitute (2.5) into (5.6), and obtain

f − 1

λ − λ∗

p∑

i=1

φiT (i)1 f + T3 f = 0. (5.8)

In view of the estimate (5.7) the operator (I + T3(λ, l)) is invertible, and theoperator (I + T3)

−1 is bounded and holomorphic w.r.t. λ in a small neighbour-hood of λ∗. Applying this operator to the last equation gives rise to one moreequation,

f = 1

λ − λ∗

p∑

i=1

�iT (i)1 f , (5.9)

where �i = �i(·, λ, l) := (I + T3(λ, l))−1φi(·, l). We denote ki = ki(λ, l) :=(λ − λ∗)−1T (i)

1 f , and apply the functionals T ( j)1 , j = 1, . . . , p, to the equation

(5.9) that leads us to (2.12), where k := (k1 . . . kp)t.

Given a non-trivial solution of (5.6), the associated vector k is non-zero, since otherwise the definition of ki, and (5.9) would imply that f = 0.


Therefore, if λ �= λ∗ is an eigenvalue of Hl, the system (2.12) has a non-trivialsolution. It is true, if and only if (2.7) holds true. Thus, each eigenvalue of Hl

converging to λ∗ and not coinciding with λ∗ should satisfy this equation.Let us show that if λ∗ is an eigenvalue of Hl, it satisfies (2.7) as well. In this

case λ∗ the associated eigenfunction is given by (5.3) that is due to Lemma 5.1.The self-adjointness of H± implies

(f±, ψ±

i

)L2(�±)

= (u±, (H± − λ∗) ψ±

i

)L2(�±)

= 0, (5.10)

i = 1, . . . , p±. Therefore, the functions u± can be represented as

u−(·, l) = T −4 (λ∗) f− −

p−∑

i=1

kiψ−i , u+(·, l) = T +

4 (λ∗) f+ −p+∑

i=1

ki+p−ψ+i ,

(5.11)where ki are numbers to be found. Employing the relation (4.13) and substi-tuting (5.11) into (5.4), we obtain

f + T3(λ∗, l) f =p∑

i=1

kiφi(·, l), f =p∑

i=1

ki�i(·, λ∗, l). (5.12)

The relations (5.10) can be rewritten as T (i)1 f = 0, i = 1, . . . , p, that together

with the second identity in (5.12) implies the system (2.12) for λ = λ∗. Thevector k is non-zero since otherwise the second relation in (5.12) and (5.11)would imply f = 0, u± = 0, ψ = 0. Thus, det A(λ∗, l) = 0, which coincides with(2.7) for λ = λ∗.

Let λ be a root of (2.7), converging to λ∗ as l → +∞. We are going to provethat in this case (5.1) has a non-trivial solution, i.e., λ is an eigenvalue of Hl. Theequation (5.1) being satisfied, it follows that the system (2.12) has a nontrivialsolution k. We specify this solution by the requirement

‖k‖Cp = 1, (5.13)

and define f :=p∑

i=1ki�i(·, λ, l) ∈ L2(�−) ⊕ L2(�+). The system (2.12) and the

definition of T (i)1 give rise to the identities

( f−, ψ−i )L2(�−) = T (i)

1 f = (λ − λ∗)ki, i = 1, . . . , p−,

( f+, ψ+i )L2(�−) = T (i+p−)

1 f = (λ − λ∗)ki+p− , i = 1, . . . , p+. (5.14)

Taking these identities into account and employing (4.1), in the case λ �= λ∗ wearrive at the formulas

u− = −p−∑

i=1

kiψ−i + T −

4 (λ) f−, u+ = −p+∑

i=1

ki+p−ψ+i + T +

4 (λ) f+. (5.15)

In the case λ = λ∗ we adopt these formulas as the definition of the functionsu± that is possible due to (4.4) and (5.14) with λ = λ∗.

174 D. Borisov

If λ �= λ∗, we employ (2.12) to check by direct calculations that f solves(5.9), and thus (5.4). If λ = λ∗, the identity (4.13) implies

(L−S(−2l)u+;L+S(2l)u−

) = T3(λ∗, l) f −p∑

i=1

kiφi(·, l) = T3(λ∗, l) f −

−p∑

i=1

ki(I + T3(λ∗, l))�i(·, λ∗, l)

= T3(λ∗, l) f − (I + T3(λ∗, l)) f = − f .

Hence, (5.4) holds true. With Lemma 5.1 in mind we therefore conclude thatthe function ψ defined by (5.3) solves (5.1).

Let us prove that ψ �≡ 0; it will imply that λ is an eigenvalue of Hl.Lemma 2.1 implies that the functions ϕ±

i obey the estimate

‖ϕ±i ‖L2(�∓) � Ce−2ls1(λ∗), (5.16)

where the constant C is independent of l. This estimate together with (5.7)gives rise to the similar estimates for �i:

‖�i‖L2(�−)⊕L2(�+) � Ce−2ls1(λ∗),∥∥∥∥

∂�i∂λ

∥∥∥∥

L2(�−)⊕L2(�+)

� Cl2e−2l(s1(λ∗)+s1(λ)), (5.17)

where the constant C is independent of λ and l. The latter inequality is basedon the formula

∂

∂λ(I + T3)

−1 = −(I + T3)−1 ∂T3

∂λ(I + T3)

−1.

Hence, the asymptotics (2.11) is valid. The vector k being non-zero, it impliesψ �≡ 0. We summarize the results of the section in

Lemma 5.2 The eigenvalues of Hl converging to a (p− + p+)-multiple λ∗ ∈ σ∗coincide with the roots of (2.7) converging to λ∗. The associated eigenfunctionsare given by (5.3), (5.11), (5.15), where the coefficients ki form non-trivialsolutions to (2.12). If λ(l) is an eigenvalue of Hl converging to λ∗ as l → +∞, itsmultiplicity coincides with the number of linear independent solutions of (2.12)taken for λ = λ(l). The associated eigenfunctions satisfy (2.11).

6 Proof of Theorem 2.3

Throughout this and next sections the parameter λ is assumed to belong to asmall neighbourhood of λ∗, while l is supposed to be large enough. We beginwith the proof of Lemma 2.1.

Proof of Lemma 2.1 We will prove the lemma for ψ−i only, the case of ψ+

i iscompletely similar. According to Lemma 4.2 the functions ψ−

i can be repre-sented as the series (4.5) in �+

a− . Let � be the space of the L2(�)-functions


spanned over ψ−i . Each function from this space satisfies the representation

(4.5) in �+a− . We introduce two quadratic forms in this finite-dimensional space,

the first being generated by the scalar product in L2(�), while the other isdefined as q(u, v) := α1[u]α1[v], where the α1[u], α1[v] are the first coefficientsin the representations (4.5) for u and v in �+

a− . By the theorem on simultaneousdiagonalization of two quadratic forms, we conclude that we can choose thebasis in � so that both these forms are diagonalized. Denoting this basis asψ−

i , we conclude that these functions are orthonormalized in L2(�), and

q

(ψ−

i , ψ−j

)= 0, if i �= j. (6.1)

Suppose that for all the functions ψ−i the coefficient α1[ψ−

i ] is zero. In thiscase we arrive at (2.17), where β− = 0. If at least one of the functions ψ−

i has anonzero coefficient α1, say ψ−

1 , the identity (6.1) implies that α1[ψ−i ] = 0, i � 2,

and we arrive again at (2.17). ��

The definition of Aij and (5.17) imply the estimates

|Aij(λ, l)| � Ce−2ls1(λ∗),

∣∣∣∣∂ Aij

∂λ(λ, l)

∣∣∣∣ � Cl2e−2l(s1(λ∗)+s1(λ)), (6.2)

where the constant C is independent of λ and l. The holomorphy of T3 w.r.t. λ

yields that the functions Aij are holomorphic w.r.t. λ. This fact and the estimate(6.2) allow us to claim that the right hand side of (2.7) reads as follows,

F(λ, l) := det((λ − λ∗)E + A(λ, l)

) = (λ − λ∗)p +p−1∑

i=0

Pi(λ, l)(λ − λ∗)i,

where the functions Pi are holomorphic w.r.t. λ, and obey the estimate

|Pi(λ, l)| � Ce−2(p−i)ls1(λ∗) (6.3)

with the constant C independent of λ and l. Given δ > 0, this estimate implies

∣∣∣∣

p−1∑

i=0

Pi(λ, l)(λ − λ∗)i

∣∣∣∣ < |λ − λ∗|p as |λ − λ∗| = δ,

if l is large enough. Now we employ Rouché theorem to infer that the functionF(λ, l) has the same amount of the zeroes inside the disk {λ : |λ − λ∗| < δ}as the function λ �→ (λ − λ∗)p does. Thus, the function F(λ, l) has exactly pzeroes in this disk (counting their orders), if l is large enough. The numberδ being arbitrary, we conclude that (2.7) has exactly p roots (counting theirorders) converging to λ∗ as l → +∞.

Lemma 6.1 Suppose that λ1(λ) and λ2(λ) are different roots of (2.7), andk1(l) and k2(l) are the associated non-trivial solutions to (2.12) normalized by(5.13). Then

(k1(l), k2(l)

)Cp = O

(le−2ls1(λ∗)

), l → +∞.

176 D. Borisov

Proof According to Lemma 5.2, the numbers λ1(l) �= λ2(l) are the eigenvaluesof Hl, and the associated eigenfunctions ψi(x, l), i = 1, 2, are generated by(5.3), (5.11), (5.15), where ki are components of the vectors k1, k2, respectively.

Using the representations (4.5) for ψ+i in �±

±a+ and for ψ−i in �±

±a− , by directcalculations one can check that

(ψ+

i ,S(2l)ψ−j

)L2(�)

= O(le−2ls1(λ∗)

), l → +∞.

The operator being self-adjoint, the eigenfunctions ψi(x, l) are orthogonal inL2(�). Now by Lemma 5.2 and the last identity we obtain

0 = (ψ1, ψ2)L2(�) =p∑

i=1

k(1)

i k(2)

i + O(e−2ls1(λ∗)

), l → +∞,

that completes the proof. ��

For each root λ(l) −−−−→l→+∞

λ∗ of (2.7) the system (2.12) has a finite number of

linear independent solutions. Without loss of generality we assume that thesesolutions are orthonormalized in C

p. We consider the set of all such solutionsassociated with all roots of (2.7) converging to λ∗ as l → +∞, and indicatethese vectors as ki = ki(l), i = 1, . . . , q. In view of the assumption for ki justmade and Lemma 6.1 the vectors ki satisfy (2.13).

For the sake of brevity we denote B(λ, l) := (λ − λ∗)E − A(λ, l).

Lemma 6.2 Let λ(l) −−−−→l→+∞

λ∗ be a root of (2.7) and ki, i = N, . . . , N + m, m �0, be the associated solutions to (2.12). Then for any h ∈ C

p the representation

B−1(λ, l)h =N+m∑

i=N

T (i)8 (l)h

λ − λ(l)ki(l) + T9(λ, l)h

holds true for all λ close to λ(l). Here T (i)8 : C

p → C are functionals, while thematrix T9(λ, l) is holomorphic w.r.t. λ in a neighbourhood of λ(l).

Proof The matrix B being holomorphic w.r.t. λ, the inverse B−1 is mero-morphic w.r.t. λ and has a pole at λ(l). The residue at this pole is a linearcombination of ki, and for any h ∈ C

p we have

B−1(λ, l)h = 1

(λ − λ(l))s

N+m∑

i=N

kiT (i)8 (l)h + O

((λ − λ(l))−s+1

), λ → λ(l),

(6.4)where s � 1 is the order of the pole, and T (i)

8 : Cp → C are some functionals.

We are going to prove that s = 1. Let g± = g±(x) ∈ L2(�, �±) be arbitraryfunctions. We consider the equation

(Hl − λ)u = g := S(−l)g+ + S(l)g−, (6.5)


where λ is close to λ(l) and λ �= λ(l), λ �= λ∗. The results of [17, Ch. VII,Sec. 3.5] imply that for such λ the function u can be represented as

u = − 1

λ − λ(l)

N+m∑

i=N

(g, ψi)L2(�)ψi + T10(λ, l)g, (6.6)

where ψi are the eigenfunctions of Hl associated with λ(l) and orthonormalizedin L2(�), while the operator T10(λ, l) is bounded and holomorphic w.r.t. λ closeto λ(l) as an operator in L2(�). The eigenvalue λ(l) of the operator Hl is (m +1)-multiple by the assumption and Lemma 5.2. Completely by analogy with theproof of Lemma 5.1 one can make sure that (6.5) is equivalent to

f + T2(λ, l) f = g, (6.7)

where g := (g−, g+) ∈ L2(�−) ⊕ L2(�+), and the solution of (6.5) is given by

u = S(l)u− + S(−l)u+, u± = (H± − λ)−1 f±. (6.8)

Proceeding as in (5.6), (5.8), (5.9), one can solve (6.7),

f =p∑

i=1

Ui�i + ˜f , U = B−1(λ, l)h, ˜f := (I + T3)−1 g, (6.9)

where Ui := T (i)1 f , and the vector U := (U1 . . . U p)

t is a solution to

B(λ, l)U = h, (6.10)

h = (h1 . . . hp)t, hi := T (i)

1 (λ)(I + T3(λ, l))−1 g (6.11)

Knowing the vectors U and f , we can restore the functions u± by (4.1),

u−(·, λ, l) = −p−∑

i=1

Ui(λ, l)ψ−i −

p∑

i=1

Ui(λ, l)T −4 (λ, l)�−

i + T −4 (λ) f−,

u+(·, λ, l) = −p+∑

i=1

Ui+p−(λ, l)ψ+i −

p∑

i=1

Ui(λ, l)T +4 (λ, l)�+

i + T +4 (λ) f+,

where �±i and f± are introduced as �i = (�−

i ; �+i ), ˜f = ( f−; f+). In these

formulas we have also employed (6.10) in the following way:

( f−, ψ−j )L2(�) = T ( j)

1 f = h j +p∑

i=1

A jiUi = (λ − λ∗)U j, j = 1, . . . , p−,

( f+, ψ+j )L2(�) = T ( j+p−)

1 f =h j+p− +p∑

i=1

A jiUi =(λ−λ∗)U j+p− , j=1, . . . , p+.

The estimates (5.17) allow us to infer that

‖T ±4 (λ, l)�±

i ‖L2(�) = O(e−2ls1(λ∗)

),

178 D. Borisov

while in view of holomorphy of T ±4 and (5.7) we have

‖T ±4 (λ) f±‖L2(�) � C‖g‖L2(�−)⊕L2(�+),

where the constant C is independent of λ. Now we use the first of the relations(6.8) and obtain

u(·, λ, l) = −p−∑

i=1

Ui(λ, l)(S(l)ψ−

i + O(e−2ls1(λ∗)

))

−p+∑

i=1

Ui+p−(λ, l)(S(−l)ψ+

i + O(e−2ls1(λ∗)

)) + O(‖g‖L2(�−)⊕L2(�+)

).

(6.12)

Now we compare (6.6) and (6.12) and conclude that the coefficients Ui(λ, l),i = 1, . . . , p have a simple pole at λ(l). By (6.9) it implies that the vectorB−1(λ, l)h has a simple pole at λ(l), where h is defined by (6.11). It followsfrom (5.7) and (6.11) that for each h ∈ C

p there exists g ∈ L2(�−) ⊕ L2(�+)

so that h is given by (6.11). Together with (6.4) it completes the proof. ��

Lemma 6.3 The number λ(l) −−−−→l→+∞

λ∗ is a m-th order zero of F(λ, l) if and

only if it is a m-multiple eigenvalue of Hl .

Proof Let λ(i)(l) −−−−→l→+∞

λ∗, i = 1, . . . , M, be the different zeroes of F(λ, l), and

ri, i = 1, . . . , M, be the orders of these zeroes. By Lemma 5.2, each zero λ(i)(l)is an eigenvalue of Hl; its multiplicity will be indicated as mi � 1. To prove thelemma it is sufficient to show that mi = ri, i = 1, . . . , M.

Let us prove first that mi � ri. In accordance with Lemma 5.2 the multiplicitymi coincides with a number of linear independent solutions of (2.12) with λ =λ(i)(l). Hence,

rank B(λ(l), l) = p − mi. (6.13)

By the assumption

∂ j

∂λ jdet B(λ, l) = 0, j = 1, . . . , ri − 1,

∂ri

∂λridet B(λ, l) �= 0,

as λ = λ(i)(l). Let B j = B j(λ, l) be the columns of the matrix B, i.e., B =(B1, . . . , Bp). Employing the well-known formula

∂

∂λdet B = det

(∂ B1

∂λB2 . . . Bp

)+ det

(B1

∂ B2

∂λ. . . Bp

)+

+ det

(B1 B2 . . .

∂ Bp

∂λ

),


one can check easily that for each 0 � j � mi − 1

∂ j

∂λ jdet B(λ, l)

∣∣∣λ=λ(i)(l)

=∑

ς

cς det Bς ,

where cς are constants, and at least (p − mi + 1) columns of each matrix Bς arethose of B. In view of (6.13) these columns are linear dependent, and thereforedet Bς = 0 for each ς . Thus, mi − 1 � ri − 1 that implies the desired inequality.

It is sufficient to check that q =M∑

i=1mi =

M∑

i=1ri = p to prove that mi = ri.

Lemma 6.2 yields that for a given fixed δ small enough and l large enough

B−1(λ, l)h =q∑

i=1

T (i)8 (l)h

λ − λ(i)(l)ki(l) + T9(λ, l)h, (6.14)

for any h ∈ Cp and λ so that |λ − λ∗| = δ. It is also assumed that |λ(i)(l) − λ∗| <

δ for the considered values of l. We integrate this identity to obtain

1

2π i

∫

|λ−λ∗|=δ

B−1(λ, ls)h dλ =q∑

i=1

kiT (i)8 (l)h. (6.15)

Due to (6.2) we conclude that

1

2π i

∫

|λ−λ∗|=δ

B−1(λ, l) dλ −−−−→l→+∞

1

2π i

∫

|λ−λ∗|=δ

E dλ

λ − λ∗= E. (6.16)

Hence, the right hand side of (6.15) converges to h. By (2.13) it implies thatq = p for l large enough. ��

The statement of Theorem 2.3 follows from the proven lemma.

7 Asymptotics for the Eigenelements of Hl

In this section we prove Theorems 2.4–2.7. Throughout the section the hypoth-esis of Theorem 2.4 is assumed to hold true.

Theorem 2.3 implies that the number of the vectors ki introduced in theprevious section equals p. Let S = S(l) be the matrix with columns ki(l), i =1, . . . , p, i.e., S(l) = (k1(l) . . . kp(l)). Without loss of generality we can assumethat det S(l) � 0.

Lemma 7.1 det S(l) = 1 + O(le−2ls1(λ∗)), as l → +∞.

Proof The relations (2.13) yield S2(l) = E + O(le−2ls1(λ∗)), l → +∞, thatimplies det2 S(l) = 1 + O(le−2ls1(λ∗)). The last identity proves the lemma. ��

180 D. Borisov

Lemma 7.1 implies that there exists the inverse matrix S−1(l) for l largeenough.

Lemma 7.2 The matrix R(λ, l) := S−1(l)A(λ, l)S(l) reads as follows

R =

⎛

⎜⎜⎜⎝

λ1 − λ∗ + (λ − λ1)r11 (λ − λ2)r12 . . . (λ − λp)r1p

(λ − λ1)r21 λ2 − λ∗ + (λ − λ2)r22 . . . (λ − λp)r2p...

......

(λ − λ1)rp1 (λ − λ2)rp2 . . . λp − λ∗ + (λ − λp)rpp

⎞

⎟⎟⎟⎠

.

where λi = λi(l), while the functions rij = rij(λ, l) are holomorphic w.r.t. λ closeto λ∗ and obey the uniform in λ and l estimates

|rij(λ, l)| � Cl2e−2l(s1(λ∗)+s1(λ)). (7.1)

Proof The system (2.12) implies

A(λ, l)ki = A(λi(l), l)ki + (A(λ, l) − A(λi(l), l)

)ki

= (λi(l) − λ∗)ki + (A(λ, l) − A(λi(l), l)

)ki.

The matrix A(λ, l) − A(λi(l), l) is holomorphic w.r.t. λ, and

A(λ, λi(l), l) := A(λ, l) − A(λi(l), l) =λ∫

λi(l)

∂A∂λ

(z, l) dz. (7.2)

Due to (6.2) and (2.12) the last identity implies that

A(λ, l)ki = (λi(l) − λ∗)ki + (λ − λi)Ki(λ, l), (7.3)

where the vectors Ki(λ, l) are holomorphic w.r.t. λ close to λ∗ and satisfy theuniform in λ and l estimate

‖Ki‖Cp � Cl2e−2l(s1(λ∗)+s1(λ)).

By Lemma 6.3 the vectors ki, i = 1, . . . , p, form a basis in Cp. Hence,

Ki(λ, l) =p∑

j=1

rij(λ, l)k j(l),(ri1(λ, l) . . . rip(λ, l)

)t = S−1(l)Ki(λ, l). (7.4)

Due to Lemma 7.1, the relations (2.13), and the established properties of Ki

we infer that the functions rij are holomorphic w.r.t. λ and satisfy (7.1). Takinginto account (7.3) and (7.4), we arrive at the statement of the lemma. ��

Lemma 7.3 The polynomial det(τE − A(λ∗, l)

)has exactly p roots τi = τi(l),

i = 1, . . . , p counting multiplicity which satisfy (2.9).


Proof Since det(τE − A(λ∗, l)

)is a polynomial of p-th order, it has p roots

τi(l), i = 1, . . . , p, counting multiplicity. It is easy to check that

det(τE − A(λ∗, l)

) = τ p +p−1∑

j=0

P j(λ∗, l)τ j,

where the functions P j(λ∗, l) satisfy (6.3). We make a change of variable τi =zie−2ls1(λ∗), and together with the representation just obtained it leads us to theequation for zi,

zp +p−1∑

j=0

e−2l( j−p)s1(λ∗) P j(λ∗, l)z j = 0. (7.5)

Due to (6.3) the coefficients of this equation are bounded uniformly in l. ByRouché theorem it implies that all the roots of (7.5) are bounded uniformly inl. This fact yields (2.9). ��

In what follows the roots τi are supposed to be ordered in accordance with(2.10). We denote μi(l) := λi(l) − λ∗, i = 1, . . . , p.

Proof of Theorem 2.4 The formulas (2.9), (2.11) were established in Lemma7.3. Let us prove (2.8). Namely, let us prove that for each l large enough theroots of (2.7) can be ordered so that

τi(l) = μi(l)(

1 + O(

l2p e− 4l

p s1(λ∗)))

, l → +∞. (7.6)

Assume that this not true on a sequence λs → +∞. We introduce an equiva-lence relation ∼ on {μi(ls)}i=1,...,p saying that μi ∼ μ j, if

μi(ls) = μ j(ls)

(1 + O

(l

2p

s e− 4lsp s1(λ∗)

)), ls → +∞.

This relation divides all μi(ls) into disjoint groups,

{λ1(l), . . . , λp(l)} =q⋃

i=1

{λmi(l), . . . , λmi+1−1(l)},

where 1 = m1 < m2 < . . . < mq+1 = p + 1, λk ∼ λt, k, t = mi, . . . , mi+1 − 1,i = 1, . . . , q, and λk �∼ λt, if mi � k � mi+1 − 1, m j � t � m j+1 − 1, i �= j.Extracting if needed a subsequence from {ls}, we assume that mi and q areindependent of {ls}. Given k ∈ {1, . . . , p}, there exists i such that mi � k �mi+1 − 1. For the sake of brevity we denote m := mi, m := mi+1, m := m − m.To prove (2.8) it is sufficient to show that m roots τi, i = m, . . . , m − 1, countingmultiplicity of det

(τE − A(λ∗, l)

)satisfy (7.6).

Since

det(τE − A(λ∗, l)

) = det(S−1(l)(τE − A(λ∗, l))S(l)

) = det(τE − R(λ∗, l)

),

182 D. Borisov

due to Lemma 7.2 the equation for τi can be rewritten as

∣∣∣∣∣∣∣∣∣

τ − μ1(1 − r11) −μ2r12 . . . −μpr1p

−μ1r21 τ − μ2(1 − r22) . . . −μpr2p...

......

−μ1rp1 −μ2rp2 . . . τ − μp(1 − rpp)

∣∣∣∣∣∣∣∣∣

= 0, (7.7)

where μ j = μ j(l), r jk = r jk(λ∗, l). Assume first that μk(ls) = 0. In view of (2.6)it implies that m = m1, m = m2, and μ j(ls) = 0, j = m, . . . , m − 1. In this case(7.7) becomes

∣∣∣∣∣∣∣∣∣∣∣∣

τ . . . 0 −μmr1m . . . −μpr1p...

......

...

0 . . . τ −μmrm−1m . . . −μprm−1p...

......

...

0 . . . 0 −μmrpm . . . τ − μp(1 − rpp)

∣∣∣∣∣∣∣∣∣∣∣∣

= 0,

and it implies that zero is the root of det(τE − A(λ∗, l)

)of multiplicity at least

m. In this case the identities (7.6) are obviously valid.Assume now that μk(ls) �= 0. We seek the needed roots as τ = μk(ls)(1 + z).

We substitute this identity into (7.7) and divide then first (m − 1) columnsby −μk(ls), while the other columns are divided by the functions −μ j(ls)

corresponding to them. This procedure leads us to the equation

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

z1 − z . . .μm−1

μkr1m−1 r1m . . . r1p

......

......

μ1

μkrm−11 . . . zm−1 − z rm−1m . . . rm−1p

μ1

μkrm1 . . .

μm−1

μkrmm−1 zm − μk

μmz . . . rmp

......

......

μ1

μkrp1 . . .

μm−1

μkrpm−1 rpm . . . zp − μk

μpz

∣∣∣∣∣∣∣∣∣∣∣∣∣∣∣

= 0,

zi = zi(l) := μi(l)μk(l)

(1 − rii(λ∗, l)) − 1, i = 1, . . . , m − 1,

zi = zi(l) := 1 − μk(l)μi(l)

− rii(λ∗, l), i = m, . . . , p,

where the arguments of all the functions are l = ls, λ = λ∗. Due to (2.6) all thefractions in this determinant are bounded uniformly in ls. Using this fact and


(7.1), we calculate this determinant and write the multiplication of the diagonalseparately,

F1(z, ls)F2(z, ls)F3(z, ls) − F4(z, ls) = 0, (7.8)

F1(z, ls) :=m−1∏

i=1

(zi(ls) − z) , F2(z, ls) :=m−1∏

i=m

(zi(ls) − z) ,

F3(z, ls) :=p∏

i=m

(zi(ls) − μi(ls)

μk(ls)z)

, F4(z, ls) :=p−2∑

i=0

zici(ls)

where the coefficients ci obey the estimate

ci(ls) = O(

l2s e−4lss1(λ∗)

), ls → +∞. (7.9)

As it follows from the definition of the equivalence relation,

|μi − μk||μk| � C0ζ, ls → +∞, i = m, . . . , m − 1, ζ = ζ(ls) := l2

s e−4lss1(λ∗),

|μi − μk||μk| � θζ, i �∈ {m, . . . , m − 1}, θ = θ(ls) → +∞, ls → +∞,

where the constant C0 is independent of ls. These estimates and (7.1), (2.6)imply

|F1(z, l)| � C(θζ )m−1, |F3(z, l)| � C(θζ )p−m+1, |z| � 2C0ζ. (7.10)

The zeroes zi(ls) of F2(z, ls) satisfy |zi(ls)| � C0ζ , and hence |F2(z, ls)| � Cm0 ζ m

as |z| = C0ζ . Employing this estimate, (7.10) and rewriting (7.8) as

F2(z, ls) − F4(z, ls)

F1(z, ls)F3(z, ls)= 0,

by Rouché theorem we conclude that the last equation has exactly m rootscounting multiplicity in the disk {|z| < 2C0ζ }. We denote these roots as z( j)(ls),j = m, . . . , m − 1. It follows from (7.8), (7.9), (7.10) that

(min

i=m,...,m−1|z( j) − zi|

)m

� |F2(z( j), ls)| =∣∣∣∣

F4(z( j), ls)

F1(z( j), ls)F3(z( j), ls)

∣∣∣∣

� Cζ m(ls)

θ p−m(ls)� Cζ m(ls),

where z( j) = z( j)(ls), and the constant C is independent of ls. Hence, for each jthere exists index i, depending on ls, such that

z( j) = zi + O(ζ ), ls → +∞.

This identity, the definition of the equivalence relation and (7.1) imply

τ ( j) = μk(1 + z( j)) = μ j(1 − r jj) + O(μkζ ) = μ j + O(μ jζ )

that yields (7.6). ��

184 D. Borisov

In the proof of Theorem 2.5 we will employ the following lemma.

Lemma 7.4 For λ close to λ∗ and h ∈ Cp the representation

B−1(λ, l)h =p∑

i=1

T (i)10 (λ, l)hλ − λi(l)

ki

holds true. Here T (i)10 (λ, l) : C

p →C are functionals bounded uniformly in λ

and l.

Proof Lemma 6.2 implies that for λ close to λ∗ the identity (6.14) holds true forany h ∈ C

p, where ls should be replaced by l, q = p and λ(i) = λi. We introducethe vectors

k⊥i (l) := ki(l)

‖ki(l)‖, ki := ki −

p∑

j=1j�=i

(ki, k j

)Cp k j.

The relations (2.13) implies that the vectors k⊥i satisfy these relation as well.

Moreover, the vectors k⊥i form the orthogonal basis for {ki}. Bearing this fact

in mind, we multiply the relation (6.15) by k⊥j with ls replaced by l, q = p and

obtain

1

2π i

⎛

⎜⎝

∫

|λ−λ∗|=δ

B−1(λ, l)h dλ, k⊥j

⎞

⎟⎠

Cp

= T ( j)8 (l)h.

Due to (6.16) we conclude that the functionals T ( j)8 are bounded uniformly in l.

Let us prove that the matrix T9(λ, l) is bounded uniformly in λ and l. Due to(6.16) and the convergences λi(λ) → λ∗ we have

‖T9(λ, l)h‖Cp =∥∥∥∥∥

B−1(λ, l)h −p∑

i=1

T (i)8 (l)h

λ − λi(l)ki(l)

∥∥∥∥∥

Cp

� C‖h‖Cp,

as |λ − λ∗| = δ, if l is large enough. The constant C here is independent ofh and λ such that |λ − λ∗| = δ. The matrix T9 being holomorphic w.r.t. λ, bythe maximum principle for holomorphic functions the estimate holds true for|λ − λ∗| < δ, too. Thus, the matrix T9 is bounded uniformly in λ and l. We canexpand T9h in terms of the basis {ki},

T9(λ, l)h =p∑

i=1

kiT (i)11 (λ, l)h,

(T (1)

11 h . . . T (p)

11 h)t = S−1(l)T9(λ, l)h, (7.11)

where the functionals T (i)11 : C

p → C are bounded uniformly in λ and l. Substi-tuting (7.11) into (6.14) with ls replaced by l, q = p, λ(i) = λi, we arrive at thedesired representation, where T (i)

10 (λ, l) = T (i)8 (l) + (λ − λi(l))T (i)

11 (λ, l). ��


Proof of Theorem 2.5 Since the matrix A0(l) satisfies the condition (A), ithas p eigenvalues τ

(0)

i , i = 1, . . . , p, counting multiplicity. By hi = hi(l), i =1, . . . , p, we denote the associated eigenvectors normalized in C

p. Completelyby analogy with the proof of (2.9) in Lemma 7.3 one can establish (2.16). It iseasy to check that the vectors hi satisfy the identities

B(λ∗ + τ(0)

i , l)hi(l) = −A1(l)hi(l) − τ(0)

i A(λ∗, λ∗ + τ

(0)

i , l)

hi(l) := hi(l),

where, we remind, the matrix A was introduced in (7.2). Now we employ (6.2)to obtain

‖hi(l)‖Cp � ‖A1(l)‖ + C|τ (0)

i |l2e−2l(s1(λ∗)+s1 (λi(l)))

� ‖A1(l)‖ + C|τ (0)

i |l2e−4ls1(λ∗), (7.12)

where the constant C is independent of λ and l. Here we have also used theidentities

λi(l) = λ∗ + O(e−2ls1(λ∗)

), e−2ls1 (λi(l)) = O

(e−2ls1(λ∗)

), l → +∞,

which are due to (2.8), (2.9).Since hi(l) = B−1(λ∗ + τ

(0)

i (l), l)hi(l), Lemma 7.4 implies

h j(l) =p∑

i=1

T (i)10 (λ∗ + τ

(0)

j (l), l)hi(l)

τ(0)

i (l) + λ∗ − λ j(l)ki(l), j = 1, . . . , p.

The fractions in these identities are bounded uniformly in λ and l since h j arenormalized and

Q ji(l) := T (i)10 (λ∗ + τ

(0)

j (l), l)h j(l)

τ(0)

i (l) + λ∗ − λi(l)= (

h j(l), k⊥i (l)

)Cp,

where, we remind, the vectors k⊥i were introduced in the proof of Lemma 7.4.

We are going to prove that the roots τ(0)

j can be ordered so that the formulas(2.15) hold true. The matrices

Q :=⎛

⎜⎝

Q11 . . . Q1p...

...

Qp1 . . . Qpp

⎞

⎟⎠ , K =

(k⊥

1 . . . k⊥p

), H = (

h1. . . hp)t

,

satisfy the identity HK = Q. The basis {k⊥i } being orthogonal to {ki}, we

conclude that Kt = S−1. Now Lemma 7.1 and the condition (A) for A0(l) implythe uniform in l estimate | det Q| � C0 > 0. Hence, there exists a permutation

η = η(l), η =(

1 . . . pη1 . . . ηp

)such that

p∏

i=1

|Qiηi | � C0

p! . (7.13)

186 D. Borisov

This fact is proved easily by the contradiction employing the definition of thedeterminant. The normalization of hi and the identities (2.13) for k⊥

i imply that|Qiηi | � ‖hi‖Cp‖k⊥

ηi‖Cp = 1. Hence, by (7.13) we obtain

C0

|Q jη j|p! �p∏

i=1i �= j

|Qiηi | � 1,C0

p! � |Q jη j|.

Substituting the definition of Qiηi in this estimate, we have

|λi − λ∗ − τ(0)

i | � C∣∣∣T (i)

10 (λ∗ + τ (0)ηi

, l)hi

∣∣∣ � C

(‖A1‖ + |τ (0)ηi

|l2e−4ls1(λ∗)).

Here we have also used the boundedness of T (i)10 (see Lemma 7.4) and (7.12).

Rearranging the roots τ(0)

i as τ(0)

i := τ (0)ηi

, we complete the proof. ��

Proof of Theorem 2.8 Let us prove first that the matrix A(λ∗, l) obeys therepresentation (2.14), where all the elements of A0 are zero except onesstanding on the intersection of the first row and of (p− + 1)-th column and(p− + 1)-th column and the first row, and these elements are given by

A0,1p−+1(l) = A0,p−+11(l) = 2s1(λ∗)β−β+e−2ls1(λ∗).

Also we are going to prove that the corresponding matrix A1 satisfies theestimate

‖A1(l)‖ = O(le−4l√

ν1−λ∗), l → +∞. (7.14)

The definition of �i, the formulas (4.11) for ϕ±j , (5.16), (5.7) imply

�i(·, λ∗, l) = φi(·, l) + O(le−4ls1(λ∗)

), l → +∞, (7.15)

in L2(�−) ⊕ L2(�+)-norm. The identity (7.15) and the definition of Aij andT (i)

1 yield that for i, j = 1, . . . , p−

Aij(λ∗, l)=T (i)1 � j(·, λ∗, l)=T (i)

1 φ j(·, l)+O(le−4ls1(λ∗)

)=O(le−4ls1(λ∗)

), (7.16)

since T (i)1 φ j(·, l) = 0, i, j = 1, . . . , p−. In the same way one can check easily the

same identity for i, j = p− + 1, . . . , p. Taking into account the definition (4.11)of ϕ−

1 , and (2.17) by direct calculations we obtain

A1p−+1(λ∗, l) = (ψ+

1 , ϕ−1 (·, l)

)L2(�+)

= (ψ+

1 ,L+S(2l)ψ−1

)L2(�+)

== β−e−2ls1(λ∗)

(ψ+

1 ,L+e−s1(λ∗)x1φ1)

L2(�+)+ O

(e−2(s1l(λ∗)+s2(λ∗))

),


as l → +∞. Now we use the definition of L+, (2.17), and integrate by parts,(ψ+

1 ,L+e−s1(λ∗)x1φ1)

L2(�+)= (

ψ+1 , (−� − λ∗ + L+)e−s1(λ∗)x1φ1

)L2(�)

= limx1→−∞

((e−s1(λ∗)x1φ1,

∂ψ+1

∂x1

)

L2(ω)

−

−(

ψ+1 ,

∂

∂x1e−s1(λ∗)x1φ1

)

L2(ω)

)

= 2s1(λ∗)β+, (7.17)

which together with previous formula implies

A1p−+1(λ∗, l) = 2β−β+s1(λ∗)e−2ls1(λ∗) + O(e−2l(s1(λ∗)+s2(λ∗))

), (7.18)

as l → +∞. In the same way one can show that

A1 j(λ∗, l) = O(e−2l(s1(λ∗)+s2(λ∗))

), j = p− + 2, . . . , p,

Aij(λ∗, l) = O(e−4ls2(λ∗)

), i = 2, . . . , p−, j = p− + 1, . . . , p,

Ap−+11(λ∗, l) = 2β−β+s1(λ∗)e−2ls1(λ∗) + O(e−2l(s1(λ∗)+s2(λ∗))

),

Ap−+1 j(λ∗, l) = O(e−2l(s1(λ∗)+s2(λ∗))

), j = 2, . . . , p−,

Aij(λ∗, l) = O(e−4ls2(λ∗)

), i = p− + 2, . . . , p, j = 1, . . . , p−,

as l → +∞. The formulas obtained and (7.16), (7.18) lead us to the represen-tation (2.14), where the matrix A0 is as described, while the matrix A1 satisfies(7.14). The matrix A0 being hermitian, it satisfies the condition (A). This factcan be proved completely by analogy with Lemma 7.1.

Let us calculate the roots of det(τE − A0). For the sake of brevity withinthe proof we denote c := −2β−β+s1(λ∗)e−2ls1(λ∗). Expanding the determinantdet(τE − A0) w.r.t. the first column, one can make sure that

det(τE − A0) = τ p−2(τ 2 − |c|2) .

Thus, τ (1)0 (l)= . . . =τ

(p−2)

0 (l)=0 is a root of multiplicity (p − 2), and τ(p−1)

0 (l)=−2|c|, τ

(p)

0 (l) = 2|c|. Applying Theorem 2.5, we arrive at (2.21). ��

Proof of Theorem 2.6 As in the proof of Theorem 2.8, we begin with theproving (2.14). Namely, we are going to show that

A0(l) = diag{A11(λ∗, l), 0, . . . , 0}, ‖A1(l)‖ = O(e−2l(s1(λ∗)+s2(λ∗))

). (7.19)

The definition of A implies

Aij(λ∗, l) = −(T −

6 (λ∗, l)(I − T +7 (λ∗, l)T −

6 (λ∗, l))−1ϕ−i (·, l), ψ−

j

)L2(�)

. (7.20)

188 D. Borisov

Employing the definition (4.9) of T −6 , for any f ∈ L2(�, �+) we check that

(T −

6 (λ∗, l) f, ψ−j

)L2(�)

= (L−S(−2l)(H+ − λ∗)−1 f, ψ−

j

)L2(�)

= (S(−2l)(H+ − λ∗)−1 f,L−ψ−

j

)L2(�)

= ((H+ − λ∗)−1 f,S(2l)(� + λ∗)ψ−

j

)L2(�)

= −((H+ − λ∗)−1 f, (H+ − λ∗ − L+)S(2l)ψ−

j

)L2(�)

= (f, (H+ − λ∗)−1L+S(2l)ψ−

j − S(2l)ψ−j

)L2(�)

.

Using this identity, (7.20), (4.10), (4.12), and Lemma 2.1, we obtain that for(i, j) �= (1, 1) the functions Aij satisfy the relation

|Aij| � ‖(I − T +7 T −

6 )−1ϕ−i ‖L2(�+)

‖(H+ − λ∗)−1L+S(2l)ψ−j − S(2l)ψ−

j ‖L2(�+) = O(e−2l(s1(λ∗)+s2(λ∗))

),

as l → +∞, where in the arguments λ = λ∗, and the operator (H+ − λ∗)−1 isbounded since p+ = 0. The identities (7.19) therefore hold true.

We apply now Theorem 2.5 and infer that the formulas (2.18) are valid forthe eigenvalues λi, i = 1, . . . , p − 1, while the eigenvalue λp satisfies

λp(l) = λ∗ + A11(λ∗, l)(1 + O(l2e−2ls1(λ∗))

) + O(e−2l(s1(λ∗)+s2(λ∗))

), (7.21)

as λ → +∞. Now it is sufficient to find out the asymptotic behaviour ofA11(λ∗, l).

The formula (7.20) for A11, Lemma 2.1, and (4.10), (4.12) yield

A11(λ∗, l) = −(T −

6 (λ∗, l)ϕ−1 (·, l), ψ−

1

)L2(�)

+ O(le−8ls1(λ∗)

)

= β−e−2ls1(λ∗)(T −

6 (λ∗, l)L+e−s1(λ∗)x1 , ψ−1

)L2(�)

++ O

(e−2l(s1(λ∗)+s2(λ∗)) + le−8ls1(λ∗)

),

(T −

6 (λ∗, l)L+e−s1(λ∗)x1, ψ−1

)L2(�)

= (L−S(−2l)(H+ − λ∗)−1L+e−s1(λ∗)x1 , ψ−

1

)L2(�−)

= (S(−2l)(H+ − λ∗)−1L+e−s1(λ∗)x1,L−ψ−

1

)L2(�−)

= (S(−2l)U+,L−ψ−

1

)L2(�−)

.

Lemma 4.2 implies that the function U satisfies (2.19) that determines theconstant β− uniquely. Employing (2.19), we continue our calculations,

(S(−2l)U+,L−ψ−

1

)L2(�−)

=β−e−2ls1(λ∗)(es1(λ∗)x1φ1(x′),L−ψ−

1

)L2(�)

++ O

(e−2ls2(λ∗)

).

Integrating by parts in the same way as in (7.17), we obtain(es1(λ∗)x1φ1(x′),L−ψ−

1

)L2(�)

= (L−es1(λ∗)x1φ1(x′), ψ−

1

)L2(�)

= −2s1(λ∗)β−.


Therefore,

A11(λ∗, l) = 2s1(λ∗)|β−|2β−e−4ls1(λ∗) + O(e−2l(s1(λ∗)+s2(λ∗)) + le−8ls1(λ∗)

).

Substituting this identity into (7.21), we arrive at the required formula for λp.��

The proof of Theorem 2.7 is completely analogous to that of Theorem 2.6.

8 Examples

In this section we provide some examples of the operators L±. In what followswe will often omit the index “±” in the notation L±, H±, �±, a±, writing L, H,�, a instead.

1. Potential. The simplest example is the multiplication operator L = V, whereV = V(x) ∈ C(�) is a real-valued compactly supported function. Although thisexample is classical one for the problems in the whole space, to our knowledge,the double-well problem in waveguide has not been considered yet.2. Second order differential operator. This is a generalization of the previousexample. We introduce the operator L as

L =n∑

i, j=1

bij∂2

∂xi∂x j+

n∑

i=1

bi∂

∂xi+ b 0, (8.1)

where the complex-valued functions bij = bij(x) are piecewise continuouslydifferentiable in �, bi = bi(x) are complex-valued functions piecewise contin-uous in �. These functions are assumed to be compactly supported. The onlyrestriction to the functions are the conditions (2.1), (2.2); the self-adjointnessof H and Hl is implied by these conditions. One of the possible way to choosethe functions in (8.1) is as follows

L = div G∇ + in∑

i=1

(bi

∂

∂xi− ∂

∂xib i

)+ b 0, (8.2)

where G = G(x) is n × n hermitian matrix with piecewise continuously dif-ferentiable coefficients, bi = bi(x) are real-valued piecewise continuouslydifferentiable functions, b 0 = b 0(x) is a real-valued piecewise continuousfunction. The matrix G and the functions bi are assumed to be compactlysupported and

(G(x)y, y)Cn � −c0‖y‖2Cn , x ∈ �, y ∈ C

n.

The constant c0 is independent of x, y and satisfies (2.3). The matrix G is notnecessarily non-zero. In the case G = 0 one has an example of a first orderdifferential operator.3. Magnetic Schrödinger operator. This is the example with a com-pactly supported magnetic field. The operator L is given by (8.2), whereG = 0. The coefficients b j form a magnetic real-valued vector-potential b =

190 D. Borisov

(b 1, . . . , b n) ∈ C1(�), and b 0 = ‖b‖2Cn + V, where V = V(x) ∈ C(�) is a com-

pactly supported real-valued electric potential. The main assumption is theidentities

∂bj

∂xi= ∂bi

∂x j, x ∈ � \ �, i, j = 1, . . . , n. (8.3)

To satisfy the conditions required for L, the magnetic vector potential shouldhave a compact support. We are going to show that one can always achieve itby employing the gauge invariance.

The operator −�(D) + L can be represented as −�(D) + L = (i∇ + b)2 + V,and for any β = β(x) ∈ C2(�) the identity

e−iβ(i∇ + b)2eiβ = (i∇ + b − ∇β)2

holds true. In view of (8.3) we conclude that there exist two functions β± =β±(x) belonging to C2(� ∩ {x : ±x1 > a}) such that ∇β± = b, x ∈ � ∩ {x : ±x1 > a}. We introduce now the function β as

β(x) =

⎧⎪⎪⎨

⎪⎪⎩

χ(a + x1 + 1)β−(x), x1 ∈ (−∞, −a), x′ ∈ ω,

0, x1 ∈ [−a, a], x′ ∈ ω,

χ(a − x1 + 1)β+(x), x1 ∈ (a, +∞), x′ ∈ ω,

where, we remind, the cut-off function χ was introduced in the proof ofLemma 3.2. Clearly, β ∈ C2(�), and ∇β = b, x ∈ � \ {x : |x1| < a + 1}. There-fore, the vector b − ∇β has compact support.

If one of the distant perturbations in the operator Hl, say, the right one,is a compactly supported magnetic field, it is sufficient to employ the gaugetransformation ψ(x) �→ eiβ(x1−l,x′)ψ(x) to satisfy the conditions for L+.4. Curved and deformed waveguide. One more interesting example is a geo-metric perturbation. Quite popular cases are local deformation of the bound-ary and curving the waveguide (see, for instance, [3, 9, 10], and referencestherein). Here we consider the case of general geometric perturbation, whichincludes in particular deformation and curving. Namely, let x = G(x) ∈ C2(�)

be a diffeomorphism, where x = (x1, . . . , xn), G(x) = (G1(x), . . . ,Gn(x)). Wedenote � := G(�), P := G−1. By P = P(x) we indicate the matrix

P :=⎛

⎜⎝

∂P1∂ x1

. . . ∂P1∂ xn

......

∂Pn∂ x1

. . . ∂Pn∂ xn

⎞

⎟⎠ ,

while the symbol p = p(x) denotes the corresponding Jacobian, p(x) :=det P(x). The function p(x) is supposed to have no zeroes in �. The mainassumption we make is

P(x) = const, PtP = E, x ∈ � \ �. (8.4)

It implies that outside � the mapping P acts as a combination of a shift and arotating. Hence, the part of � given by G((a, +∞) × ω) is also a tubular domain


being a direct product of a half-line and ω. The same is true for G((−∞, −a) ×ω). The typical example of the domain � is given on Fig. 1.

Let −�(D) be the negative Dirichlet Laplacian in L2(�) with the domainW2

2,0(�). It is easy to check that the operator U : L2(�) → L2(�) defined as

(Uv)(x) := p−1/2(x)v(P−1(x)

)(8.5)

is unitary. The operator H := −U�(D)U−1 has W22,0(�) as the domain and is

self-adjoint in L2(�). It can be represented as H = −�(D) + L, where L is asecond order differential operator

L = −p1/2 divx p−1PtP∇xp1/2 + �x. (8.6)

Indeed, for any u1, u2 ∈ C∞0 (�)

(Hu1, u2)L2(�) = − (�(D)U−1u1,U−1u2

)L2(�)

= (∇xp1/2u1, ∇xp1/2u2)

L2(�)= (

P∇xp1/2u1, p−1P∇xp1/2u2)

L2(�)

= − (p1/2 divx p−1PtP∇xp1/2u1, u2

)L2(�)

. (8.7)

The assumption (8.4) yields that p = 1 holds for x ∈ � \ �, and therefore thecoefficients of the operator L have the support inside �.

We are going to check the conditions (2.1), (2.2), (2.3) for the operator Lintroduced by (8.6). The symmetricity is obvious, while the estimates followfrom (8.6), (8.7),

(Lu, u)L2(�) = ‖p−1/2P∇xp1/2u‖2L2(�) − ‖∇u‖2

L2(�)

� C‖∇xp1/2u‖2L2(�) − ‖∇u‖2

L2(�)

= C(‖p1/2∇xu‖2

L2(�)+‖u∇xp1/2‖2L2(�) + 2(p1/2∇xu, u∇p1/2)L2(�)

) −

− ‖∇u‖2L2(�) � C

(1

2‖p1/2∇xu‖2

L2(�) − ‖u∇xp1/2‖2L2(�)

)−

− ‖∇xu‖2L2(�)

� −(

1 − C2

)‖∇xu‖2

L2(�) − C‖u‖2L2(�),

where C > 0 is a constant.

Fig. 1 Geometricperturbation

192 D. Borisov

If both the operators L± are the geometric perturbations described by thediffeomorphisms G±(x), without loss of generality we can assume that G±(x) ≡x as ±x1 < −a±. We introduce now one more diffeomorphism

Gl(x) :={G+(x1 − l, x′) + (l, 0, . . . , 0), x1 � 0,

G−(x1 + l, x′) − (l, 0, . . . , 0), x1 � 0,

This mapping is well-defined for l > max{a−, a+}. The domain �l := Gl(�) canbe naturally regarded as a waveguide with two distant geometric perturbations.Considering the negative Dirichlet Laplacian in L2(�), we obtain easily anunitary equivalent operator in L2(�). The corresponding unitary operator isdefined by (8.5) with P replaced by G−1

l and the similar replacement for p isrequired. Clearly, the obtained operator in L2(�) is the operator Hl generatedby the operators L± associated with G±.5. Delta interaction. Our next example is the delta interaction supported by amanifold. Let � be a closed bounded C3-manifold in � of the codimension oneand oriented by a normal vector field ν. It is supposed that � ∩ ∂� = ∅. Themanifold � can consist of several components. By ξ = (ξ1, . . . , ξn−1) we denotecoordinates on �, while � will indicate the distance from a point to � measuredin the direction of ν = ν(ξ). We assume that � is so that the coordinates(�, ξ) are well-defined in a neighbourhood of �. Namely, we suppose thatthe mapping (�, ξ) = P�(x) is C3-diffeomorphism, where x = (x1, . . . , xn) arethe coordinates in �. Let b = b(ξ) ∈ C3(�) be a real-valued function. Theoperator in question is the negative Laplacian defined on the functions v ∈W2

2(� \ �, ∂�) ∩ W12,0(�) satisfying the condition

[v]0 = 0,

[∂v

∂�

]

0

= 0, [u]0 := u∣∣�=+0 − u

∣∣�=−0. (8.8)

We indicate this operator as H� . An alternative way to introduce H� is viaassociated quadratic form

q�(v1, v2) := (∇xv1, ∇xv2)L2(�) + (bv1, v2)L2(�), (8.9)

where u ∈ W12,0(�) (see, for instance, [2, Appendix K, Sec. K.4.1], [7, Remark

4.1]). It is known that the operator H� is self-adjoint.We are going to show that there exists a diffeomorpism x = P (x) such

that a unitary operator U defined by (8.5) maps L2(�) onto L2(�), andthe operator UH�U−1 has W2

2,0(�) as the domain. We will also show thatUH�U−1 = −�(D) + L, where L is given by (8.1).

First we introduce an auxiliary mapping as

P (x) := P−1� (�, ξ) , � := � + 1/2 �|�|b(ξ), (�, ξ) = P�(x).

The coordinates (�, ξ) are well-defined in a neighbourhood of �, which can bedescribed as {x : |�| < δ}, where δ is small enough. Indeed,

Pi(x) = xi + �|�|Pi(x), (8.10)


where the functions Pi(x) are continuously differentiable in a neighbourhoodof �. Therefore, the Jacobian of Pi tends to one as � → 0 uniformly in ξ ∈ �.Now we construct the required mapping as follows

x = P (x) := (1 − χ (�/δ)

)x + χ (�/δ)P (x). (8.11)

The symbol χ(t) indicates an even infinitely differentiable cut-off functionbeing one as |t| < 1 and vanishing as |t| > 2. We assume also that δ is chosenso that supp χ (�/δ) ∩ ∂� = ∅. Let us prove that P is a C1-diffeomorhism andP(�) = �.

It is obvious that P ∈ C1(�). If χ (�/δ) = 0, the mapping P acts as an identitymapping, and therefore for such x the Jacobian p = p(x) of P equals one.The function χ is non-zero only in a small neighbourhood {x : |�| < 2δ} of�, where the identities (8.10) are applicable. These identities together withthe definition of P imply that p(x) = 1 + �p1(x), where the function p1(x)

is bounded uniformly in δ and x as |�| � 2δ. Hence, we can choose δ smallenough so that p � 1/2 as |�| � 2δ. Therefore, P is C1-diffeomorphism. As xclose to ∂�, the diffeomorphism P acts as the identity mapping. It yields thatP(∂�) = ∂�, P(�) = �. Since � = 0 as � = 0, it follows that P(�) = �.

The function � �→ � is continuously differentiable and its second derivativeis piecewise continuous. Thus, the second derivatives of Pi(x) are piecewisecontinuous as well. The same is true for the inverse mapping P−1.

We introduce the unitary operator U by (8.5), where P is the diffeo-morphism defined by (8.11). It is obvious that U maps L2(�) onto itself.Let us prove that it maps the domain of H� onto W2

2,0(�). In order to doit, we have to study the behaviour of p in a vicinity of �. It is clear thatp(x) ∈ C2(� \ �) ∩ C(�), and this function has discontinuities at � only. Byp� = p�(�, ξ) we denote the Jacobian corresponding to P� . It is obviousthat p� ∈ C2({(�, ξ) : |�| � δ}). Employing the well-known properties of theJacobians, we can express p in terms of p� ,

p(x) = (1 + |�|b(ξ))p�(�, ξ)

p�(�, ξ)= p�(�, ξ)(1 + |�|b(ξ))

p�(� + 1/2 �|�|b(ξ), ξ), |�| � δ.

This relation allows us to conclude that p ∈ C2({(�, ξ) : 0 � � � δ}), p ∈C2({(�, ξ) : −δ � � � 0}), and

[p1/2]0 = 0,

[∂p1/2

∂�

]

0

= b . (8.12)

Given u = u(x) ∈ W22,0(�), we introduce the function v = v(x) := (U−1u)(x) =

p1/2(x)u(P (x)). Due to the smoothness of p, v(x) ∈ W22(� \ �) ∩ W1

2,0(�). Theidentities (8.12) and the belonging u(x) ∈ W2

2(�) imply the condition (8.8)for v.

Suppose now that a function v = v(x) ∈ W22(� \ �) ∩ W1

2,0(�) satisfies (8.8).Due to the smoothness of P it is sufficient to check that the function Uv

regarded as depending on x belongs to W22,0(�), i.e., u(x) := p−1/2(x)v(x) ∈

W22,0(�). It is clear that u ∈ W2

2(� \ �) ∩ W12,0(�). Hence, it remains to check

the belonging u ∈ W22({x : |�| � δ}). The functions Pi being twice piecewise

194 D. Borisov

continuously differentiable, it is sufficient to make sure that the function utreated as depending on � and ξ is an element of W2

2({(�, ξ) : |�| < δ, ξ ∈ �}).We have u ∈ W1

2(R), u ∈ W22(R+), u ∈ W2

2(R−), R := {(�, ξ) : |�| < δ, ξ ∈ �},R± := {(�, ξ) : 0 < ±� < ±δ, ξ ∈ �}. Hence, we have to prove the existence ofthe generalized second derivatives for u belonging to L2(R). The condition(8.8) and the formulas (8.12) yield that

u∣∣�=+0 = u

∣∣�=−0,

∂u∂�

∣∣∣∣�=+0

= ∂u∂�

∣∣∣∣�=−0

.

Since u∣∣�=±0 ∈ W1

2(�), the first of these relations implies that

∂u∂ξi

∣∣∣∣�=+0

= ∂u∂ξi

∣∣∣∣�=−0

, i = 1, . . . , n − 1.

Having the obtained relations in mind, for any ζ ∈ C20(R) we integrate by parts,

(u,

∂2ζ

∂�2

)

L2(R)

=(

u,∂2ζ

∂�2

)

L2(R−)

+(

u,∂2ζ

∂�2

)

L2(R+)

=(

∂2u∂�2

, ζ

)

L2(R−)

+(

∂2u∂�2

, ζ

)

L2(R+)

,

and in the same way we obtain(

u,∂2ζ

∂�∂ξi

)

L2(R)

=(

∂2u∂�∂ξi

, ζ

)

L2(R−)

+(

∂2u∂�∂ξi

, ζ

)

L2(R+)

,

(u,

∂2ζ

∂ξi∂ξ j

)

L2(R)

=(

∂2u∂ξi∂ξ j

, ζ

)

L2(R−)

+(

∂2u∂ξi∂ξ j

, ζ

)

L2(R+)

.

Thus, the generalized second derivatives of the function u exist and coincidewith the corresponding derivatives of u regarded as an element of W2

2(R−) andW2

2(R+).Let us show that UH�U−1 = −�(D) + L, where L is given by (8.1). Proceed-

ing in the same way as in (8.7) and using (8.9), for any u1, u2 ∈ C∞0 (�) we

obtain

(UH�U−1u1, u2)L2(�) = (∇xp1/2u1, p−1PtP∇xp1/2u2)

L2(�)+ (bu1, u2)L2(�).

We have used here that p ≡ 1 as x ∈ � and P(�) = �. Employing (8.12) andhaving in mind that P

∣∣�

= E due to (8.10), we integrate by parts,

(UH�U−1u1, u2)L2(�) =([

∂

∂ρ(p1/2u1)

]

0

− bu1, u2

)

L2(�)

−

− (p1/2 divx p−1PtP∇xp1/2u1, u2

)L2(�\�)

= − (p1/2 divx p−1PtP∇xp1/2u1, u2

)L2(�\�)

.


Here � \ � means that in a neighbourhood of � we partition the domain ofintegration into two pieces one being located in the set {x : |�| < 0}, while theother corresponds to |�| > 0. Such partition is needed since the first derivativesof p have jump at � and therefore the second derivatives of p are not defined at�. At the same time, the function p has continuous second derivatives as � < 0and these derivatives have finite limit as � → −0. The same is true for � > 0.

The matrix P is piecewise continuously differentiable, and thereforeUH�U−1 = −p1/2 divx p−1PtP∇xp1/2, where the second derivatives of p aretreated in the aforementioned sense. This operator is obviously self-adjoint.Outside the set {x : |�| < 2δ} the diffeomorphism P acts as the identity map-ping. It yields that at such points P = E, p = 1. Therefore, L is a differentialoperator having compactly supported coefficients, and is a particular case of(8.1). It is also clear that the operator L satisfies (2.1). The inequality (2.2) canbe checked in the same way how it was proved in the previous subsection.6. Integral operator. The operator L is not necessary to be either a differentialoperator or reducible to a differential one. An example is an integral operator

Lu =∫

�

L(·, y)u(y) dy.

The kernel L ∈ L2(� × �) is assumed to be symmetric, i.e., L(x, y) = L(y, x).It is clear that the operator L satisfies (2.1), (2.2). It is also �(D)-compact andtherefore the operators H and Hl are self-adjoint.

In conclusion we stress that not all possible examples of L are exhausted bythe list given above. For instance, combinations of these examples are possiblelike compactly supported magnetic field with delta interaction, delta interac-tion in a deformed waveguide, integro-differential operator, etc. Moreover,the operators L− and L+ need not necessarily to be of the same nature. Forexample, L− can be a potential, while L+ may describes compactly supportedmagnetic field with delta interaction.

Acknowledgements I am grateful to Pavel Exner who attracted my attention to the problemstudied in this article. I thank him for stimulating discussion and the attention he paid to this work.I also thank the referee for useful remarks.

References

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2. Albeverio, S., Gesztesy, S., Høegh-Krohn, H. Holden, R.: Solvable Models in QuantumMechanics, 2nd edn. AMS Chelsea Publishing. Providence, Rhode Island (2005)

3. Borisov, D., Exner, P., Gadyl’shin, R., Krejcirík, D.: Bound states in weakly deformed stripsand layers. Ann. H. Poincaré. 2, 553–572 (2001)

4. Borisov, D., Exner, P.: Exponential splitting of bound states in a waveguide with a pair ofdistant windows. J. Phys. A. 37, 3411–3428 (2004)

5. Borisov, D., Exner, P.: Distant perturbation asymptotics in window-coupled waveguides. I.The non-threshold case. J. Math. Phys. 47, 2113502-1–113502-24 (2006)

196 D. Borisov

6. Borisov, D.: Distant perturbations of the Laplacian in a multi-dimensional space. Ann. H.Poincare (2007) (in press)

7. Brasche, J.F., Exner, P., Kurepin, Yu.A., Šeba, P.: Schrödinger operator with singular interac-tions. J. Math. Anal. Appl. 184, 112–139 (1994)

8. Briet, Ph., Combes, J.M., Duclos, P.: Spectral stability under tunneling. Comm. Math. Phys.126, 133–156 (1989)

9. Bulla, W., Gesztesy, F., Renger, W., Simon, B.: Weakly coupled bound states in quantumwaveguides. Proc. Amer. Math. Soc. 125, 1487–1495 (1997)

10. Chenand, B., Duclos, P., Freitas, P., Krejcirík, D.: Geometrically induced spectrum in curvedtubes. Differential Geom. Appl. 23, 95–105 (2005)

11. Combes, J.M., Duclos, P., Seiler,R.: Krein’s formula and one-dimensional multiple well.J. Funct. Anal. 52, 257–301 (1983)

12. Davies, E.B.: Spectral Theory and Differential Operators. Cambridge University Press,Cambridge (1995)

13. Gadyl’shin, R.: On local perturbations of Schrödinger operator in axis. Theor. Math. Phys.132, 976–982 (2002)

14. Harrel, E.M.: Double wells. Comm. Math. Phys. 75, 239–261 (1980)15. Harrel, E.M., Klaus, M.: On the double-well problem for Dirac operators. Ann. Inst. H.

Poincaré, Sect. A. 38, 153–166 (1983)16. Høegh-Krohn, R., Mebkhout, M.: The 1

r expansion for the critical multiple well problem.Comm. Math. Phys. 91, 65–73 (1983)

17. Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)18. Klaus, M., Simon, B.: Binding of Schrödinger particles through conspiracy of potential wells.

Ann. Inst. H. Poincaré, Sect. A. 30, 83–87 (1979)19. Kondej, S., Veselic I.: Lower bounds on the lowest spectral gap of singular potential

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operator. Math. Scand. 8, 143–153 (1960)22. Simon, B.: Semiclassical analysis of low-lying eigenvalues. I. Non-generate minima: asymptotic

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Mathematik der Universität Frankfurt (1995)


Asymptotic Inverse Problem for Almost-PeriodicallyPerturbed Quantum Harmonic Oscillator

Alexis Pokrovski

Received: 12 July 2007 / Accepted: 13 July 2007 /Published online: 13 September 2007© Springer Science + Business Media B.V. 2007

Abstract Let {μn}∞n=0 be the spectrum of − d2

dx2 + x2 + q(x) in L2(R), where q isan even almost-periodic complex-valued function with bounded primitive andderivative. It is known that μn = μ0

n + O(n− 14 ), where {μ0

n}∞n=0 is the spectrumof the unperturbed operator. Suppose that the asymptotic approximation tothe first asymptotic correction �μn = μn − μ0

n + o(n− 14 ) is given. We prove the

formula that recovers the frequencies and the Fourier coefficients of q in termsof �μn.

Keywords Almost-periodic perturbation · Inverse problem ·Quantum harmonic oscillator · Spectral asymptotics

Mathematics Subject Classifications (2000) 34L20 · 81Q15

1 Introduction and Main Result

Consider the operator describing perturbed quantum harmonic oscillator

A = − d2

dx2+ x2 + q(x) in L2(R) (1.1)

with the perturbation q(x) from the class B = {q : ‖q′‖∞ + ‖Q‖∞ < ∞}, whereQ(x) = ∫ x

0 q dt and ‖ · ‖∞ denotes the norm in L∞(R). It was proved in [1] thatthe spectrum {μn}∞n=0 of A has the asymptotics μn = μ0

n + μ1n + O(n− 1

3 ), whereμ0

n = 2n + 1 and μ1n = O(n− 1

4 ).

A. Pokrovski (B)Laboratory of Quantum Networks, Institute for Physics,St-Petersburg State University, St. Petersburg 198504, Ulyanovskaya 1, Russiae-mail: [email protected]

198 A. Pokrovski

For the perturbations that are sum of almost-periodic and decaying termswe study the problem of recovering of the almost-periodic part from the firstasymptotic correction μ1

n. Specifically, we consider the perturbations

q = p+r∈B, p∈ B1, p(−x)= p(x) and

‖r‖B1 ≡ limT→∞

1

2T

∫ T

−T|r(x)|dx = 0, (1.2)

where B1 is the Besikovitch space of almost-periodic functions [2] (closureof trigonometric polynomials

∑Nk=0 akeiνkx, νk real, in the norm ‖ · ‖B1 ). It is

sufficient to recover p in terms of its Fourier transform [2]. Here is the mainresult.

Theorem 1.1 Let {�μn}∞n=N approximates the first asymptotic correction to thespectrum of the operator (1.1), (1.2):

�μn = μn − μ0n + o

(n− 1

4

). (1.3)

Then the spectrum and the Fourier coefficients of the almost-periodic part pcan be recovered from the relation

limL→∞

1

xL

L−1∑

n=N

�μnGν(xn, xL)(xn+1−xn)

= limT→∞

1

T

∫ T

0p(t) cos νt dt, ν �0, (1.4)

where xn = √μ0

n =√2n + 1, Gν(x, T) = −x

∫ Tx

ϕ′ν,T (t)dt√t2−x2 , ϕν,T(t) = η(t −T) cos νt

and η ∈ C2(R) is a smoothed step function such that η(t) = 1 for x ∈ (−∞, −1],η(t) = 0 for x ∈ [0, ∞) and η′(0) = 0.

Asymptotic inverse spectral problem for quantum harmonic oscillator withslowly decaying perturbation was considered by Gurarie [3]. He studied theoperator (1.1) with real q(x) ∼ |x|−α

∑am cos ωmx for |x| → ∞, where α > 0

and the sum in the numerator is finite. The approach in [3] is based on thespectral asymptotics

μn = q(√

2n)

n1/4+α/2+ O

(1√n

)

, q(x) = const∑ am√

ωmcos(ωmx − π/4)

which exhibits linear relation between the leading asymptotic terms of q andμn. However, the technique of [3] does not cover the case α = 0.

We consider just this case in a slightly more general setting (almost-periodicfunctions vs. finite trigonometric sums). Our method also allows complex-valued q. Technically, the result is based on the recent proof [1] of the spectralasymptotics

μn = μ0n + 1

2π

∫ π

−π

q(√

μ0n sin ϑ

)

dϑ + O(

n− 13

), for q ∈ B. (1.5)

Asymptotic inverse problem 199

Thus the proof of Theorem 1.1 follows from the asymptotic behavior of theintegral in (1.5), which is analyzed in Lemmas 2.1 and 2.3.

2 Properties of the Schlömilch Integral

The integral in the spectral asymptotics (1.5) is the Schlömilch integral [4]

gq(x)= 2

π

∫ π/2

0q+(x sin ϑ)dϑ = 2

π

∫ x

0

q+(t)dt√x2 − t2

, q+(x)= (q(x) + q(−x))/2

(2.6)

evaluated at the points xn = √2n + 1. In the next Lemma we estimate the

integral and its derivatives. Then in Lemma 2.3 we prove similar estimates forthe inverse Schlömilch integral. (We could not find in the literature the resultsof these Lemmas for the specific class B.) Using the two Lemmas, we proveTheorem 1.1.

Everywhere below C denotes an absolute constant.

Lemma 2.1 Let f ∈ B and g(x) = ∫ π/20 f (x sin ϑ)dϑ . Then

|g(x)| � C‖F‖∞ + ‖ f‖∞√

1 + x, |g′(x)| � C

‖ f‖∞ + ‖ f ′‖∞√1 + x

, x > 0, (2.7)

where F(x) = ∫ x0 f dt.

Proof For x � 1 the result is evident, so we consider only the case x > 1. Usingthe change of variables t = sin ϑ , we write g(x) = ∫ 1

0f (xt)dt√

1−t2 and split it as

g = I1 + I2, I1 =∫ 1−ε

0

f (xt)dt√1 − t2

, I2 =∫ 1

1−ε

f (xt)dt√1 − t2

, (2.8)

where ε = 1/x. Using the notation (. . .)′t for ∂∂t (. . .) and choosing the prim-

itive F of f , satisfying F(x(1 − ε)) = 0, we have I1 = 1x

1−ε∫

0

(F(xt)

)′t

dt√1−t2 =

1x

(F(xt)√

1−t2

∣∣∣t=1−ε

t=0+

1−ε∫

0

F(xt)t dt(1−t2)3/2

)

. Therefore,

|I1| � 2‖F‖∞

x

⎛

⎝1 +1∫

ε

dtt3/2

⎞

⎠ � C‖F‖∞

x

(

1 + 1√ε

)

. (2.9)

200 A. Pokrovski

Now we substitute the estimate |I2| � ‖ f‖∞∫ ε

0dt√

t= 2‖ f‖∞/

√x and (2.9)

into (2.8). This gives the first inequality in (2.7). We prove the second one in asimilar way, writing

g′ = I′1 + I′

2, I′1 =

∫ 1−ε

0

t f ′(xt)dt√1 − t2

, I′2 =

∫ 1

1−ε

t f ′(xt)dt√1 − t2

, ε = 1/x.

(2.10)

We integrate by parts in I′1, choosing the primitive f (xt) = f (xt) − f (x(1 − ε)).

This gives I′1 = − 1

x

∫ 1−ε

0f (xt) dt

(1−t2)3/2 , hence

|I′1| � C

‖ f‖∞x

(

1 + 1√ε

)

. (2.11)

We substitute the estimate |I′2| � C‖ f ′‖∞/

√x and (2.11) in (2.10). This gives

the second inequality in (2.7). �

Remark 2.2 The rate of decay x−1/2 as x → ∞ in (2.7) cannot be improved.The example f (x) = cos x gives the Bessel function J0.

Lemma 2.3 Let T > 2, ϕ, ϕ′′ ∈ L∞([0, T]) and ϕ(T) = 0. Then the equationϕ(t) = 2

π

∫ Tt

g(x)dx√x2−t2 has the unique solution g(x) = −x

∫ Tx

ϕ′(t)dt√t2−x2 for x ∈ [0, T],

such that|g(x)| � C(‖ϕ‖∞ + ‖ϕ′‖∞)

√x, for x > 1. (2.12)

If, in addition, ϕ′(T) = 0, then

|g′(x)| � C(‖ϕ′‖∞ + ‖ϕ′′‖∞)√

x, for x > 1. (2.13)

Proof In terms of g(x) = g(√

x)

2√

x and ϕ(t) = ϕ(√

t) the equation on g becomes

the Abel equation ϕ(t) = 2π

∫ T2

tg(s) ds√

s−t. Its solution for absolutely continuous ϕ

is g(s) = ϕ(T2)√T2−s

− ∫ T2

sϕ′(u)du√

u−s(see Chap. 1, paragraph 2 of [5]). Using ϕ(T) = 0,

we obtain the required formula for g.Consider (2.12). For x ∈ [T − 1, T] the inequality follows from the direct

estimate | g(x)

x | � ‖ϕ′‖∞√2x

∫ TT−1

dt√t−(T−1)

. For x ∈ [0, T − 1] write

−g(x)

x= I1 + I2, I1 =

∫ x+1

x

ϕ′(t)dt√t2 − x2

, I2 =∫ T

x+1

ϕ′(t)dt√t2 − x2

(2.14)

and integrate I2 by parts. We have I2 = ϕ(t)√t2−x2

∣∣∣t=T

t=x+1− ∫ T

x+1 ϕ(t) ∂∂t

1√t2−x2 dt.

Therefore,

|I2| � ‖ϕ‖∞√1 + 2x

+ ‖ϕ‖∞(−1)√t2 − x2

∣∣∣t=∞t=x+1

� 2‖ϕ‖∞√1 + 2x

. (2.15)

Now we substitute the estimate |I1| � ‖ϕ′‖∞∫ x+1

xdt√

t2−x2 � ‖ϕ′‖∞√

2/x and(2.15) into (2.14). This gives (2.12), as required.


Next consider (2.13). By g′(x) = x(g(x)/x)′ + g(x)/x and (2.12), it is suffi-cient to estimate x(g(x)/x)′. Using ϕ′(T) = 0, we obtain

x(g(x)/x)′ = − Tϕ′(T)√T2 − x2

+∫ T/x

1

xsϕ′′(xs) ds√s2 − 1

=∫ T

x

tϕ′′(t) dt√t2 − x2

. (2.16)

For x ∈ [T − 1, T] we have |x(g(x)/x)′| � 2x√2x

∫ TT−1

|ϕ′′(t)| dt√t−(T−1)

� 2√

2‖ϕ′′‖∞√

x.For x ∈ [0, T − 1] write

x(g(x)/x)′ = I′1 + I′

2, I′1 =

∫ x+1

x

tϕ′′(t)dt√t2 − x2

, I′2 =

∫ T

x+1

tϕ′′(t)dt√t2 − x2

(2.17)

and take the integral for I′2 by parts. We have I′

2 = − (x+1)ϕ′(x+1)√(x+1)2−x2

+∫ T

x+1 ϕ′(t) ∂∂t

t√t2−x2 dt. Hence, using ∂

∂tt√

t2−x2 � 0 we obtain

|I′2| � (1 + x)‖ϕ′‖∞√

1 + 2x+ ‖ϕ′‖∞

∫ ∞

x+1

∂

∂t(−t)√t2 − x2

dt � 2‖ϕ′‖∞√

1 + x. (2.18)

Now we substitute the estimate |I′1| � ‖ϕ′′‖∞

∫ x+1x

tdt√t2−x2 � ‖ϕ′′‖∞

√1 + 2x and

(2.18) into (2.17). This gives (2.13). �

Remark 2.4 Note that the condition ϕ(T) = 0 is necessary for (2.12). If ϕ(T) �=0, then g(x) is unbounded due to the non-integral term in the inversion formulafor the Abel equation. The term is O(

ϕ(T)√T−x

) for x ↑ T. Similarly, the conditionϕ′(T) = 0 is necessary for (2.13).

Proof of Theorem 1.1 Compare (1.3) with the asymptotis (1.5). It is clear that

�μn = gq(xn) + o(

n− 14

), xn =

√μ0

n = √2n + 1, (2.19)

where the Schlömilch integral gq is given by (2.6). The proof is basedon the fact that the set {xn}∞n=0 becomes arbitrarily dense as n → ∞,so that Riemann sums 1

T

∑xn+1�T Gν(xn, T)gq(xn)(xn+1 − xn) approximate

1T

∫ T0 Gν(x, T)gq(x)dx, provided the integrand is smooth enough. Using the

inversion formula for the Schlömilch integral, we choose Gν such that the lastexpression tends to 1

T

∫ T0 q+(t) cos νt dt as T → ∞.

By Lemma 2.3, we have∫ T

0Gν(x, T)gq(x)dx =

∫ T

0q+(t)

(2

π

∫ T

t

Gν(x, T)dx√x2 − t2

)

dt

=∫ T

0q+(t)ϕν,T(t) dt. (2.20)

202 A. Pokrovski

Let us show that the integrand in the left-hand side of (2.20) is sufficientlysmooth. By Lemma 2.1,

|gq(x)| � C‖Q‖∞ + ‖q‖∞√

1 + x, |g′

q(x)| � C‖q‖∞ + ‖q′‖∞√

1 + x, x � 0,

(2.21)

where C is an absolute constant. Similarly, since ϕν,T satisfies the hypothesis ofLemma 2.3, uniformly in T � x

|Gν(x, T)| � C(1 + ν)√

x,

∣∣∣ ∂∂x Gν(x, T)

∣∣∣ � C(1 + ν)2√x, x � 1,

(2.22)

where we used ‖ϕν,T‖∞ � C, ‖ϕ′′ν,T‖∞ � C(1 + ν)2. Therefore, for any fixed

ν the function hT(x)def= Gν(x, T)gq(x) and its x-derivative are uniformly

bounded for x � 1 and T � x. Hence, for T → ∞ we have

1

T

∑

xn+1�T

hT(xn)(xn+1 − xn)− 1

T

∫ T

0hT(x) dx= 1

T

∫ T

0O

(1

x

)

dx= O(

ln TT

)

→0,

(2.23)

where we used xn+1 − xn = O(x−1n ). Now, by (2.23), (2.19) and the first esti-

mate in (2.22),

limT→∞

1

T

∑

xn+1�T

�μnGν(xn, T)(xn+1 − xn) = limT→∞

1

T

∫ T

0Gν(x, T)gq(x) dx. (2.24)

Next we divide (2.20) by T and take the limit T → ∞. By (2.24) andlim

T→∞1T

∫ T0 q+ϕν,T dt = lim

T→∞1T

∫ T0 p(t) cos νt dt, this gives (1.4). �

Remark 2.5 We have (νx)−1Gν(x, ∞) = 2π

∫ ∞x

sin νt dt√t2−x2 = J0(νx), where J0 is the

Bessel function (see e.g. [6]).

Acknowledgements The author is thankful to E. Korotyaev and S. Naboko for fruitful discus-sions and valuable advice.

References

1. Klein, M., Korotyaev, E., Pokrovski, A.: Spectral asymptotics of the harmonic os-cillator perturbed by bounded potentials. Ann. Henri Poincare 6, 747–789 (2005)(arxiv.org/math-ph/0312066)

2. Besikovitch, A.S.: Almost Periodic Functions. Dover Publ Inc. (1954)3. Gurarie, D.: Asymptotic inverse spectral problem for anharmonic oscillators. Comm. Math.

Phys. 112, 491–502 (1987)



4. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, vol. 1. Cambridge UniversityPress (1927)

5. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Integrals and derivatives of fractional order andtheir applications (in Russian), Nauka i Tehnika, Minsk (1987)

6. Gradstein, I.S., Ryzhik, I.M.: Tables of integrals, sums, series and products (in Russian),GIFML, Moscow (1963)


Cuspons and Smooth Solitonsof the Degasperis–Procesi EquationUnder Inhomogeneous Boundary Condition

Guoping Zhang · Zhijun Qiao

Received: 16 January 2007 / Accepted: 3 August 2007 /Published online: 26 September 2007© Springer Science + Business Media B.V. 2007

Abstract This paper is contributed to explore all possible single peakon solu-tions for the Degasperis–Procesi (DP) equation mt + mxu + 3mux = 0, m =u − uxx. Our procedure shows that the DP equation either has cusp soliton andsmooth soliton solutions only under the inhomogeneous boundary conditionlim|x|→∞ u = A �= 0, or possesses the regular peakon solutions ce−|x−ct| ∈ H1 (cis the wave speed) only when lim|x|→∞ u = 0 (see Theorem 4.1). In particular,we first time obtain the stationary cuspon solution u = √

1 − e−2|x| ∈ W1,1loc of

the DP equation. Moreover we present new cusp solitons (in the space ofW1,1

loc ) and smooth soliton solutions in an explicit form. Asymptotic analysisand numerical simulations are provided for smooth solitons and cusp solitonsof the DP equation.

Keywords Soliton · Integrable system · Analysis · Traveling wave

Mathematics Subject Classifications (2000) 35D05 · 35G30 · 35Q53 · 37K10 ·37K40 · 76B15 · 76B25

G. Zhang · Z. Qiao (B)Department of Mathematics, The University of Texas – Pan American,1201 W University Drive, Edinburg, TX 78541, USAe-mail: [email protected]

G. ZhangDepartment of Mathematics, Morgan State University,1700 E Cold Spring Ln, Baltimore, MD 21239, USAe-mail: [email protected]

206 G. Zhang, Z. Qiao

1 Introduction

The b -weight-balanced wave equation

mt + mxu + bmux = 0, m = u − uxx (1.1)

was proposed in [8] in 2003, and recently has arisen a lot of attractive attention.This family is proved to be integrable when b = 2, 3 by using symmetryapproach [11]. More deeper mathematical attributes on the CH equationwas shown by the constraint method and r-matrix structure [14, 15], inversespectral theory [3], Riemann–Hilbert problem [12], and the global conservativesolution [1].

In an earlier paper [18], Qiao and Zhang discussed the traveling wavesolutions for the b = 2 equation – the Camassa–Holm (CH) equation [2]under the inhomogeneous boundary condition lim|x|→∞ u = A (A is a non-zero constant), and found new soliton solutions both smooth and cusped. Inthe paper [19], Vakhnenko and Parkes presented the periodic and loop solitonsolutions for the b = 3 equation – the Degasperis–Procesi (DP) equation [6]from the mathematical point of view. Their solutions were expressed in aimplicit form. Later in [10], Lenells also studied the traveling wave solitarysolutions of the DP equation, which decay to zero at both infinities, but did notgive explicit soliton solutions, either.

An important issue to study both the CH equation and the DP equation isto find their new solutions through investigating their intrinsic mathematicalstructures. The DP equation has Lax pair [7] (therefore is integrable), andis able to be extended to a whole integrable hierarchy of equations withparametric solutions under some constraints [13, 17]. Also, the DP equationadmits the global weak solution and blow-up structure [9].

In this paper, we give all possible single peak soliton solutions of the DPequation

mt + mxu + 3mux = 0, m = u − uxx, (1.2)

through setting the traveling wave mode under the boundary condition u →A (A is a constant) as x → ±∞. Our procedure shows that the DP equationeither has cusp soliton and smooth soliton solutions only under the boundarycondition lim|x|→∞ u = A �= 0, or possesses the regular peakon solutions onlywhen lim|x|→∞ u = 0 (see Theorem 4.1). In particular, we first time obtain astationary cuspon solution of the DP equation. Moreover we present new cuspsolitons and smooth soliton solutions in an explicit form [see formulas (3.7),(5.11) and (5.16)]. Asymptotic analysis and numerical simulations are providedfor smooth solitons and cusp solitons of the DP equation. In the literature [4],cusp soliton was also called breaking wave. Due to some complex notationsand definitions, we shift the statement of our main result to Section 4 (seeTheorem 4.1).

Cuspons and smooth solitons of the Degasperis–Procesi equation 207

2 Traveling Wave Setting

Let us consider the traveling wave solution of the DP equation (1.2) througha generic setting u(x, t) = U(x − ct), where c is the wave speed. Let ξ = x − ct,then u(x, t) = U(ξ). Substituting it into the DP equation (1.2) yields

(U − c)(U − U ′′)′ + 3U ′(U − U ′′) = 0, (2.1)

where U ′ = Uξ , U ′′ = Uξξ , U ′′′ = Uξξξ .If U − U ′′ = 0, then (2.1) has general solutions of U(ξ) = c1eξ + c2e−ξ with

any real constants c1, c2. Of course, they are the solutions of the DP equation(1.2). This result is not so interesting for us. On the other hand, the DPequation has the well-known peakon solution [8] u(x, t) = U(ξ) = ce−|x−ct−ξ0|(ξ0 = x0 − ct0) with the following properties

U(ξ0) = c, U(±∞) = 0, U ′(ξ0−) = c, U ′(ξ0+) = −c, (2.2)

where U ′(ξ0−) and U ′(ξ0+) represent the left-derivative and the right-derivative at ξ0, respectively.

Let us now assume that U is neither a constant function nor satisfies U −U ′′ = 0. Then (2.1) can be changed to

(U − U ′′)′

U − U ′′ = 3U ′

c − U. (2.3)

Taking the integration twice on both sides leads to

(U ′)2 = U2 − C2

(c − U)2+ C1, (2.4)

where C2 �= 0, C1 ∈ R are two integration constants.Let us solve (2.4) with the following boundary condition

limξ→±∞ U = A, (2.5)

where A is a constant. Equation (2.4) can be cast into the following ODE:

(U ′)2 = (U − A)2[(U − c + A)2 − cA](U − c)2

. (2.6)

The fact that both sides of (2.6) are nonnegative implies

(U − c + A)2 − cA � 0. (2.7)

If cA � 0, we denote

B1 = c − A + √cA, B2 = c − A − √

cA. (2.8)

Apparently, B1 � B2.

3 Smooth Solution and Weak Solution

Let Ck(�) denote the set of all k times continuously differentiable functionson the open set �. Lp

loc(R) refers to be the set of all functions whose restriction


on any compact subset is Lp integrable. H1loc(R) and W1,1

loc (R) stand forH1

loc(R) = {u∈ L2loc(R)| u′ ∈ L2

loc(R)} and W1,1loc (R) = {u∈ L1

loc(R)| u′ ∈ L1loc(R)},

respectively.

Definition 3.1 A function u(x, t) = U(x − ct) is said to be a single peak solitonsolution for the DP equation (1.2) if U satisfies the following conditions

(C1) U(ξ) is continuous on R and has a unique peak point, denoted by ξ0,where U(ξ) attains its global maximum or minimum value;

(C2) U(ξ) ∈ C3(R − {ξ0}) satisfies (2.6) on R − {ξ0};(C3) U(ξ) satisfies the boundary condition (2.5).

Without losing the generality, from now on we assume ξ0 = 0.

Lemma 3.2 Assume that u(x, t) = U(x − ct) is a single peak soliton solution ofthe DP equation (1.2) at the peak point ξ0 = 0. Then we have

a) if cA < 0, then U(0) = c;b) if cA � 0, then U(0) = c or U(0) = B1 or U(0) = B2.

Moreover, we have the following solutions classification:

(1) if U(0) �= c, then U(ξ) ∈ C∞(R), and u is a smooth soliton solution.(2) if U(0) = c and A �= 0, then u is a cusp soliton and U has the following

asymptotic behavior

U(ξ) − c = λ|ξ |1/2 + O(|ξ |), ξ → 0;

U ′(ξ) = 1/2λ|ξ |−1/2sign(ξ) + O(1), ξ → 0;

where λ = ±√

2|c − A|√A2 − cA. Thus U(ξ) /∈ H1loc(R).

(3) if U(0) = c and A = 0, then u gives the regular peaked soliton ce−|x−ct|.

Proof (3) is obvious. Let us prove (1), a), b) and (2) in order.(1) If U(0) �= c, then U(ξ) �= c for any ξ ∈ R since U(ξ) ∈ C3(R − {0}).

Differentiating both sides of (2.4) yields U ∈ C∞(R).a) When cA < 0, if U(0) �= c, by (1) we know that U is smooth and U ′(0) =

0. However, by (2.6) we must have U(0) = A, which contradicts the fact that 0is the unique peak point.

b) When cA � 0, if U(0) �= c, by (2.4) we know U ′(0) exists. So, U ′(0) = 0since 0 is a peak point. But, by (2.6) we obtain U(0) = B1 or U(0) = B2, sinceU(0) = A contradicts the fact that 0 is the unique peak point.

(2) If U(0) = c and A �= 0, then by the definition of the single peak solitonwe have A �= c, thus (U − c + A)2 − cA doesn’t contain the factor U − c.From (2.6) we obtain

U ′ = sign(c − A)U − AU − c

√(U − c + A)2 − cAsign(ξ). (3.1)


Let h(U) = sign(c−A)

(U−A)√

(U−c+A)2−cA, then h(c) = 1

|c−A|√A2−cA, and

∫h(U)(U − c)dU =

∫sign(ξ)dξ. (3.2)

Inserting h(U) = h(c) + O(U − c) into (3.2) and using the initial conditionU(0) = c, we obtain

h(c)2

(U − c)2(1 + O(U − c)) = |ξ |, (3.3)

thus

U − c = ±√

2

h(c)|ξ |1/2(1 + O(U − c))−1/2 = ±

√2

h(c)|ξ |1/2(1 + O(U − c))

(3.4)which implies U − c = O(|ξ |1/2). Therefore we have

U(ξ) = c ±√

2

h(c)|ξ |1/2 + O(|ξ |) = c + λ|ξ |1/2 + O(|ξ |), ξ → 0,

λ = ±√

2

h(c)= ±

√2|c − A|

√A2 − cA,

and

U ′(ξ) = (1/2)λ|ξ |−1/2sign(ξ) + O(1), ξ → 0.

So, U /∈ H1loc(R). �

Let us rewrite (2.6) in the following form

[U ′(U − c)]2 = (U − A)2[(U − c + A)2 − cA]. (3.5)

Then, we have the following proposition.

Proposition 3.3 If u(x, t) = U(x − ct) is a single peak soliton for the DP equa-tion (1.2), then U must be a weak solution of (3.5) in the distribution sense. Inthis sense we say u is a weak solution for the DP equation (1.2).

Proof By the asymptotic estimates in Lemma 3.2, we have U ′(U − c) isbounded, which implies [U ′(U − c)]2 ∈ L1

loc, thus the left hand side of (3.5)does make sense. Since U is bounded, the right hand side of (3.5) is alsoin L1

loc. Thus we may define the distribution function L(U) = [U ′(U − c)]2 −(U − A)2[(U − c + A)2 − cA]. By the definition condition (C2), we know thatsuppL(U) ⊂ {0}. Thus L(U) must be a linear combination of Dirac functionδ(ξ) and its derivatives. However the previous analysis shows L(U) ∈ L1

loc(R).Therefore L(U) = 0. �


If u ∈ H1, the DP equation (1.2) can be equivalently cast into the followingnonlocal conservation law form [1, 4]

L(u) ≡ ut + uux + ∂x(1 − ∂2x)−1

(3

2u2

)= 0. (3.6)

However, if u /∈ H1, (3.6) is no longer equivalent to the DP equation (1.2).Actually, through a direct calculation we can verify that the following function(a stationary cusp soliton)

u(x, t) =√

1 − e−2|x| ∈ W1,1loc (but /∈ H1) (3.7)

satisfies the DP equation (1.2) for any x �= 0. However, the solution (3.7) doesnot solve the nonlocal conservation system (3.6) because

L(u) = e−|x|sign(x).

Therefore it is impossible to find the cusp soliton of the DP equation (1.2)through investigating (3.6). Therefore, we conclude this section with thefollowing remark.

Remark 3.4 The nonlocal DP equation (3.6) is equivalent to the standard DPequation (1.2) under the H1-norm instead of W1,1

loc norm. In fact, the regularpeakon u(x, t) = ce−|x−ct| ∈ H1 (c is the wave speed), which is a stable solutionof the CH equation [5], also satisfies both the nonlocal equation (3.6) andthe DP equation (1.2). But u(x, t) = √

1 − e−2|x| ∈ W1,1loc is a solution of the DP

equation (1.2) in the sense of our definition, but does not satisfy the nonlocalDP equation (3.6).

4 New Single Peak Solitons

Lemma 3.2 (3) gives a classification for all single peak soliton solutions forthe DP equation (1.2). In this section we will present all possible single peaksoliton solutions and find some explicit solution in the case of specific c and A.

We will discuss three cases: cA = 0, cA > 0 and cA < 0.

4.1 Case I: cA = 0

(1) If A = 0, then the only possible single peak soliton is the regular peakonsoliton.

(2) If A �= 0 and c = 0, there is the following stationary cusp soliton solution

u(x, t) = A√

1 − e−2|x| ∈ W1,1loc .


4.2 Case II: cA > 0

By virtue of Lemma 3.2 any single peak soliton for the DP equation (1.2) mustsatisfy the following initial and boundary values problem (IBVP)

⎧⎪⎪⎪⎨⎪⎪⎪⎩

(U ′)2 = g(U) = (U − A)2(U − B1)(U − B2)

(U − c)2;

U(0) ∈ {c, B1, B2};lim|ξ |→∞ U(ξ) = A.

(4.1)

g(U) � 0 and the boundary condition (2.5) imply

U � B1, or U � B2,

and

(A − B1)(A − B2) � 0. (4.2)

By (2.8), (4.2) is equivalent to

(c − A)(c − 4A) � 0. (4.3)

Since A �= 0, introducing the constant α = c/A yields

(α − 1)(α − 4) � 0, (4.4)

which implies:

0 < α < 1; α > 4; α = 1; α = 4.

From the standard phase analysis and Lemma 3.2 we know that if U is a singlepeak soliton of the DP equation, then

U ′ = −U − AU − c

(U − B1)

√U − B2

U − B1sign(ξ), (4.5)

and

U(0) ={

max(c, B1), if U(0) is a minimum,

min (c, B2), if U(0) is a maximum.(4.6)

Let

h(U) = U − c(U − A)(U − B1)

√U − B1

U − B2, (4.7)

then taking the integration of both sides of (4.5) leads to∫

h(U)dU = −|ξ |.After a lengthy calculation of integral, we obtain (α �= 4, i.e. c �= 4A)

∫h(U)dU = sign(A − c)INT1(U) −

√c − Ac − 4A

INT2(U) − K ≡ H(U) − K,

(4.8)


where

INT1(x) = ln∣∣∣ B1 + B2

2− x − √

(x − B1)(x − B2)

∣∣∣, (4.9)

INT2(x)= ln∣∣∣ (A − B1)(x − B2)+(A − B2)(x − B1)+2

√(A − B1)(A − B2)

√(x − B1)(x − B2)

x − A

∣∣∣,

(4.10)

and K is an arbitrary integration constant. Thus we obtain the implicit solutionU defined by

H(U) = −|ξ | + K, (4.11)

where H(U) is very complicated. But its derivative h(U) is simple so that wemay get all single peak soliton through our monotonicity analysis [18].

Apparently,

INT1(B1)=INT1(B2)= ln∣∣∣ B1−B2

2

∣∣∣, INT2(B1)=INT2(B2)= ln|B1−B2|.So, for U(0) = B1 or B2, the constant K0 = H(U(0)) is defined by

K0 = −⎛⎝sign(A) +

√c − A

c − 4A

⎞⎠ ln

√cA −

√c − Ac − 4A

ln2 ∈ R, (4.12)

and for U(0) = c,

K0 = sign(A − c)INT1(c) −√

c − Ac − 4A

INT2(c) ∈ R. (4.13)

4.2.1 Case II.1: 0 < α < 1

1. If A > 0, then B2 < B1 < c < A and U � B1. By standard phase analysis,we have

U(0) = c, c < U < A,

and

H(U) = INT1(U) −√

c − Ac − 4A

INT2(U). (4.14)

H(U) is strictly decreasing on the interval [c, A), thus

H1(U) = H|[c,A)(U) (4.15)

will give an single peak soliton. Apparently,

H1(c) = K0, limU→A

H1(U) = −∞. (4.16)

Therefore U1(ξ) = H−11 (−|ξ | + K0) is the solution satisfying

U1(0) = c, limξ→±∞ U1(ξ) = A,


and

U ′1(±0) = ±∞.

So, U1(ξ) is a kind of cusp soliton solution.2. If A < 0, then B1 > B2 > c > A and U � B2. A similar analysis gives

H(U) = −INT1(U) −√

c − Ac − 4A

INT2(U) (4.17)

is strictly decreasing on the interval (A, c]. Thus

H1(U) = H|(A,c](U) (4.18)

has the inverse denoted by U1(ξ) = H−11 (−|ξ | + K0). U1(ξ) gives a kind of

cusp soliton solution satisfying

U1(0) = c, limξ→±∞ U1(ξ) = A, U ′

1(±0) = ∓∞.

4.2.2 Case II.2: α > 4

1. If A > 0, then c > 4A and

A < B2 < c < B1, U(0) = B2, A < U � B2.

Since H(U) is strictly increasing on the interval (A, B2],H2(U) = H|(A,B2](U) (4.19)

gives a smooth soliton solution. Moreover, U2(ξ) = H−12 (−|ξ | + K0) is the

unique soliton satisfying the IBVP (4.1) with U2(0) = B2 and U ′2(0) = 0.

2. If A < 0, then c < 4A and

B2 < c < B1 < A, U(0) = B1, B1 � U < A.

Through a similar analysis, we know that the strictly increasing on theinterval [B1, A)

H2(U) = H|[B1,A)(U) (4.20)

gives a smooth soliton solution U2(ξ) = H−12 (−|ξ | + K0) satisfying the

IBVP (4.1) with U2(0)= B2 and U ′2(0)= 0.

4.2.3 Case II.3: α = 1

In this case A = c, (2.6) becomes

U ′ = −√

U2 − A2sign(A)sign(ξ), U(±∞) = A.

A direct calculation shows that there is no solution for the above boundarycondition.


4.2.4 Case II.4: α = 4

If A > 0, then B1 = 5A, B2 = A, B2 < c < B1 and if A < 0, then B1 =A, B2 = 5A, B2 < c < B1. For both subcases, there is no single peak soliton.

4.3 Case III: cA < 0

In this case, U(0) = c (see Lemma 3.2). Let us separate two subcases to discuss:(1) c < 0 < A and (2) c < 0 < A.

(1) If A < 0 < c, we have c � U < A and

U ′ = −U − AU − c

√(U − c + A)2 − cAsign(ξ). (4.21)

Let

X = U − c + A, p = A, q = 2A − c, r = √−cA,

then (4.21) becomes

f (X)dX ≡ X − pX − q

dX√X2 + r2

= −sign(ξ)dξ. (4.22)

Integration of both sides of (4.22) gives

F(X) = −|ξ | + K (4.23)

where

F(X)= ln(X+√

X2 − cA) −

−√

(c− A)

(c−4A)

[ln

∣∣∣ (2A−c)X−cA+√(c− A)(c−4A)

√X2−cA

X+c−2A

∣∣∣+ln2]

(4.24)

F(X) is strictly decreasing on the interval [A, 2A − c) and limX→2A−c

F(X) = −∞. Define

F1(X) = F|[A,2A−c)(X). (4.25)

Then

F1(X) = K0 − |ξ |, (4.26)

where

K0 = F(X(0)) = F(A) = ln(A +√

A2 − cA ) ++ c − A√

(c − A)(c − 4A)

[ln(2A +

√4A2 − cA ) + ln2

]∈ R. (4.27)

Since F1 is a strictly decreasing function from [A, 2A − c) onto (−∞, K0] wecan solve for X uniquely from (4.26) and obtain

U(ξ) = F−11 (K0 − |ξ |) + c − A. (4.28)


It is easy to check that U satisfies

U(0) = c, lim|ξ |→∞ U(ξ) = A, U ′(0+) = ∞, U ′(0−) = −∞.

Therefore, the solution U defined by (4.28) is a cusp soliton solution for theDP equation.

(2) If A < 0 < c, we have A < U � c and

U ′ = U − AU − c

√(U − c + A)2 − cAsign(ξ). (4.29)

Similarly, we get a strictly decreasing function F(X) on the interval (2A −c, A] satisfying:

F(X) = |ξ | + K (4.30)

where F(X) is defined by equation (4.24). Let

F1(X) = F|(2A−c,A](X), (4.31)

then F1 is a strictly decreasing function from (2A − c, A] onto [K0, ∞) so thatwe can solve for X and obtain

U(ξ) = F−11 (|ξ | + K0) + c − A. (4.32)

It is easy to check that U satisfies

U(0) = c, lim|ξ |→∞ U(ξ) = A, U ′(0+) = −∞, U ′(0−) = +∞.

Therefore, the solution U defined by (4.32) is also a cusp soliton solution forthe DP equation.

Let us summarize our results in the following theorem.

Theorem 4.1 Assume that the single peak soliton u(x, t) = U(x − ct) (withoutlosing the generality, we assume 0 is the unique peak point of U) of the DPequation (1.2) satisfies the boundary condition (2.5). Then we have

(1) if A = 0, the single peak soliton u(x, t) is only the following peakon

u(x, t) = U(x − ct) = ce−|x−ct|,

with the properties:

U(0) = c, U(±∞) = 0, U ′(0+) = −c, U ′(0−) = c;(2) if A �= 0, let α = c/A, then

(a) if 1 � α � 4, there is no soliton for the DP equation (1.2);(b) if α < 0 (cA < 0), the single peak soliton can be uniquely expressed

as (see Figs. 1 and 2)

u(x, t) = U(x − ct) = F−11 (K0 − sign(A)|x − ct|),


Fig. 1 2D graphic for a cusp soliton with A = 1, c = −1

with the property:

U(0) = c, U ′(0+) = sign(A)∞,

U(±∞) = A, U ′(0−) = −sign(A)∞,

where F1 and K0 are defined by (4.25) (if A > 0), (4.31) (if A < 0)and (4.27) respectively. In this case, the single peak soliton is a cuspsoliton.

Fig. 2 2D graphic for a cusp soliton with A = −1, c = 1


Fig. 3 2D graphic for a cusp soliton with A = 8, c = 5

(c) if α = 0 (c = 0), there is the following stationary cusp soliton

u(x, t) = A√

1 − e−2|x| ∈ W1,1loc ;

(d) if 0 < α < 1, the single peak soliton (see Figs. 3 and 4) of the DPequation (1.2) can be uniquely expressed as

u(x, t) = U(x − ct) = H−11 (−|x − ct| + K0),

with the property:

U(0) = c, U ′(0+) = sign(A)∞,

U(±∞) = A, U ′(0−) = −sign(A)∞,

where H1 and K0 are defined by (4.15) (if A > 0), (4.18) (if A < 0)and (4.13) respectively. In this case, the single peak soliton is a cuspsoliton.

(e) if α > 4, the DP equation (1.2) has the following traveling soliarywave solutions (see Figs. 5 and 6)

u(x, t) = U(x − ct) = H−12 (−|x − ct| + K0)

with the properties:

U(0) = c − A + sign(A)√

cA, U(±∞) = A, U ′(0) = 0,

where H2 and K0 are defined by (4.19)(if A > 0), (4.20)(if A > 0)and (4.12) respectively. In this case, the soliton solution is smooth.


Fig. 4 2D graphic for a cusp soliton with A = −8, c = −5

5 Explicit Solutions

In the previous section we constructed all possible single peak soliton solutionsin our main theorem (Theorem 4.1). But, usually it is very hard to find anexplicit formula of the solution based on the implicit functions H(U) we

Fig. 5 2D graphic for a smooth soliton with A = 1, c = 5


Fig. 6 2D graphic for a smooth soliton with A = −1, c = −5

obtained. However, in some specific cases, we do have explicit solutions. Forthis purpose, let us rewrite (4.5) as follows

U − c(U − A)(U − B1)

√U − B1

U − B2dU = −sign(ξ)dξ. (5.1)

Let

X =√

U − B1

U − B2, a =

√A − B1

A − B2,

then

U = B2 − B1 − B2

X2 − 1, dU = 2(B1 − B2)XdX

(X2 − 1)2,

and (5.1) is converted to

r(

dXX − a

− dXX + a

)+ dX

X + 1− dX

X − 1= −sign(ξ)dξ, (5.2)

where

r = c − Aa(B2 − A)

.

Taking integration on both sides, we arrive at∣∣∣∣

X − aX + a

∣∣∣r∣∣∣ X + 1

X − 1

∣∣∣∣ = C0e−|ξ |, (5.3)


where C0 is a positive constant.In the following, we try to find explicit formulas of cusp soliton and smooth

soliton for some specific number r.

Case 1 0 < α = c/A < 1 In this case, the solution U is a cusp soliton.

1. If A > 0, then we have

0 < c < A, B2 < B1 < c < U < A, U(0) = c,

therefore

0 < X < a < 1.

Equation (5.3) becomes(

a − Xa + X

)r (1 + X1 − X

)= C0e−|ξ |, (5.4)

where

C0 =(

a − X(0)

a + X(0)

)r (1 + X(0)

1 − X(0)

), X(0) =

√A − √

cA

A + √cA

.

For general r, (5.4) is not algebraically solvable. But, a specific value of rmay make (5.4) solvable for X. To this end let us write r as follows

r = A − c√(c − A)(c − 4A)

=√

1 − α

4 − α, 0 < α < 1.

A simple analysis shows that the range of r is 0 < r < 1/2.

2. If A < 0, a similar analysis yields

X > a > 1,

and (X − aX + a

)r (X + 1

X − 1

)= C0e−|ξ |, (5.5)

where

C0 =(

X(0) − aX(0) + a

)r (X(0) + 1

X(0) − 1

), X(0) =

√A − √

cA

A + √cA

The range of r is 0 < r < 1/2.

Notice that r = 1/2 corresponds α = 0 (c = 0). In this case, there is anexplicit stationary cusp soliton

u(x, t) = A√

1 − e−2|x|.

If taking r = 1/3, then we obtain

c = 5/8A, a = 11 − 2√

10sign(A)

9, X(0) = 2

√6 + √

15sign(A)

3,


and

(X − a)(X + 1)3 = b(ξ)(X + a)(X − 1)3, (5.6)

where

b(ξ) = C30e−3|ξ |.

Equation (5.6) is able to be algebraically solvable. But that is a very compli-cated procedure. We omit it here.

Case 2 α = c/A > 4 In this case, the solution U is a smooth soliton.

1. If A > 0, then we have

c > 4A > 0, A < U < B2 < c < B1, U(0) = B2,

X > a > 1, X(0) = ∞,

and

(X − aX + a

)r (X + 1

X − 1

)= e−|ξ |, r =

√α − 1

α − 4, 1 < r < ∞. (5.7)

Let us choose r = 2, then

c = 5A, a = 3 + √5

2,

and (5.7) becomes

(X − a)2(X + 1) = e−|ξ |(X + a)2(X − 1). (5.8)

Substituting a = 3+√5

2 into (5.8), we obtain

X3 − (2 + √5)bX2 + 1 + √

5

2X + 7 + 3

√5

2b = 0, b = 1 + e−|ξ |

1 − e−|ξ | .

(5.9)


With the help of Maple it is easy to check (5.9) has the following real root

X(ξ)=⎡⎣− 7 + 3

√5

3b+

+ 38 +17√

5

27b 3+

√2 +√

5

27+ 517+231

√5

54b 2− 521+233

√5

54b 4

⎤⎦

1/3

+

+⎡⎣− 7 + 3

√5

3b + 38 + 17

√5

27b 3−

−√

2 + √5

27+ 517 + 231

√5

54b 2− 521 + 233

√5

54b 4

⎤⎦

1/3

+ 2 + √5

3b .

(5.10)

Hence, we obtain an explicit formula of the smooth soliton solution(see Fig. 7)

U(ξ) = A

[(4 − √

5) − 2√

5

X(ξ)2 − 1

], A > 0, (5.11)

where X(ξ) is defined by (5.10).

Fig. 7 2D graphic for a smooth soliton with A = 1, c = 5


2. If A < 0, then we have

c < 4A < 0, A > U > B1 > c > B2, U(0) = B1,

0 < X < a < 1, X(0) = 0,

and

(a − Xa + X

)r (1 + X1 − X

)= e−|ξ |, r =

√α − 1

α − 4, 1 < r < ∞. (5.12)

Let us take r = 2, then

c = 5A, a = 3 − √5

2.

In a similar way, we obtain

X3 + (√

5 − 2)bX2 + 1 − √5

2X + 7 − 3

√5

2b = 0, (5.13)

where

b = 1 − e−|ξ |

1 + e−|ξ | . (5.14)

Fig. 8 2D graphic for a smooth soliton with A = −1, c = −5


Solving (5.13) leads to

X(ξ) = ω2

⎡⎣− 7 − 3

√5

3b + 38 − 17

√5

27b 3+

+√

2 − √5

27+ 517 − 231

√5

54b 2 − 521 − 233

√5

54b 4

⎤⎦

1/3

+

+ ω

⎡⎣− 7 − 3

√5

3b + 38 − 17

√5

27b 3−

−√

2 − √5

27+ 517 − 231

√5

54b 2 − 521 − 233

√5

54b 4

⎤⎦

1/3

+

+2 − √5

3b ,

(5.15)

where b is defined by (5.14) and ω = −1+√3i

2 .Thus we obtain another explicit form of smooth soliton solution (see Fig. 8)

U(ξ) = A

[(4 + √

5) + 2√

5

X(ξ)2 − 1

], A < 0, (5.16)

where X(ξ) is defined by equation (5.15).If we take r = 3, then

c = 35

8A, a = 19 + 2

√70sign(A)

9.

We can repeat the above procedure to get explicit soliton solutions corre-sponding to r = 3. This is left for reader’s practice.

6 Conclusions

In this paper, we investigate the DP equation under the inhomogeneousboundary condition. Through the traveling wave setting, the DP equation isconverted to the ODE (2.6), which we solve for all possible single solitonsolutions of the DP equation. Actually, the ODE (2.6) has a physical meaningand can be cast into the Newton equation U ′2 = V(U) − V(A) of a particlewith a new potential V(U)

V(U) = U2 + 4cA(c − A)

U − c+ A(c + A)(c − A)2

(U − c)2.


In the paper, we successfully solve the Newton equation U ′2 = V(U) − V(A)

and give a single peak cusp [including a stationary cusp soliton, see (3.7)] andsmooth soliton solutions in an explicit formula [see (5.11) and (5.16)]. Oursmooth solutions (5.11) and (5.16) are orbitally stable, but we do not knowif our new cuspon (4.32) and the cuspon, defined by equation (4.18), are stable.Very recently, we found a new integrable equation with no classical (smooth)soliton, only possessing weak solutions, such as cuspons and W/M-shape peaksolitons, see the details in paper [16].

Acknowledgement Qiao’s work was partially supported by the UTPA-FRC and theUTPA-URI.

References

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3. Constantin, A., Gerdjikov, V., Ivanov, R.: Inverse scattering transform for the Camassa-Holmequation. Inverse Problems 22, 2197–2207 (2006)

4. Constantin, A., Strauss, W.: Stability of the Camassa–Holm solitons. J. Nonlinear Sci. 12,415–522 (2002)

5. Constantin, A., Strauss, W.: Stability of peakons. Comm. Pure Appl. Math. 53, 603–610 (2000)6. Degasperis, A., Procesi, M.: Asymptotic integrability. In: Degasperis, A., Gaeta, G. (eds.)

Symmetry and Perturbation Theory, World Scientific, pp. 23–37 (1999)7. Degasperis, A., Holm, D.D., Hone, A.N.W.: A new integrable equation with peakon solutions.

Theoret. and Math. Phys. 133, 1463–1474 (2002)8. Holm, D.D., Staley, M.F.: Nonlinear balance and exchange of stability in dynamics of solitons,

peakons, ramps/cliffs and leftons in a 1+1 nonlinear evolutionary PDE. Phys. Lett. A 308,437–444 (2003)

9. Escher, J., Liu, Y., Yin, Z.: Global weak solutions and blow-up structure for the Degasperis–Procesi equation. J. Funct. Anal. 241, 457–485 (2006)

10. Lenells, J.: Traveling wave solutions of the Degasperis–Procesi equation. J. Math. Anal. Appl.306, 72–82 (2005)

11. Mikhailov, A.V., Novikov, V.S.: Perturbative symmetry approach. J. Phys. A, Math. Gen. 35,4775–4790 (2002)

12. Boutet de Monvel, A., Shepelsky, D.: Riemann–Hilbert approach for the Camassa-Holmequation on the line. C. R. Math. Acad. Sci. Paris 343, 627–632 (2006)

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14. Qiao, Z.J.: The Camassa–Holm hierarchy, N-dimensional integrable systems, and algebro-geometric solution on a symplectic submanifold. Comm. Math. Phys. 239, 309–341 (2003)

15. Qiao, Z.J.: Generalized r-matrix structure and algebro-geometric solutions for integrablesystems. Rev. Math. Phys. 13, 545–586 (2001)

16. Qiao, Z.J.: A new integrable equation with cuspons and W/M-shape-peaks solitons. J. Math.Phys. 47, 112701–9 (2006)

17. Qiao, Z.J., Li, S.: A new integrable hierarchy, parametric solution, and traveling wave solution.Math. Phys. Anal. Geom. 7, 289–308 (2004)

18. Qiao, Z.J., Zhang, G.: On peaked and smooth solitons for the Camassa–Holm equation.Europhys. Lett. 73, 657–663 (2006)

19. Vakhnenko, V., Parkes, E.: Periodic and solitary-wave solutions of the Degasperis–Procesiequation. Chaos Solitons Fractals 20, 1059–1073 (2004)


Topological Classification of Morse Functionsand Generalisations of Hilbert’s 16-th Problem

Vladimir I. Arnold

Received: 30 August 2007 / Accepted: 18 September 2007 /Published online: 3 January 2008© Springer Science + Business Media B.V. 2007

Abstract The topological structures of the generic smooth functions on asmooth manifold belong to the small quantity of the most fundamental objectsof study both in pure and applied mathematics. The problem of their study hasbeen formulated by A. Cayley in 1868, who required the classification of thepossible configurations of the horizontal lines on the topographical maps ofmountain regions, and created the first elements of what is called today ‘MorseTheory’ and ‘Catastrophes Theory’. In the paper we describe this problem, andin particular describe the classification of Morse functions on the 2 sphere andon the torus.

Keywords Classification of maps · Morse functions

Mathematics Subject Classifications (2000) 57R99 · 58D15 · 58E05

The topological structures of the generic smooth functions on a smoothmanifold belong to the small quantity of the most fundamental objects of studyboth in pure and applied mathematics. The problem of their study has beenformulated by A. Cayley in 1868, who required the classification of the possibleconfigurations of the horizontal lines on the topographical maps of mountainregions, and created the first elements of what is called today ‘Morse Theory’and ‘Catastrophes Theory’.

V. I. Arnold (B)Steklov Mathematical Institut, Moscow, Russiae-mail: [email protected]

V. I. ArnoldUniversité Paris Dauphine, Paris, France

228 V.I. Arnold

M. Morse has told me, in 1965, that the problem of the description of thepossible combinations of several critical points of a smooth function on amanifold looks hopeless to him. L.S. Pontrjagin and H. Whitney were of thesame opinion.

I formulate below some recent results in this domain. Thus the classificationof the Morse functions on a circle S1 leads to the Taylor coefficients of thetangent function. On the two-dimensional sphere the number of topologicallydifferent Morse functions with T saddle points (that is, having 2T + 2 criticalpoints) grows with T as T2T . The tangent function is replaced in this study bysome elliptic integral (discovered by L. Nicolaescu, while he was continuingArnold’s calculation of the number of topologically different Morse functionshaving 4 saddle points on the sphere S2—that number is equal to 17 746).

Replacing the 2-sphere by the two-dimensional torus, one obtains an infinitenumber of topologically different Morse functions with a fixed number ofcritical points, provided that the topological equivalence is defined by theaction of the identity connected component of the group of diffeomorphisms(or homeomorphisms), that is, if the corresponding mappings of the torus aresupposed to be homotopic to the identity map.

However, if one accepts mappings permuting the parallels and the meridiansof the torus, the classification becomes finite and similar to that of the Morsefunctions on the 2-sphere.

The topological classification of the functions on the 2-sphere is related toone of the questions of the 16-th Hilbert’s problems (on the arrangementsof the planar algebraic curves of fixed degree, defined on the real plane R

2).A real polynomial of fixed degree, defined on the real plane R

2, generates asmooth function on S2 (with one more critical point at infinity), and the topo-logical structure of these functions on the 2-sphere influences the topologicalproperties of the arrangements of the real algebraic curves where the initialpolynomials vanish.

It seems that only a small part of the topological equivalence classesof the Morse functions having T saddle points on S2, contains polynomialrepresentatives of corresponding degree. For instance, in the case of 4 saddles(T = 4) the polynomials are of degree 4 in two real variables, and provide lessthat 1000 topological types, from the total number of different topologic types,which is 17 746: the majority of the classes of smooth Morse functions with 4saddle points have no polynomial representatives.

The classification of the Morse functions on the torus is related similarlyto the topological investigation of the trigonometric polynomials. The degreeof a polynomial is replaced in the case of the functions on T2 by the Newtonpolygon of a trigonometric polynomial (which is the convex hull of the set ofthe wave-vectors of the harmonics forming the trigonometric polynomial).

When such a convex polygon is fixed, its trigonometric polynomials (formedby the harmonics whose wave-vectors belong to the Newton polygon) realizeonly a finite subset of the infinite set of the topologically different classesof functions, classified up to those transformations of the torus which arehomotopic to the identity map. This finite subset depends on the Newton

Morse functions and Hilbert’s 16-th problem 229

polygon, and is small when the polygon is small. It is finite for the followingreason:

In the torus transformations needed to reduce the initial trigonometricpolynomial to the finite set of classes, discussed above, parallels and meridiansare sent only to linear combinations of the parallel and meridian classes whosecoefficients are not large. The number of the homotopy classes of such trans-formations of the torus is finite, making finite the resulting set of topologicalequivalence classes of trigonometric polynomials with fixed Newton polygon(classified up to the torus transformations homotopic to the identity map).

Definition 1 The graph of a Morse function f : M → R on a manifold M is thespace whose points are the connected components of the level hypersurfacesf −1(c) ⊂ M. This space is a finite complex (at least when M is compact).

Example 1 The twin peak mountain Elbrouz (whose height function f is con-sidered continued to the sphere S2 with a minimum D at the antipodal point)generates, as the graph of the height function f , the character ‘Y’ (Fig. 1).

The topographical ‘bergshtrechs’ at the horizontals map show the antigradi-ent vector directions (followed by the grass of the mountain after the rain).

Example 2 The volcanic Vesuvius mountain has a maximum height point A,a height local minimum point B (crater), and a saddle point C, defining thegraph of Fig. 2.

The vertices of the graphs (the end-points A, B, D and the triple point C)represent the critical points of the function f . They form a finite set for a Morsefunction on a compact manifold M.

We see that the graphs of the two preceding examples are topologicallyequivalent, while the two corresponding functions are not. Taking this intoaccount, we will distinguish the graphs, ordering the vertices by the criticalvalues of the height.

D D D

Fig. 1 The twin peak mountain Elbrouz

230 V.I. Arnold

DDD

Fig. 2 The volcanic Vesuvius mountain

For instance, for a Morse function with n critical points, we may fix then critical values to be 1 < 2 < · · · < n. Then the first (Elbrouz) graph wouldobtain the numbering of its vertices

{ f (D) = 1, f (C) = 2, f (B) = 3, f (A) = 4}and the second (Vesuvius) graph’s vertex numbering is

{ f (D) = 1, f (B) = 2, f (C) = 3, f (A) = 4}Therefore we will consider the graphs of the Morse functions to be labeled by theabove ‘height’ numbering (1, 2, . . . , n) of the vertices (ordering then maxima,minima, and saddle points).

Examples 1 and 2 show that the heights of the neighbours of a triple vertexcan’t be all three higher or all three lower than the triple vertex itself (Fig. 3).

The ordering of the vertices verifying this restriction, will be called regular.The regularly ordered graph is a topological invariant of a Morse function.

For Morse functions f on the spheres (M = Sm, m > 1, f : M → R) the graphsare trees. For Morse functions on a surface of genus g, the graph has gindependent cycles (1 for the torus surface T

2, two for the sphere with twohandles, three for the bretzel surface and so on).

In the case of the sphere of dimension 2 this regularly ordered graph isthe only topological invariant of a Morse function f : S2 → R (up to diffeo-morphisms or up to homeomorphisms of S2, if we fix the critical values to be{1, 2, . . . , n}). All regular orderings (of any tree) with n vertices are realized asthe graphs of such Morse functions.

Counting (in 2005) the number ϕ(T) of such regularly ordered trees withT triple points (and of diffeomorphism classes of Morse functions having

Fig. 3 Possible, impossiblecases

Possible cases Impossible cases


values {1, 2, . . . , n} at their 2T + 2 critical points, including the T saddle points)Arnold proved the following:

Theorem 1 There exists positive constants a and b such that for any T theinequalities aTT < ϕ(T) < b T2T hold.

The article [1] claims that the upper bound of Theorem 1 is closer to thegenuine asymptotics of the function ϕ, mentioning some arguments for thisconjecture, based on (unproved) ergodic properties of random graph ordering.

This conjecture was then proved by L. Nicolaescu [6] who replaced therecurrent inequality used by Arnold to prove his upper bound theorem by anexact nonlinear recurrent relation (similar to the Givental’s mirror symmetryproof in quantum field theory).

Nicolaescu’s recurrent relation involves 43 terms of which Arnold hadused only the first leading term to prove his inequality. Using the computer,Nicolaescu solved this nonlinear recurrence explicitly, representing ϕ in termsof the coefficients of the power series expansion of some elliptic integral.

Studying the behaviour in the complex domain of these integrals, Nicolaescusucceeded to prove the Arnold conjecture on the growth of ϕ (leaving un-proved however, the conjecture on the ergodic theory of random graphs whichhad led Arnold to his conjecture on the asymptotics of ϕ).

The first values of the function ϕ (calculated in Arnold [1]) are

T 1 2 3 4

ϕ(T) 2 19 428 17 746

Already the easy computation of ϕ(2) = 19 provides some of the ideas ofthe general structure theory.

According to Nicolaescu’s computer ϕ(5) = 1178792. Arnold was unable todraw all the shapes of functions with T = 5 saddles, drawing only those withT = 4.

In the case T = 5 the upper bound T2T = 102

10 ≈ 1010−3 is only 10 timeshigher than the exact value provided by Nicolaescu.

A Morse polynomial f of degree m in two variables has at most (m − 1)2

critical points. Let m = 2k be even. Suppose that the leading form at infinity(of degree m) is positive. In this case the polynomial provides a Morse function˜f : S2 → R having exactly 4k2 − 4k + 2 critical points (taking that at infinityinto account).

The relation 2T + 2 = 4k2 − 4k + 2 provides the value T = 2k(k − 1) forthe number T of the saddles. For the polynomials of degree m=4 we find k=2and T = 4. Therefore, the topological types of these 4-th degree polynomials(with 9 critical points in R

2) are included in the set of 17 746 classes oftopologically equivalent smooth Morse functions on the sphere S2, with T = 4saddles.

The topological equivalence is defined here by the actions of the homeo-morphisms (or diffeomorphisms) both of the sphere, where the functions are

232 V.I. Arnold

defined, and the orientation preserving homeomorphisms of the axis of values.One also needs the mappings of the axis of values, if the critical values are notfixed e.g. as {1, 2, . . . , 9}, but may move.

According to my calculations, among the 17 746 topologically differentclasses of the Morse functions on S2 with 4 saddles at most, one thousand ofclasses have a polynomial representative of degree 4.

I have no conjecture on the growth rate of the number of classes realizableby polynomials of the corresponding degree, for a growing value of the numberT of saddles.

A trigonometric Morse polynomial of degree n has at most 2n critical pointson the circle S1. For the case where all the 2n critical values are different,the table (from ‘serpent’ theory [4]) provides the following numbers ϕ of thetopologically different functions f : S1 → R with 2n critical points (consideredup to the orientation preserving homeomorphisms or diffeomorphisms of S1

and of R)

2n 2 4 6 8 10

ϕ 1 2 16 272 7 936

Every such class of smooth functions includes some trigonometric polyno-mial of degree n.

Theorem 2 The numbers ϕ(n) of classes of the oriented topological equivalenceof Morse functions with 2n critical points and 2n critical values on the circle,provide the Taylor series of the tangent function (as of the exponential generatingfunction):

tan t = 1

1! t + 2

3! t3 + 16

5! t5 + 272

7! t7 + 7936

9! t9 + . . .

We consider next the trigonometric polynomials of two variables as somesmooth functions on the 2-torus, f : T

2 → R.

Example 3 The ˜A2 affine Coxeter group of trigonometric polynomials ofdegree 1 form the following 6-parameter vector space of trigonometric poly-nomials in two variables x and y:

(∗) f = a cos x + bsin x + c cos y + d sin y + p cos(x + y) + q sin(x + y).

This family of trigonometric polynomials has been studied in the articles byArnold [2, 3]. The number of critical points of such Morse polynomials (∗)

on T2 does not exceed 6. In the general case of the arbitrary trigonometric

polynomials the upper bound of the number of critical points on T2 is provided

by the doubled area of the Newton polygon (n! the volume of the Newtonpolyhedron for the n-torus case T

n) (Fig. 4).


Fig. 4 Newton polygon

Newton polygon

T∗T2area = 3

saddles number, T = 3

Definition 2 The Diff-equivalence of smooth functions on T2 is the belonging

to the same orbit of the natural action on functions on the torus of the groupDiff (T2) of the diffeomorphisms of the torus, accompanied by the orientationpreserving diffeomorphisms of the axis of the values.

The Diff0-equivalence is defined similarly, replacing, however, the groupDiff(T2) by its connected subgroup Diff0(T

2), formed by the diffeomorphismshomotopic to the identity, which is the connected component of the identity inDiff(T2) (whose elements may interchange the meridian and parallel classes).

The articles by Arnold [2, 3] contain the proof of

Theorem 3 The number of equivalence classes of the C∞ Morse functionswith 6 critical points on the two-dimensional torus T

2 and of the trigonometricpolynomials (∗) of the affine Coxeter group ˜A2 have the following values:

Diff Diff0

C ∞ 16 ∞˜A2 2 6

The 16 topologically different types of the smooth Morse functions with 6critical points and values on the torus are described in Arnold [3] by the 16regularly ordered graphs having 3 triple points (�), three end points (©) andone cycle (for each graph). These 16 graphs, ordered by the natural heightfunction, are shown in Fig. 5.

Among all these topological equivalence classes only two classes (markedby the sign (∗)) contain some of the trigonometric polynomials.

All the other 14 cases are eliminated by the algebraic geometry of the ellipticlevel curves of functions (∗), as is explained in articles by Arnold [2, 3].

This finite set of the 16 Diff -equivalence classes provides an infinity ofdifferent Diff0-classes for the following reason.

A generic point P on the cycle of the graph does not separate the graph.Hence the level line component, represented by P, does not separate the torusT

2 into two parts, and therefore this closed curve is not homologous to zero onthe torus.

Its homology class does not depend on the choice of the point P on thecycle, since any pair of such points {P, P}′ separates the graph into two parts,and hence the cycle P − P′ is the boundary of a domain on the torus.

This 1-dimensional homology class is an invariant of the Diff0-equivalence(at least up to the choice of its sign).

234 V.I. Arnold

Fig. 5 The 16 graphes

For suitable choices of Morse functions with 6 critical points and val-ues on the torus, it can take an infinity of different values. All these 1-homology classes are Diff -equivalent, each of them being represented by anon-bounding, non-self-intersecting closed curve on the torus.

Thus we obtain an infinite set of the Diff 0-equivalence classes of the Morsefunctions having 6 critical points and 6 critical values on the 2-torus T

2.However the trigonometric polynomials (∗) provide only a finite part of this

infinite set of the Diff0-equivalence classes. Namely, in Arnold [2] it is provedthat the algebraic geometry of real elliptic curves restricts the homology classof the level line component P on the torus: it may attain only 3 values (or 6 ifwe take the orientation into account).

These 6 realizable classes form in the one-dimensional homology group ofthe torus the Dynkin diagram of the Coxeter group A2 (whence the name ˜A2

of the class (∗) of trigonometric polynomials) (Fig. 6).For the other classes (say defined by arbitrary Newton polygons) the

number of Diff -equivalence and Diff0-equivalence classes are unknown. I donot know how fast the number of Diff -equivalence classes grows when thenumber T of saddle points is large (neither for the smooth Morse functions,nor for the trigonometric polynomials having a fixed Newton polygon). Even


Fig. 6 H1(T2, Z)

the finiteness proof for the number of Diff -equivalence classes (for each valueof T) is not published.

This abundance of unsolved problems was the reason to choose thesequestions of real algebraic geometry and of Morse functions statistics for thispaper: I hope that the readers will go further than me in this rapidly evolvingdomain at the intersection of all the branches of mathematics.

It is rather strange that the computer contribution to real algebraic geometryis still almost negligible while theoretical mathematicians have made a lot.

The only real contribution that I know, is the recent result of a formerstudent of the Université Paris-Jussieu, Adriana Ortiz-Rodriguez, working atMexico (and having started these works in Paris). This result describes thetopology of the parabolic curves on algebraic surfaces of the projective space,and belongs to the intersection of real algebraic geometry and symplecticgeometry.

The graph {z = f (x, y)} of a polynomial f in two real variables x and y is asmooth surface in R

3. The parabolic lines of this surface are projected to the{(x, y)}-plane as algebraic curves of degree 2n − 4.

The question studied by A. Ortiz-Rodriguez is to evaluate the possiblenumbers of closed parabolic curves for polynomials of a given degree: howlarge may this number of parabolic curves be? And how large may be thenumber of connected components of the corresponding algebraic curve ofdegree 2n − 4 in the real projective plane?

The classical Harnack theorem of real algebraic geometry implies that forthe polynomial f of degree n = 4 the number M of closed parabolic curvescannot exceed 4.

It is not too difficult to construct examples of polynomials f of degree 4, forwhich the number of closed parabolic curves attains the value M = 3. But theproblem of wether the case of 4 closed parabolic curves is realizable by somepolynomial of degree 4 resisted to all the attempts of mathematicians, and onlythe computer helped to solve it.

Namely, in a year of uninterrupted calculations, the Mexico computer ofA. Ortiz-Rodriguez has studied 50 millions of polynomials of degree 4.

Among all these polynomials, just 3 polynomials have, each of them, 4closed parabolic curves on their graphs.

236 V.I. Arnold

For the polynomials f of degree n, A. Ortiz-Rodriguez has proved (in herthesis, preceding the experiment described above) the inequalities

an2 � M � bn2

for the maximal number M(n) of closed parabolic curves on the graph of apolynomial of degree n: for any such polynomial, the number of paraboliccurves is at most bn2, and there exist polynomials of degree n for which theparabolic curve number is at least an2.

Unfortunately, b is higher than a, and thus the question of the genuineasymptotics of M(n) is waiting for the courageous researchers (and computerexperiments).

The situation is even more difficult with another similar problem (alsostudied by A. Ortiz-Rodriguez). Consider in the 3-dimensional projectivespace RP3 a smooth algebraic surface of degree n. How large may be thenumber of its closed parabolic curves?

Here the Ortiz-Rodriguez boundaries are

An3 � M � Bn3,

but the coefficient B is a dozen of times higher than A and the genuine asymp-totic behaviour of the maximal number M(n) of the connected componentsof the parabolic curves on smooth real projective hypersurfaces of degree nremains unknown.

These real algebraic geometry versions of the higher dimensional Plückerformulae remain a challenge for modern algebraic geometry, which seems,however, to be unable to study the real things. Of course the celebratedTarski-Seidenberg theorem implies the existence of an algorithm providing theneeded answers, but the required time for the real application is usually muchlarger than the whole life span of the universe.

References

1. Arnold, V.I.: Smooth functions statistics. Funct. Anal. Other Math. 1(2), 125–133 (2006) (ICTPPreprint IC2006/012, 1–9 (2006))

2. Arnold, V.I.: Topological classifications of trigonometric polynomials, related to affineCoxeter group A2. ICTP Preprint IC2006/039, 1–15 (2006)

3. Arnold, V.I.: Dynamical systems: modeling, optimization, and control. Proc. Steklov Inst. Math.Suppl. 1, 13–23 (2006)

4. Arnold, V.I.: Snake calculus and the combinatorics of the Bernoulli, Euler and Springer num-bers. Russian Math. Surveys 47(1), 1–51 (1992)

5. Arnold, V.I.: Topological classification of real trigonometric polynomials and cyclic serpentspolyhedron. The Arnold-Gelfand Seminars, pp. 101–106. Boston, Birkhäuser (1996)

6. Nicolaescu, L.I.: Morse functions statistics. Funct. Anal. Other Math. 1(1), 97–103 (2006)(Counting Morse functions on the 2-sphere, Preprint math.GT/0512496)


Limit Cycles Bifurcating from a k-dimensionalIsochronous Center Contained in R

n with k � n

Jaume Llibre · Marco Antonio Teixeira ·Joan Torregrosa

Received: 12 January 2007 / Accepted: 16 October 2007 /Published online: 7 December 2007© Springer Science + Business Media B.V. 2007

Abstract The goal of this paper is double. First, we illustrate a method forstudying the bifurcation of limit cycles from the continuum periodic orbits of ak-dimensional isochronous center contained in R

n with n � k, when we perturbit in a class of C2 differential systems. The method is based in the averagingtheory. Second, we consider a particular polynomial differential system in theplane having a center and a non-rational first integral. Then we study thebifurcation of limit cycles from the periodic orbits of this center when weperturb it in the class of all polynomial differential systems of a given degree.As far as we know this is one of the first examples that this study can be madefor a polynomial differential system having a center and a non-rational firstintegral.

Keywords Limit cycle · Periodic orbit · Center · Isochronous center ·Averaging method · Generalized Abelian integral

Mathematics Subject Classifications (2000) 34C29 · 34C25 · 47H11

The first and third authors are partially supported by a MCYT/FEDER grantMTM2005-06098-C01, and by a CIRIT grant number 2005SGR-00550. The second authoris partially supported by a FAPESP–BRAZIL grant 10246-2. The first two authors are alsosupported by the joint project CAPES–MECD grant HBP2003-0017.

J. Llibre · J. Torregrosa (B)Departament de Matemàtiques, Universitat Autònoma de Barcelona,08193 Bellaterra, Barcelona, Spaine-mail: [email protected]

J. Llibree-mail: [email protected]

M. A. TeixeiraDepartamento de Matemática, Universidade Estadual de Campinas,Caixa Postal 6065, 13083-970, Campinas, São Paulo, Brazile-mail: [email protected]

238 J. Llibre et al.

1 Introduction and Statement of the Results

We say that a singular point p ∈ Rn of a differential system is a k-dimensional

center if there exists a k-dimensional submanifold M of Rn with k � n such

that p ∈ M, M is invariant under the flow of the differential system, and all theorbits M \ {p} are periodic. Moreover, we say that the k-dimensional center pis isochronous if all its periodic orbit have the same period.

In the first part of this paper we illustrate a method for studying the limitcycles bifurcating from the periodic orbits of a k-dimensional isochronouscenter contained in R

n with k � n, by studying with all the details an examplewith k = 2 and n = 4.

In recent years equations of the form

xIV + bxII + ax = ψ(t, x, xI, xII, xIII)

arise in many contexts. For example, the simplest cases when ψ = x2 andψ = x3 describe the travelling waves solutions of some Korteweg–de Vriesequations (KdV) and nonlinear Schrödinger equations, see [3, 8, 9]. On theother hand, the existence of periodic solutions is discussed in [4] for equationsmodeling undamped oscillators and having the form xII + ω2

0x = φ(t) whereω0 > 0, and φ is a continuous periodic function whose period is normalized to2π . In this paper we deal with a particular differential equation of order fourof the form

d4xdt4

+ α x + ψ(x, t) = 0.

This class of equations have been studied in Peletier and Troy [10] and Sanchez[12]. Here we will analyze the particular differential equation

d4xdt4

− x − ε sin(x + t) = 0,

or equivalently the differential system

x = y,

y = z,

z = w,

w = x + ε sin(x + t), (1)

where the dot denotes the derivative with respect to the time variable t. Ourmain result is the content of the following theorem.

Theorem 1 For |ε| �= 0 sufficiently small the differential system (1) has anarbitrary number of limit cycles bifurcating from the continuum of the periodicorbits of the 2-dimensional isochronous center that the system has for ε = 0.

The proof of Theorem 1 is given in Section 3, and uses the averaging theory,more precisely the proof uses Theorem 3.

Limit cycles bifurcating from an isochronous center 239

In the second part we deal with the homogeneous polynomial differentialsystem

x = −y(3x2 + y2),

y = x(x2 − y2), (2)

of degree 3 that has the non-rational first integral

H(x, y) = (x2 + y2

)exp

(− 2x2

x2 + y2

).

Theorem 2 The homogeneous polynomial differential system (2) has a globalcenter at the origin (i.e. all the orbits contained in R

2 \ {(0, 0)} are periodic).Let P(x, y) and Q(x, y) be two polynomials of degree at most m. Then, forconvenient polynomials P and Q, the polynomial differential system

x = −y(3x2 + y2

) + εP(x, y),

y = x(x2 − y2

) + εQ(x, y), (3)

has [(m − 1)/2] limit cycles bifurcating from the periodic orbits of the globalcenter (2), where [·] denotes the integer part function.

As far as we know this is one of the first examples for which the limit cyclesbifurcating from the periodic orbits of a 2-dimensional center of a polynomialdifferential system having a non-rational first integral have been studied. Theunique other example that we know was given recently in [6].

The proof of Theorem 2 is given in Section 4. Again we use averaging.More precisely, we will apply Theorem 4, which gives a method to determinebifurcation of periodic solutions from isochronous centers. We note that thecenter of system (2) is not isochronous; but, after a change of variables, it canbe transformed to an isochronous center.

We also show in Section 4 that Theorem 2 can be proved using the theorybased on the generalized Abelian integrals, see a definition of these integralsat the end of Section 2.

2 Basic Results

In this section first we present the basic results from the averaging theory andAbelian integrals that we shall need for proving the main results of this paper.

We consider the problem of the bifurcation of T-periodic solutions from thedifferential system

x′(t) = F0(t, x) + εF1(t, x) + ε2 F2(t, x, ε), (4)

with ε = 0 to ε �= 0 sufficiently small. Here, the functions F0, F1 : R × � → Rn

and F2 : R × � × (−ε0, ε0) → Rn are C2 functions, T-periodic in the first


variable, and � is an open subset of Rn. One of the main assumptions is that

the unperturbed system

x′(t) = F0(t, x), (5)

has a submanifold of periodic solutions. A solution of this problem is givenusing the averaging theory. For a general introduction to the averaging theorysee the books of Sanders and Verhulst [13], and of Verhulst [14].

Let x(t, z) be the solution of the unperturbed system (5) such that x(0, z)=z.We write the linearization of the unperturbed system along the periodicsolution x(t, z) as

y′ = Dx F0(t, x(t, z))y. (6)

In what follows we denote by Mz(t) some fundamental matrix of the lineardifferential system (6), and by ξ : R

k × Rn−k → R

k the projection of Rn onto

its first k coordinates; i.e. ξ(x1, . . . , xn) = (x1, . . . , xk).

Theorem 3 Let V ⊂ Rk be open and bounded, and let β0 : Cl(V) → R

n−k be aC2 function. We assume that

(i) Z = {zα = (α, β0(α)) , α ∈ Cl(V)} ⊂ � and that for each zα ∈ Z the solu-tion x(t, zα) of (5) is T-periodic;

(ii) for each zα ∈ Z there is a fundamental matrix Mzα(t) of (6) such that the

matrix M−1zα

(0) − M−1zα

(T) has in the upper right corner the k × (n − k)

zero matrix, and in the lower right corner a (n − k) × (n − k) matrix �α

with det(�α) �= 0.

We consider the function F : Cl(V) → Rk

F(α) = ξ

(∫ T

0M−1

zα(t)F1(t, x(t, zα))dt

). (7)

If there exists a ∈ V with F(a) = 0 and det ((dF/dα) (a)) �= 0, then there is aT-periodic solution ϕ(t, ε) of system (4) such that ϕ(0, ε) → za as ε → 0.

Theorem 3 goes back to Malkin [7] and Roseau [11], for a shorter proof seeBuica et al. [2].

We assume that there exists an open set V with Cl(V) ⊂ � such that foreach z ∈ Cl(V), x(t, z, 0) is T-periodic, where x(t, z, 0) denotes the solution ofthe unperturbed system (5) with x(0, z, 0) = z. The set Cl(V) is isochronousfor the system (4); i.e. it is a set formed only by periodic orbits, all of themhaving the same period. Then, an answer to the problem of the bifurcation ofT–periodic solutions from the periodic solutions x(t, z, 0) contained in Cl(V) isgiven in the following result.


Theorem 4 (Perturbations of an isochronous set) We assume that there existsan open and bounded set V with Cl(V) ⊂ � such that for each z ∈ Cl(V), thesolution x(t, z) is T-periodic, then we consider the function F : Cl(V) → R

n

F(z) =∫ T

0M−1

z (t, z)F1(t, x(t, z))dt. (8)

If there exists a ∈ V with F(a) = 0 and det ((dF/dz) (a)) �= 0, then there exists aT-periodic solution ϕ(t, ε) of system (4) such that ϕ(0, ε) → a as ε → 0.

For a proof of Theorem 4 see Corollary 1 of Buica et al. [2].Now we summarize the results on generalized Abelian integrals that we

shall use.Suppose that the unperturbed system

x = f (x, y),

y = g(x, y), (9)

has a first integral H(x, y) with an integrating factor 1/R(x, y). Assume thatthe origin of this system is a center and that the periodic orbits of this centerare given by the family of ovals γh contained in the level curves {H(x, y) = h}.

Now we consider the perturbed system

x = f (x, y) + εP(x, y),

y = g(x, y) + εQ(x, y),

which can be written into the form

x = −∂ H∂y

(x, y)R(x, y) + εP(x, y),

y = ∂ H∂x

(x, y)R(x, y) + εQ(x, y). (10)

Then the generalized Abelian integral associated to this system is

I(h) =∫

γh

P(x, y)dy − Q(x, y)dxR(x, y)

. (11)

Since I(h) gives the first order approximation in ε of the displacement function,we get the following result.

Theorem 5 The simple zeros of the function I(h) provide limit cycles for theperturbed system (10) which bifurcate from the periodic orbits of the unper-turbed system (9).

For more details about (generalized) Abelian integrals and the proof ofTheorem 5 see Li [5].


3 Perturbation of a 2-Dimensional Isochronous Center in R4

In this section we prove Theorem 1.The linear part at the origin of the differential system (1) is given by the

matrix⎛

⎜⎜⎝

0 1 0 00 0 1 00 0 0 11 0 0 0

⎞

⎟⎟⎠ , (12)

and its eigenvalues are ±1 and ±i. Doing the change of variables (x, y, z, w) �→(X, Y, Z , W) given by

⎛

⎜⎜⎝

XYZW

⎞

⎟⎟⎠ =

⎛

⎜⎜⎝

1 −1 −1 1−1 −1 1 1

1 1 1 1−1 1 −1 1

⎞

⎟⎟⎠

⎛

⎜⎜⎝

xyzw

⎞

⎟⎟⎠ ,

system (1) becomes

X = −Y + ε sin((4t + X − Y + Z − W)/4),

Y = X + ε sin((4t + X − Y + Z − W)/4),

Z = Z + ε sin((4t + X − Y + Z − W)/4),

W = −W + ε sin((4t + X − Y + Z − W)/4). (13)

Note that the differential of this system at the origin is the real normal Jordanform of the matrix (12).

Now we shall apply Theorem 3 to the differential system (13) taking

x = (X, Y, Z , W),

F0(t, x) = (−Y, X, Z , −W),

F1(t, x) = (A, A, A, A),

F2(t, x, ε) = 0,

� = R4, (14)

where A = sin((4t + X − Y + Z − W)/4).Clearly system (13) with ε = 0 has a linear center at the origin in the (X, Y)-

plane. We remark that all linear centers are isochronous. Using the notation ofSection 2 (mainly the notation related with the statement of Theorem 3), theperiodic solution x(t, z) of this center with z = (X0, Y0, 0, 0) is

X(t) = X0 cos t − Y0 sin t,

Y(t) = Y0 cos t + X0 sin t,

Z (t) = 0,

W(t) = 0, (15)


with period T = 2π . The V and α of Theorem 3 are

V = {(X, Y, 0, 0) : 0 < X2 + Y2 < ρ},for some real number ρ > 0, and α = (X0, Y0) ∈ V.

For the function F0 given in (14) and the periodic solution x(t, z, 0) given in(15) the fundamental matrix M(t) of the differential system (6) such that M(0)

is the identity matrix of R4 is

M(t) =

⎛

⎜⎜⎝

cos t − sin t 0 0sin t cos t 0 0

0 0 et 00 0 0 e−t

⎞

⎟⎟⎠ .

We remark that for system (13) with ε = 0 the fundamental matrix does notdepend on the particular periodic orbit x(t, z); i.e. it is independent of the initialconditions z. Therefore, an easy computation shows that

M−1(0) − M−1(2π) =

⎛

⎜⎜⎝

0 0 0 00 0 0 00 0 1 − e−2π 00 0 0 1 − e2π

⎞

⎟⎟⎠ .

Consequently all the assumptions of Theorem 3 are satisfied. Therefore,we must study the zeros in V of the system F(α) = 0 of two equations andtwo unknowns, where F is given by (7). More precisely, we have F(α) =(F1(X0, Y0),F2(X0, Y0)) where

F1 =∫ 2π

0(cos t + sin t) sin

[t + (X0 − Y0) cos t − (X0 + Y0) sin t

4

]dt,

F2 =∫ 2π

0(cos t − sin t) sin

[t + (X0 − Y0) cos t − (X0 + Y0) sin t

4

]dt.

After a tedious calculation (which can be checked using an algebraic proces-sor) and the change of variables (X0, Y0) �→ (r, s) given by

X0 − Y0 = 4r cos s,X0 + Y0 = −4r sin s,

we obtain for

gj(r, s) = Fj(2r(cos s − sin s), −2r(cos s + sin s)),

with j = 1, 2, that

g1(r, s) = π [J0(r) + J2(r)(cos 2s − sin 2s)],g2(r, s) = −π [J0(r) + J2(r)(cos 2s + sin 2s)],

where Jμ(r) is the μ-th Bessel function of first kind (see Abramowitz andStegun [1]).


Adding and subtracting the two equations g j(r, s) = 0, for j = 1, 2, we obtainthe system

h1(r, s) = J2(r) sin 2s = 0,

h2(r, s) = J0(r) + J2(r) cos 2s = 0. (16)

It is known that the zeros of the functions Jμ(r) are distinct for different μ’s,then either s = 0, or s = π/2. We are not interested in all the solutions of thissystem, we are only interested to show that it has as many solutions as we wantsatisfying the assumptions of Theorem 3. So, in what follows we only study thesolutions with s = 0. Consequently, from the second equation of (16) we obtain

J0(r) + J2(r) = 0.

Since J0(r) + J2(r) = 2J1(r)/r, and the function J1(r) has infinitely many pos-itive zeros tending to be uniformly distributed when r → ∞, because theasymptotic behavior of J1(r) is

√2/(πr) cos(r − 3π/4), it follows that system

(16) has infinitely many solutions of the form (r0, 0) being r0 a positive zero ofJ1(r). Then, (X0, Y0) = (2r0, −2r0) is a solution of the system Fj(X0, Y0) = 0for j = 1, 2. Moreover, the determinant of ∂(F1,F2)/∂(X0, Y0) at the point(2r0, −2r0) is

det(r0) = π2

8r2

0H(3, −r2

0/2)[H

(3, −r2

0/2) − H

(2, −r2

0/2)].

where H is the regularized hypergeometric function, see Abramowitz andStegun [1]. Using the formula

Jμ(z) = z2

2μ(μ + 1)! H(μ + 1, −z2/4

),

we get

det(r0) = 72 π2 J2(r0)2

r20

.

Since the zeros of J1(r) and J2(r) are different, we get that det(r0) �= 0. Hence,by Theorem 3 for each (2r0, −2r0) contained in V we have a periodic orbit ofsystem (13) with |ε| �= 0 sufficiently small.

Finally, for a given positive integer N we can fix ρ in the definition of Vin such a way that the interval (0, ρ) contains exactly N zeros of the functionJ1(r). Then taking |ε| �= 0 small enough, Theorem 3 guarantees the existenceof N periodic orbits for system (13). Moreover, choosing |ε| �= 0 smaller ifnecessary, since system (13) with ε = 0 has its periodic orbits strongly stableand unstable in the directions Z and W respectively, it follows that the Nperiodic orbits for system (13) obtained using Theorem 3 are limit cycles; i.e.they are isolated in the set of all periodic orbits. This completes the proof ofTheorem 1.


4 Perturbation of a 2-Dimensional Center having a Non-rational First Integral

First we show that the homogeneous polynomial differential system (2) has aglobal center at the origin. In polar coordinates (r, θ) defined by x = r cos θ ,y = r sin θ , system (2) becomes

r = −r3 sin 2θ,

θ = r2.

Of course, to study this system is equivalent to study the differential equation

drdθ

= −r sin 2θ, (17)

whose solution r(θ, z) satisfying r(0, z) = z is

r(θ, z) = z exp(− sin2 θ

). (18)

Therefore all the solutions of the differential equation (17) and consequentlyall the solutions of the homogeneous polynomial differential system (2) areperiodic with the exception of the origin which is a singular point. Hence it isproved that the origin of system is a global center.

Now we want to study the limit cycles of the perturbed system (3) for |ε| �=0sufficiently small, which bifurcate from the periodic orbits of the center ofsystem (2).

We write the polynomial P(x, y) of degree m of system (2) as

P(x, y) =m∑

l=0

Pl(x, y),

where Pl(x, y) is the homogeneous part of degree l of P(x, y). We do the samefor the polynomial Q(x, y).

Doing for system (13) the same changes of variables that we have done tosystem (2), we obtain that system (2) can be written as

drdθ

= −r sin 2θ + εF1(θ, r) + O(ε2

), (19)

where

F1(θ, r) =m∑

l=0

rl − 2 [(cos θ + cos 3θ)Pl(cos θ, sin θ)++ (3 sin θ + sin 3θ)Ql(cos θ, sin θ)].

Now we shall apply Theorem 4 to the differential equation (19) taking k =n = 1 and

x = r,

t = θ,

F0(θ, x) = −r sin 2θ,

� = (0, ∞). (20)


Clearly the differential equation (19) is T = 2π periodic in the variable θ .Moreover this equation for ε = 0 has all its solutions 2π -periodic and given by(18). The V and α of Theorem 4 are

V = {r : 0 < r < ρ},for some real number ρ > 0, and α = z ∈ V.

For the function F0 given in (20) and the periodic solution r(θ, z) given in(18) the 1 × 1 fundamental matrix M(θ) of the differential equation (19) withε = 0 such that M(0) = (1) is

M(θ) = (e− sin2 θ

).

We remark that for system (20) the fundamental matrix does not depend onthe particular periodic orbit r(θ, z); i.e. it is independent of the initial conditionz. Therefore

M−1(θ) = (esin2 θ

).

Since all the assumptions of Theorem 4 are satisfied, we must study the zerosin V of the function F(z), where F is given by (8). More precisely, we have

F(z) =m∑

l=0

zl−2 Il,

where

Il =∫ 2π

0e(3−l) sin2 θ [(cos θ + cos 3θ)Pl(cos θ, sin θ)+

+ (3 sin θ + sin 3θ)Ql(cos θ, sin θ)]dθ.

By symmetry the integral Il = 0 if l is even. So, if m = 2ν + 1 then

F(z) = 1

z

ν∑

l=0

z2l I2l+1. (21)

Hence the polynomial F(z) at most can have ν = [(m − 1)/2] positive realroots. If m = 2ν then

F(z) = 1

z

ν−1∑

l=0

z2l I2l+1. (22)

Therefore, again the polynomial f (z) at most can have ν − 1 = [(m − 1)/2]positive real roots. In short, using Theorem 4 we at most can get [(m − 1)/2]limit cycles of system (3) bifurcating from the periodic orbits of system (2).

We shall prove that when m = 2ν + 1 the function (21) for the perturbedsystem (3) with

P(x, y) =ν∑

l=0

a2l+1x2l+1, Q(x, y) = 0, (23)


can be chosen in order that it has exactly ν = [(m − 1)/2] positive arbitraryzeros. In a similar way it can be proved that when m = 2ν the function (22) forthe perturbed system (3) with

P(x, y) =ν−1∑

l=0

a2l+1x2l+1, Q(x, y) = 0,

can be chosen in order that it has exactly ν − 1 = [(n − 1)/2] positive arbitraryzeros. Hence Theorem 4 will be proved.

Now for m = 2ν + 1 we consider system (3) with P and Q given by (23).Then for its corresponding function (21) we have

I2l+1 = a2l+1

∫ 2π

0e2(1−l) sin2 θ (cos θ + cos 3θ) cos2l+1 θ dθ.

Again after a tedious computation (that we can help with an algebraic manip-ulator as mathematica or maple) we obtain that I2l+1 is equal to

L(l) = (12l2 − 28l + 19) H(l + 1/2, l + 3, 2l − 2) ++(l + 2)(6l − 5) H(l − 1/2, l + 2, 2l − 2), (24)

multiplied by the constant

2L − 1

(l + 2)!√

π �(l − 1/2) e2−2l,

where H(a, b, z) is the Kummer confluent hypergeometric function and �(z) isthe Gamma function, see Abramowitz and Stegun [1]. The value of L(l) is non-zero for all non-negative integer l, see the Appendix. Hence in the polynomial(21) we always can choose the coefficients al+1 conveniently and alternatingthe sign (by the Descartes rule) in order that the polynomial has the maximumpossible number of positive roots ν = [(m − 1)/2]. This completes the proof ofTheorem 2.

Finally we remark that if we compute the generalized Abelian integral (11)for the system (3) taking

x(θ) = ze− sin2 θ cos θ, y(θ) = ze− sin2 θ sin θ,

as a parametrization of the center when ε = 0 and integrating with respect tothe variable θ between 0 and 2π , we obtain that

I(z) = ze2

F(z).

Hence both functions have the same positive zeros, and in this case the methodbased in Theorem 3 and the method based in the generalized Abelian integralcoincide.


Appendix

The Kummer confluent hypergeometric function is given by the series

H(a, b, z) =∞∑

μ=0

�(a + μ)�(b)

�(a)�(b + μ)μ! zμ. (25)

We shall prove that L(l) defined in (24) is always positive for any integer l � 0.For showing that we will compute the coefficients of the series expansion ofL(l) and we will see that all of them are positive.

Using (25) we obtain that

L(l) =∞∑

μ=0

Lμ,l (2l − 2)μ,

where the coefficient Lμ,l is equal to

(36l2 − 72l + 43)μ + 3(2l − 1)(6l2 − 7l + 3)(l + 2)! �(l + μ − 1/2)

�(l + 1/2)(l + μ + 2)! μ! .

For l > 0 since 36l2 − 72l + 43 > 0, 6l2 − 7l + 3 > 0, �(l + μ − 1/2) > 0 and�(l + 1/2) > 0 it follows that Lμ,l > 0 for μ = 0, 1, 2 . . . and l > 0. ThereforeL(l) > 0 if l > 0.

Finally, from (25) we have

L(0) = 2eπ(J0(1) − 2J1(1)) ≈ 2.31849804 > 0.

References

1. Abramowitz, M., Stegun, I.A.: Bessel Functions J and Y, 9.1. In: Handbook of MathematicalFunctions with Formulas, Graphs, and Mathematical Tables, 9th printing, pp. 358–364. Dover,New York (1972)

2. Buica, A., Françoise, J.P., Llibre, J.: Periodic solutions of nonlinear periodic differentialsystems with a small parameter. Commun. Pure Appl. Anal. 6, 103–111 (2007)

3. Champneys, A.R.: Homoclinic orbits in reversible systems and their applications in mechanics,fluids and optics. Phys. D 112, 158–186 (1998)

4. Fabry, C., Mawhin, J.: Properties of solutions of some forced nonlinear oscillations at res-onance. Progress in Nonlinear Analysis. In: Proc. of the Second Conference on NonlinearAnalysis, pp 103–118. Tianjin, China (1999)

5. Li, C.: Abelian integrals and applications to weak Hilbert’s 16th problem. In:Christopher, C., Li, C. (eds.) Limit Cycles of Differential Equations. Advanced Coursesin Mathematics, pp. 91–162. CRM Barcelona, Birkhaüser, Basel (2007)

6. Li, J.: Limit cycles bifurcated from a reversible quadratic center. Qual. Theory Dyn. Syst. 6,205–216 (2005)

7. Malkin, I.G.: Some problems of the theory of nonlinear oscillations. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (1956) (Russian)

8. Ostrovski, L., et al.: On the existence of stationary solitons. Phys. Lett. A 74, 177–170 (1979)9. Peletier, L.A., Troy, W.C.: Spatial patterns described by the extended Fisher–Komolgorov

equation: Kinks. Differential Integral Equations 8, 1279–1304 (1995)10. Peletier, L.A., Troy, W.C.: Spatial Patterns. Higher Order Models in Physics and Mechanics.

Progress in Nonlinear Differential Equations and their Applications, vol. 5. Birkhaüser,Boston (2001)


11. Roseau, M.: Vibrations non linéaires et théorie de la stabilité, (French) Springer Tracts inNatural Philosophy, vol.8. Springer, Berlin Heidelberg New York (1966)

12. Sanchez, L.: Boundary value problems for some fourth order ordinary differential equations.Appl. Anal. 38, 161–177 (1990)

13. Sanders, J.A., Verhulst, F.: Averaging methods in nonlinear dynamical systems. Appl. Math.Sci. 59, 1–247 (1985)

14. Verhulst, F.: Nonlinear Differential Equations and Dynamical Systems, Universitext.Springer, Berlin Heidelberg New York (1991)


On the Uniqueness of Gravitational Centre

Irmina Herburt

Received: 25 October 2006 / Accepted: 10 December 2007 /Published online: 3 January 2008© Springer Science + Business Media B.V. 2007

Abstract The dual volume of order α of a convex body A in Rn is a functionwhich assigns to every a ∈ A the mean value of α-power of distances of a fromthe boundary of A with respect to all directions. We prove that this functionis strictly convex for α > n or α < 0 and strictly concave for 0 < α < n (forα = 0 and for α = n the function is constant). It implies that the dual volumeof a convex body has the unique minimizer for α > n or α < 0 and has theunique maximizer for 0 < α < n. The gravitational centre of a convex body inR3 coincides with the maximizer of dual volume of order 2, thus it is unique.

Keywords Gravitational centre · Gravitational potential · Convex body ·Dual volume · Radial centre

Mathematics Subject Classifications (2000) 52A20 · 52A40 · 51P05 ·85A25 · 86A20

1 Introduction

A point mass M in R3 placed at a distance r from the reference point x is thesource of the gravitational potential at x, which is given by

�(x) = −GMr

,

I. Herburt (B)Department of Mathematics and Information Science,Warsaw University of Technology, Pl. Politechniki 1, 00-661, Warsaw, Polande-mail: [email protected]

252 I. Herburt

where G is the gravitational constant. Hence, a uniformly distributed mass ina region A ⊂ R3 will create the gravitational potential at the point x accordingto the following formula:

�A(x) = −G∫

A

γ dλ3

r,

where r is the distance from the point x and γ is the mass density (hereconstant).

The minimum of the gravitational potential defines the place where the testparticle can stay at rest, i.e., a stable equilibrium.

For A being a convex body we can apply techniques for convex sets(comp. [6]).

The class Kn0 of convex bodies consists of compact, convex subsets of Rn with

nonempty interiors. Let us recall that for every convex body A with 0 ∈ A theradial function �A : Sn−1 → R is defined by

�A(u) := sup{λ � 0 | λu ∈ A}.Evidently, if x ∈ int A, then for every u ∈ Sn−1

�A−x(u) = ‖x − a‖,where a is the point of bd A ∩ {x + {λu | λ > 0} (comp. [3] (A.55) or [10]). Thusfor A ∈ K3

0 we obtain

�A(x) = −Gγ

2

∫S2

�2A−x(u)dσu,

where σ is the spherical Lebesgue measure (see Section 2 or [6]).Hence the minimum of the function �A coincides with the maximum of the

function �αA : A −→ R given by

�αA(a) :=

∫Sn−1

�αA−a(u)dσ(u) (1)

for n = 3 and α = 2.

Functions defined by (1) are called dual volumes of order α of a convex bodyA (compare [3, 10]).

Therefore, the dual volume of order α of a convex body A is a function�α

A : A −→ R which assigns (up to a constant factor) to every a ∈ A the meanvalue of α-power of distances of a from the boundary of A with respect toall directions. These important functionals have been studied by many authors(see [1, 2, 4, 7–9]).

In [6] it is proved that for every A ∈ Kn0 , if α ∈ (0, 1], then the function �A

defined by (1) is strictly concave. Extremizers of dual volumes of order α arecalled radial centers of order α. Therefore, for any convex body a radial centreof order α for α ∈ (0, 1] is unique. In this paper we extend this result to eachα ∈ R. We shall prove that the dual volume of order α is strictly convex forα > n or α < 0 and strictly concave for 0 < α < n. Obviously, the dual volume

On the uniqueness of gravitational centre 253

of order 0 is a constant function. The dual volume of order n is equal to n timesthe volume of A, for every A ∈ Kn

0 and every a ∈ A (see [11], Example 3.2).Since the gravitational centre of a convex body A is the radial centre of

order 2 of A it is unique.Radial centres have other interesting physical interpretations. In [6] it is also

proved that

– The light intensity radiated by a transparent medium (like stars uniformlyfilling some region in the universe) attains its maximum at the radial centreof order 1.

– Results analogous to gravitational centre hold if instead of gravitationalinteractions one considers electric forces.

In [5] the notion of dual volume of order α is used to describe the prop-agation of electromagnetic waves from a transmitter, chosen randomly andaccording to the uniform distribution in a convex body A, to a receiver fixedat the origin. The authors investigate the case when the origin coincides withthe centre of an inscribed ball, which in many cases is not the optimal choice.To maximize the mean received power, the receiver should be located at theradial centre of suitable order.

We follow, in principle, terminology and notation used in [13]. In particular:

– The k-dimensional Lebesgue measure in Euclidean k-dimensional space isλk (in integrals with respect to λ1 we shall abbreviate dλ1(x) to dx).

– Bn is the unit ball in Rn and Sn−1 is its boundary in Rn.

– [x, y] is the closed segment with the end points x and y in Rn.

2 Convexity of Dual Volumes

For every nonnegative measurable function f in Rn

∫Rn

f (x)dλn(x) =∫

Sn−1

∫ ∞

0f (tu)tn−1dtdσ(u).

(comp. [14], formula (5.2.3) in Theorem 5.2.2). Thus for every α > 0, A ∈ Kn0

and a ∈ A we get

∫Sn−1

�αA−a(u)dσ(u) = α

∫Sn−1

∫ �αA−a(u)

0tα−1dtdσ(u) = α

∫A

1

‖x − a‖n−αdλn(x).

(2)

Hence, for α > 0, the convexity of �αA(·) is equivalent to the convexity of∫

A1

‖x−(·)‖n−α dλn(x).

For every α ∈ R and a ∈ int A the integral �αA(a) is finite. However the

integral∫

A1

‖x−a‖n−α dx is divergent for α < 0. Thus, the case α < 0 needs morecareful treatment.

254 I. Herburt

Lemma 2.1 Let A ∈ Kn0 . If for every x0 ∈ int A and u ∈ Sn−1 there exists ε > 0

such that the function �αA|{x0 + tεu : −1 < t < 1} is convex, then the function

�αA is convex on int A.

Proof The proof is based on two facts concerning convex functions:

– A function is convex on int A if and only if for any a, b ∈ int A its restrictionto relint[a, b ] is convex

– A function of one variable is convex on an open interval I if and only if itis locally convex on I. �

Lemma 2.2 Let A ∈ Kn0 and let f be a similarity of Rn. If �α

A is convex(concave) on A, then �α

f (A)is convex (concave) on f (A).

Take an arbitrary x0 ∈ int A and ε such that εBn + x0 ⊂ int A. Then

�αA(a) = α

∫A\(εBn+x0)

1

‖x − a‖n−αdλn(x) + �α

(εBn+x0)(a). (3)

By Lemma 2.1, to investigate the convexity of �αA it is enough to investigate the

convexity of α∫

A\(εBn+x0)1

‖x−(·)‖n−α dλn(x) and �α(εBn+x0)

restricted to {x0 + tεu :−1 < t < 1}, for every u ∈ Sn−1.

Let FsK be defined by

FsK(a) =

∫K

‖x − a‖sdλn(x).

In case s + n > 0 we take K = A and a ∈ A. In case s + n < 0 we take K =A \ (εBn + x0) and a ∈ εBn + x0.

We are going to prove that the function FsK is strictly convex for s > 0 and

strictly concave for s < 0.We shall need the following lemmas. Easy technical parts of their proofs will

be omitted.

Lemma 2.3 Let A ∈ Kn0 . For every isometry f of Rn and a in the domain of Fs

K

FsK(a) = Fs

f (K)( f (a)).

Lemma 2.4 Let A be a convex set in Rn. Let μ be a measure on a σ -field F andB ∈ F . Let f (a, ·) : B −→ R be integrable with respect to μ for every a in A. Iff (·, b) is convex (concave) on A for almost all b ∈ B then

∫B f (·, b)dμ(b) is

convex (concave) on A.

Lemma 2.5 Let A ∈ Kn0 and a, b in the domain of Fs

K. Then for every λ ∈ [0, 1]Fs

K(λa + (1 − λ)b) > (<) λFsK(a) + (1 − λ)Fs

K(b)


if and only if there exists an isometry f : Rn −→ Rn such that f (a) =(a

′1, a

′2, . . . , a

′n), f (b) = (b

′1, a

′2, . . . , a

′n) and

Fsf (K)(λa

′1 + (1 − λ)b

′1, a

′2, . . . , a

′n) > (<)

λFsf (K)(a

′1, a

′2, . . . , a

′n) + (1 − λ)Fs

f (K)(b′1, a

′2, . . . , a

′n).

Proof =⇒ By Lemma 2.3 the function FsK is equivariant under the isometries

of Rn, thus it is enough to take an isometry f which maps [a, b ] to [a′, b

′ ]parallel to the first axis, and so a

′i = b

′i for i = 2, 3, ..., n.

⇐= Isometry f −1 maps (λa′1 + (1 − λ)b

′1, a

′2, . . . , a

′n) onto λa + (1 − λ)b . �

Lemma 2.6 Let A ∈ Kn0 . Then there exist functions fi, hi, for i = 1, . . . , n − 1,

and numbers α1, α2 such that for every x ∈ A

fi(xi+1, xi+2, . . . , xn) � xi � hi(xi+1, xi+2, . . . , xn)

and

α1 � xn � α2.

Proof The proof is by induction with respect to the number of coordinates andbased on the observation that if we take the natural ordering of a line in anydirection, then there exists the first and the last element in the intersection ofthe line with the convex body A. �

Theorem 2.7 FsK is strictly convex for s > 0 and strictly concave for s < 0.

Proof By the definition,

FsK(a) =

∫K

‖x − a‖sdλn(x)

=∫

K

((x1 − a1)

2 + . . . + (xn − an)2) s

2 dλn(x).

By Lemma 2.5, it is enough to prove the convexity with respect to the firstcoordinate a1 with a2, a3, . . . , an being fixed. For K = A and s + n > 0, byLemma 2.6, we obtain

FsK(a) =

∫ α2

α1

∫ hn−1

fn−1

· · ·∫ h1

f1

((x1 − a1)

2 + . . . + (xn − an)2)p

dx1 . . . dxn, (4)

where p = s2 . By Lemma 2.4, to prove the convexity of Fs

K it is enough to provethe convexity of the function F given by

F(a1) =∫ h1

f1

((x1 − a1)

2 + c)p

dx1,

256 I. Herburt

where c = (x2 − a2)2 + . . . + (xn − an)

2 is fixed. After a substitution t = x1−a1√c

we obtain (up to a constant factor) the integral∫ h1−a1√

cf1−a1√

c

(t2 + 1)pdt.

By standard calculations we can check that the sign of its second derivativewith respect to a1 is equal to the sign of p. Therefore, for s > 0 the function Fs

Kis strictly convex and for −n < s < 0 it is strictly concave.

For K = A \ (εBn + x0) and s + n < 0 we have to replace the integral in (4)by the finite sum of integrals of such form, and then the proof follows. �

Lemma 2.8 The function f : [−1, 1] −→ R defined, for k > 0, by the formula

f (t) =∫ π

2

− π2

(−t cos ϕ +

√1 − t2(sin ϕ)2

)α

(cos ϕ)kdϕ

is strictly convex for α < 0.

Proof By standard calculations we show that d2 fdt2 is positive. �

Lemma 2.9 For every v ∈ Sn−1 and every α < 0 the function �αBn |(Bn ∩ lin(v))

is strictly convex.

Proof Let (e1, e2, . . . , en) be the canonical basis in Rn. We may assume v = en.Then for u ∈ Sn−1 we have

u = (cos ϕn−1) · u + (sin ϕn−1) · en,

where u ∈ Sn−1 ∩ lin(e1, . . . , en−1) and ϕn−1 in [−π2 , π

2 ] is the measure of theangle between u and its projection onto lin(e1, e2, . . . , en−1). Since Sn−1 ∩lin(e1, . . . , en) = Sn−2 ⊕{0} we get, by induction a parametrization p of Sn−1

p : (ϕ1, ϕ2, . . . , ϕn−1) −→ u ∈ Sn−1.

Let pi = δpδϕi

for i = 1, . . . , n − 1 and let G := (pi ◦ pj)i, j=1,...,n−1. Moreover,

let p : (ϕ1, ϕ2, . . . , ϕn−2) −→ u, pi = δ pδϕi

for i = 1, . . . , n − 2 and let G := ( pi ◦p j)i, j=1,...,n−2.

Then√

detG = (cos ϕn−1)n−2 ·

√detG.

In general√

detG = 1 for n = 2,

√detG = cos ϕ2 for n = 3

and√

detG = (cos ϕn−1)n−2 · . . . · (cosϕ3)

2 · cos ϕ2 for n > 3.


For every t ∈ [−1, 1], a = tv and ϕ = ϕn−1 we get

�αBn(a) =

∫Sn−1

�αBn−a(u)dσ(u)

=∫

Sn−2⊕{0}

∫ π2

− π2

(−t cos ϕ +

√1 − t2(sin ϕ)2

)α

(cos ϕ)n−2√

detGdϕdσ(u).

Thus, by Lemma 2.8 and Lemma 2.4 we get the claim. �

Theorem 2.10 Let A ∈ Kn0 . The dual volume �α

A is strictly convex for α > n orα < 0 and strictly concave for 0 < α < n.

Proof Let s = α − n. By (2) and Theorem 2.7 the dual volume of order α isstrictly convex for α > n and strictly concave for 0 < α < n. By (3), Theorem2.7, Lemma 2.9 and Lemma 2.2, the dual volume of order α is strictly convexfor α < 0 with values ∞ for points in the bd A. �

By Theorem 2.10 we immediately obtain

Corollary 2.11 Let A ∈ Kn0 . The dual volume �α

A has the unique minimizer forα > n or α < 0 and has the unique maximizer for 0 < α < n.

Example 2.12 Let As be a triangle with vertices (1, 0), (0, −s), (0, s) in R2

for some s ∈ R+. We shall calculate approximate coordinates of radial centerrα(As) of order α of As for some α and s. Since L = {(t, 0) : t ∈ R} is thesymmetry line of the triangle As and rα(As) is equivariant under the isometriesof R2 it follows that rα(As) ∈ L. Thus we calculate �α

As(a) only for a = (t, 0)

and t ∈ [0, 1]. Let γ1(t, s) = π2 − arctan t

s , γ2(t, s) = π2 − (

arctan 1s − arctan t

s

),

γ3(t, s) = π2 − arctan s. Then, as is easy to check, �α

As(a) = 2 f α

s (t), where

f αs (t) = tα

∫ γ1(t,s)

0

1

(cos φ)αdφ

+ ((1 − t) sin arctan(s))α∫ γ2(t,s)

0

1

(cos φ)αdφ

+ ((1 − t) sin arctan(s))α∫ γ3(t,s)

0

1

(cos φ)αdφ.

In Fig. 1 there are graphs of t �→ f αs (t) for chosen values of parameters α and

s and for t close to the argument of extremum. The values of the function t �→f αs (t) are found by means of numerical methods.

258 I. Herburt

Fig. 1 Graphs of t �→ 12 �α

As((t, 0)) for various α and s

Finally let us mention that the notion of dual volumes can be generalized onstar bodies (see [11] or [12]). In the first version of the present paper we gavean example of star body whose radial center of order 1 is not unique. Howeverthere was a mistake in calculations and the example is wrong. This examplewas cited in [12], p. 195.

For non-convex bodies the problem of uniqueness of extremizers of dualvolumes remains open.


References

1. Gardner, R.J., Jensen, E.B., Volcic, A.: Geometric tomography and local stereology. Adv. inAppl. Math. 30, 397–423 (2003)

2. Gardner, R.J.: Geometric tomography. Notices Amer. Math. Soc. 42(4), 422–429 (1995)3. Gardner, R.J.: Geometric Tomography. Cambridge University Press, Cambridge (1995)4. Gardner, R.J., Soranzo, A., Volcic, A.: On the determination of star and convex bodies by

section functions. Discrete Comput. Geom. 21(1), 69–85 (1999)5. Hansen, J., Reitzner, M.: Electromagnetic wave propagation and inequalities for moments of

chord lengths. Adv. in Appl. Probab. 36(4), 987–995 (2004)6. Herburt, I., Moszynska, M., Peradzynski, Z.: Remarks on radial centres of convex bodies.

MPAG 8(2), 157–172 (2005)7. Klain, D.A.: Star valuations and dual mixed volumes. Adv. Math. 121, 80–101 (1996)8. Klain, D.A.: Invariant valuations on star-shaped sets. Adv. Math. 125, 95–113 (1997)9. Ludwig, M.: Dual mixed volumes. Pacific J. Math. 58, 531–538 (1975)

10. Lutwak, E.: Intersection bodies and dual mixed volumes. Adv. Math. 71(2), 232–261 (1988)11. Moszynska, M.: Looking for selectors of star bodies. Geom. Dedicata 81, 131–147 (2000)12. Moszynska, M.: Selected Topics in Convex Geometry, Birkhäuser (2005)13. Schneider, R.: Convex Bodies: the Brunn–Minkowski theory. Cambridge University Press,

Cambridge (1993)14. Stroock, D.W.: A Concise Introduction to the Theory of Integration, Birkhäuser (1990)


Estimating Eigenvalue Moments via Schatten NormBounds on Semigroup Differences

M. Hansmann

Received: 10 August 2007 / Accepted: 11 December 2007 /Published online: 4 January 2008© Springer Science + Business Media B.V. 2007

Abstract We derive new bounds on the moments of the negative eigenvaluesof a selfadjoint operator B. The moments of order 0 < γ � 1 are estimated interms of Schatten-norm bounds on the difference of the semigroups generatedby B and a reference operator A which is assumed to be nonnegative andselfadjoint. The estimate in the case γ = 1 is sharp.

Keywords Eigenvalues · Moments · Schatten ideals · Selfadjoint operators ·Semigroup difference · Spectral shift function

Mathematics Subject Classifications (2000) 47A10 · 47A75 · 81Q10

1 Introduction

The spectral analysis of selfadjoint operators plays an important role in manyareas of mathematical physics. Topics such as the stability of the absolutelycontinuous spectrum, absence of singular continuous spectrum or perturba-tions of eigenvalues have been investigated in great detail and a lot of resultsare known, not only in the context of Schrödinger operators.

Recently, Demuth and Katriel [5] proposed a new method to derive boundson the moments of the negative eigenvalues of a selfadjoint operator. Giventwo selfadjoint operators A � 0 and B � −cB, cB > 0 in a Hilbert spaceH and assuming that D = e−B − e−A ∈ S1, where S1 denotes the ideal of

M. Hansmann (B)Institute of Mathematics, Technical University of Clausthal,38678 Clausthal-Zellerfeld, Germanye-mail: [email protected]

262 M. Hansmann

trace-class operators, they showed that the following inequality is valid forevery γ > 1

∑

λ∈σ(B)∩(−∞,0)

|λ|γ � Ctr(γ )‖D‖S1 . (1)

Here σ(B) means the spectrum of B, Ctr(γ ) is some suitable constant withCtr(γ ) → ∞ as γ → 1 and each eigenvalue of B is repeated according to itsmultiplicity. A similar result has been shown in case that D is a Hilbert–Schmidt operator, see [5].

The method used in [5] involves complex function theory (e.g. Jensen’sidentity) and is rather general, in particular it is possible to extend this resultto other Schatten-classes Sp, p � 1 (Demuth and Katriel, private communica-tion). Recall that D ∈ Sp if

‖D‖Sp :=( ∞∑

k=1

|μk(D)|p

)1/p

< ∞, 0 < p < ∞,

where μk(D) is the k−th singular value of the compact operator D. Note thatwe write ‖D‖Sp even in the case p < 1 where ‖.‖Sp does not fulfill the triangleinequality.

There exists an alternative way to proof estimates similar to (1) suggestedby M. Solomyak.

Theorem 1 [Solomyak, private communication] Let A and B be as aboveand suppose that D = e−B − e−A ∈ Sp where 1 < p < ∞. Then the followingestimates are valid

∑

λ∈σ(B)∩(−∞,0)

|λ|p �[2Cp‖D‖Sp

]p (2)

and in particular∑

λ∈σ(B)∩(−∞,0)

|λ|2 � ‖D‖2S2

. (3)

Here Cp → ∞ as p → 1 or p → ∞.

Since the idea of proof will be used in the sequel, we provide a short sketch.

Proof Define the Lipschitz continuous function

h(x) ={

0 , x � 1ln(x) , x > 1

.

Note that this function admits the representation

h(x) = h(0) +∫ x

0η(s) ds

Estimating eigenvalue moments via Schatten norm bounds 263

with a suitable left-continuous function η of bounded total variation. Using thetheory of double operator integrals developed by Birman and Solomyak, onecan show that

∥∥h(e−B) − h

(e−A)‖Sp � 2Cp‖e−B − e−A

∥∥Sp

,

where Cp is a suitable constant, see [2, Theorem 8.6]. By the spectral theorem

h(e−B) − h

(e−A) = B−,

where B− = 12 (|B| − B) and this implies the validity of (2). In the same manner

(3) follows from the inequality ‖B−‖S2 � ‖D‖S2 which is a consequence of [2,Theorem 8.1]. ��

Note that in the case of Hilbert-Schmidt perturbations inequality (3) issharp, i.e. there can not be a constant c < 1 such that

∑

λ∈σ(B)∩(−∞,0)

|λ|2 � c‖D‖2S2

. (4)

This can be seen by considering a compact operator B such that B =−diag(bn) in some suitable basis and A = 0. Then the inequality

∑

n

b2n � c ·

∑

n

(ebn − 1

)2

holds in general only if c � 1.Regarding the estimates (1) and (2) which give estimates on

∑

λ∈σ(B)∩(−∞,0)

|λ|γ

in the case γ >1, it is quite natural to ask what can be said in the case 0<γ �1.The aim of this article is to show that estimates like (1) and (2) are valid in

this case as well. To be more explicit we will proof the following two theorems.

Theorem 2 Let A and B be as above and suppose that D = e−B − e−A ∈ S1.Then for every γ � 1 the following estimate holds

∑

λ∈σ(B)∩(−∞,0)

|λ|γ � C1(γ )‖D‖S1 , (5)

where C1(γ ) = γ (γ − 1)(γ−1)e1−γ . In particular∑

λ∈σ(B)∩(−∞,0)

|λ| � ‖D‖S1 . (6)

Note that as in the case of (3) the estimate in (6) is sharp.

264 M. Hansmann

Theorem 3 Let A and B be as above and suppose that D = e−B − e−A ∈ Sp

where 0 < p < 1. Then for every γ > p the following estimate holds∑

λ∈σ(B)∩(−∞,0)

|λ|γ � 2cγ−pB

γ

γ − p‖D‖p

Sp, (7)

where B � −cB.

The main ingredient in the proof of Theorem 3 is the following propositionwhich is of interest in its own.

Proposition 1 Let A and B be as above and suppose that D = e−B − e−A ∈ Sp

where 0 < p < 1. Then

NB(−s) � 2 s−p ‖D‖pSp

, s > 0

where NB(−s) gives the number of eigenvalues of B in (−∞, −s] and eacheigenvalue is counted according to its multiplicity.

Our proofs will neither depend on complex function theory nor on estimatesof the type used by Solomyak. Instead we will apply the theory of Krein’sspectral shift function (SSF) which has been developed by M. G. Krein withfurther contributions by Birman, Solomyak, Peller and other authors. See [3,Birman et al.] for an extensive review including references to the originalliterature.

In the next section we will provide some basic facts about the SSF necessaryto understand the proofs of Theorem 2 and Theorem 3 given in Sections 3and 4 respectively. Finally, in Section 5 we gather some remarks concerningpossible applications and extensions of the given results.

2 Some Information on Krein’s SSF

We refer to [8, Yafaev, Chapter 8] and [3, Birman et al.] as general referencesfor the results in this section.

The SSF for the pair e−B, e−A is defined via the corresponding perturbationdeterminant from scattering theory, that is

ξ(x) := ξ(x, e−B, e−A) := π−1 limε→0+

arg det[1 + D

(e−A − x − iε

)−1],

where D = e−B − e−A ∈ S1. Given this definition the following holds∫

R

ξ(x) dx = TrD (8)

and∫

R

|ξ(x)| dx � ‖D‖S1 . (9)


Furthermore, for suitable f , D ∈ S1 implies that f (e−B) − f (e−A) ∈ S1 and

Tr[

f(e−B) − f

(e−A)] =

∫

R

f ′(x)ξ(x) dx. (10)

This equality is known as Krein’s trace formula. In particular, it is valid fordifferentiable f with Hölder continuous derivative, see [1, Birman et al.].

We will also need some information from the Lp-theory of the SSF, see [4,Combes et al.]. Namely

∫

R

|ξ(x)|r dx � ‖D‖S 1r, r � 1. (11)


Modifying the idea of Solomyak’s proof we define a function h in the follow-ing way

h(x) ={

0 , x � 1ln(x)γ , x > 1

, γ > 1.

It is not difficult to see that h ∈ C1(R) and that h′ is Hölder continuous.Hence, using (10) we get h

(e−B

) − h(e−A

) ∈ S1 and

Tr[h

(e−B) − h

(e−A)] =

∫

R

h′(x)ξ(x) dx. (12)

By the spectral theorem h(e−B

) = (B−)γ and h(e−A

) = 0. Thus (12) can berewritten in the following way

∑

λ∈σ(B)∩(−∞,0)

|λ|γ =∫

R

h′(x)ξ(x) dx

= γ

∫ ∞

1

ln(x)(γ−1)

xξ(x) dx.

Since maxx∈[1,∞) x−1 ln(x)(γ−1) = (γ−1)(γ−1)

e(γ−1) , we get for γ > 1

∑

λ∈σ(B)∩(−∞,0)

|λ|γ � γ (γ − 1)(γ−1)e(1−γ )

∫ ∞

1|ξ(x)| dx. (13)

In particular we can take the limit γ → 1+ on both sides such that

∑

λ∈σ(B)∩(−∞,0)

|λ| �∫ ∞

1|ξ(x)| dx. (14)

The desired result now follows from (13) and (14) using estimate (9).

266 M. Hansmann


We take some arbitrary function g ∈ C1(R) with Hölder continuous derivativesuch that 0 � g � 1, g(x) = 0 if x � 0 and g(x) = 1 if x � 1. Set

ha,b (t) := g(

t − ab − a

), t ∈ R, a < b .

Then ha,b inherits the smoothness properties of g and

ddt

ha,b (t) =

⎧⎪⎨

⎪⎩

0 , t ∈ (−∞, a] ∪ [b , ∞)

1

b − ag′

(t − ab − a

), t ∈ (a, b).

(15)

Next we select some specific values for the constants a and b . Let s > 0 and0 < ε < 1 and set

h(t) := he(1−ε)s,es(t) =

⎧⎪⎨

⎪⎩

0 , t � e(1−ε)s

h(t) , t ∈ (e(1−ε)s, es)

1 , t � es.

(16)

By the spectral theorem h(e−A) = 0 and furthermore

(h

(e−B)

f, f) =

∫ −s

−∞d(EB(x) f, f ) +

∫ −(1−ε)s

−sh(e−x) d(EB(x) f, f )

�∫ −s

−∞d(EB(x) f, f )

= (EB ((−∞, −s]) f, f ) , f ∈ H. (17)

Let NB(−s) be the number of eigenvalues of B smaller than −s, s > 0. Bythe min-max-principle (17) implies

NB(−s) = Tr(E(−∞,−s](B)

)

� Tr(h(e−B))

= Tr(h(e−B) − h

(e−A))

.

Hence, using (10) we get

NB(−s) � Tr(h(e−B) − h

(e−A))

=∫

R

h′(x)ξ(x) dx

=∫ es

e(1−ε)sh′(x)ξ(x) dx. (18)

Let q, r > 1 and q−1 + r−1 = 1. By Hölder’s inequality

NB(−s) �[∫ es

e(1−ε)s|h′(x)|q dx

]1/q [∫

R

|ξ(x)|r dx]1/r

.


To simplify the following computations we substitute a = e(1−ε)s and b = es.We have already seen in (15), that

h′(x) = 1

b − ag′

(x − ab − a

).

Hence we can compute

∫ b

a|h′(x)|q dx = 1

(b − a)q

∫ b

a

∣∣∣∣g′(

x − ab − a

)∣∣∣∣q

dx

= 1

(b − a)q−1

∫ 1

0|g′(u)|q du

= ‖g′‖qLq(0,1)

(b − a)q−1.

In conclusion we get

NB(−s) � ‖g′‖Lq(0,1)

(es − e(1−ε)s)1− 1

q

[∫

R

|ξ(x)|r dx]1/r

� ‖g′‖Lq(0,1)

(εs)1r

[∫

R

|ξ(x)|r dx]1/r

,

where we have used that es − e(1−ε)s � εs and r−1 = 1 − q−1.Using (11) we arrive at

NB(−s) � ‖g′‖Lq(0,1)

(εs)1r

‖D‖ 1rS 1

r

.

Choosing r = 1p and taking the limit ε → 1− it follows that

NB(−s) �‖g′‖L(1−p)−1

(0,1)

sp‖D‖p

Sp, 0 < p < 1.

The number of negative eigenvalues NB(−s) is connected to the corre-sponding moments via

∫ cB

0sγ−1 NB(−s) ds = 1

γ

∑

λ∈σ(B)∩(−∞,0)

|λ|γ , γ > 0. (19)

Using (19) we can finally conclude that for every γ > p

∑

λ∈σ(B)∩(−∞,0)

|λ|γ � cγ−pB

γ

γ − p‖g′‖L(1−p)−1

(0,1)‖D‖p

Sp, (20)

where B � −cB. Note that the choice of g is free and that it is elementary toconstruct a function g with ‖g′‖∞ � 2.

268 M. Hansmann

5 Remarks

1. In applications it is often more usual to posses bounds on the Schattennorms of differences of powers of the resolvents corresponding to A andB. It is not difficult to see that our method can be adjusted to this case. Forexample in the case γ = 1 one can derive

∑

λ∈σ(B)∩(−∞,0)

|λ| � (1 + r)r−1

ra1+r‖(B + a)−r − (A + a)−r‖S1 , r � 1,

(21)

provided a > cB. The moments of order 0 < γ < 1 can be handled as well.2. The method described above translates the problem of estimating the

moments of a semibounded selfadjoint operator B to the problem offinding bounds on the p-th Schatten norm of the semigroup difference D =e−B − e−A where A is some arbitrary nonnegative selfadjoint operator.For concrete operators B this opens the opportunity to find an optimaloperator A = A(B) adjusted to B, e.g. to improve constants, etc.

3. For the important class of Schrödinger operators, i.e. B = − + V whereV is in the Kato-class, the problem of estimating the p-th Schatten normof D has already been subject of intensive study and we would like tomention just a few known results. For A = − and B as above tracenorm estimates are well known and we refer to [6, Demuth et al.] wherea thorough account on this topic can be found. In the case p < 1 it hasrecently been shown in [7, Hundertmark et al.] that supp(V) < ∞ impliesthat D ∈ Sp for every p > 0 and this result can be generalized to potentialsV with unbounded support given some suitable decay conditions on V atinfinity.

4. For Schrödinger operators it is interesting to compare our estimates withestimates of the Lieb–Thirring type. We refer to [5, Demuth et al.] wheresome results on this topic can be found.

5. In some situations moment estimates in terms of Schatten norm boundscan be inadequate and our estimates can be improved. In particular, thisis possible in case that more information on the decay rate of the singularvalues of D is available. As an example we consider the results of the aforementioned paper [7], where A = − and B = − + V with V of boundedsupport (note that in [7] the case of magnetic Schrödinger operators wasconsidered).The authors of [7] showed that the singular values μn(D) decay almostexponentially, i.e. μn(D) � Ce−cn1/d

, and they used this result to derive an


integral bound on the SSF for the pair A, B. In the following we shortlysketch the proof of the corresponding result in [7]

(with ξ(s, A, B) replaced

by ξ(s, e−A, e−B

)). Define the convex function Ft : [0, ∞) → [0, ∞)

Ft(x) =∫ x

0

(exp

(ty1/d) − 1

)dy, t > 0

and note that F ′t (0) = 0. Using the decay rate of μn(D) it can be shown that

for t small enough∫

R

Ft(|ξ (

s, e−B, e−A) |) ds � Kt,

where Kt > 0 is a suitable constant. Let Gt be the Legendre transform ofFt. Then

Gt(y) � y(

log(1 + y)

t

)d

, y � 0.

Using Young’s inequality∫

f (x)ξ(x) dx �∫

Ft(|ξ (

x, e−A, e−B) |) dx +∫

Gt(| f (x)|) dx

� Kt + t−d{log(1 + ‖ f‖∞)}d‖ f‖L1 , (22)

where f is any bounded integrable function. Coming back to our problemwe note that in the proof of Theorem 3 we showed that

NB(−s) �∫ es

e(1−ε)sh′(x)ξ(x) dx, 0 < ε < 1, s > 0,

where h was defined in (16). Since ‖h′‖∞ � cs−1 and ‖h′‖L1 = 1 estimate(22) implies that NB(−s) � c log(1 + s−1) as s → 0. This is an improvementcompared to NB(−s) � cs−γ , γ > 0 which can be derived directly fromTheorem 3.

Acknowledgements I would like to thank M. Demuth, G. Katriel and I. Veselic for useful hintsand discussions. Further I would like to thank M. Solomyak whose ideas initiated the work on thisarticle.

References

1. Birman, M.Sh., Solomyak, M.Z.: Remarks on the spectral shift function. J. Sov. Math. 3,408–419 (1975)

2. Birman, M.Sh., Solomyak, M.Z.: Double operator integrals in a Hilbert space. Integral Equa-tions Operator Theory 47, 131–168 (2003)

3. Birman, M.Sh., Yafaev, D.: The spectral shift function. The papers of M.G. Krein and theirfurther development. (Russian) Algebra i Analiz 4(5), 1–44 (1992)

270 M. Hansmann

4. Combes, J.M., Hislop, P.D., Nakamura, S.: The Lp-theory of the spectral shift function, theWegner estimate, and the integrated density of states for some random operators. Comm.Math. Phys. 218(1), 113–130 (2001)

5. Demuth, M., Katriel, G.: Eigenvalue Inequalities in Terms of Schatten Norm Bounds onDifferences of Semigroups, and Application to Schrödinger Operators. To appear in AnnalesHenri Poincare. http://arxiv.org/abs/math/0612279

6. Demuth, M., van Casteren, J.: Stochastic Spectral Theory for Selfadjoint Feller Operators: AFunctional Integration Approach. Birkhäuser Verlag, Basel (2000)

7. Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselic, I.: Bounds on the spectralshift function and the density of states. Comm. Math. Phys. 262(2), 489–503 (2006)

8. Yafaev, D.: Mathematical Scattering Theory. General Theory. Translations of MathematicalMonographs, vol. 105. American Mathematical Society, Providence, RI (1992)

http://arxiv.org/abs/math/0612279


Weak Convergence and Vector-Valued Functions:Improving the Stability Theory of Feynman’sOperational Calculi

Lance Nielsen

Received: 3 August 2007 / Accepted: 12 December 2007 /Published online: 23 January 2008© Springer Science + Business Media B.V. 2007

Abstract In this paper we present a theorem that establishes a relationbetween continuous, norm-bounded functions from a metric space into aseparable Hilbert space and weak convergence of sequences of probabilitymeasures on the metric space. After establishing this result, it’s application tothe stability theory of Feynman’s operational calculi will be illustrated. We willsee that the existing time-dependent stability theory of the operational calculiwill be significantly improved when the operator-valued functions take theirvalues in L(H), H a separable Hilbert space.

Keywords Weak convergence · Disentangling ·Feynman’s operational calculi · Stability theory

Mathematics Subject Classifications (2000) 47A13 · 47A60 · 47A56 ·47N50 · 60F99

1 Introduction

The two topics under consideration in this paper are a relation between weakconvergence of probability measures and Hilbert space valued functions andthe consequences of this relation when applied to Feynman’s operationalcalculus. As indicated, we will first state and prove a theorem (Theorem 2.3below) that establishes a relation between weak convergence of sequencesof probability measures on a metric space S and functions f : S → H, H aseparable Hilbert space, that are continuous and norm bounded. The utility ofhaving such a theorem in hand was clear to the author during his development

L. Nielsen (B)Department of Mathematics, Creighton University, Omaha, NE 68178, USAe-mail: [email protected]

272 L. Nielsen

of the stability theory for Feynman’s operational calculi in his thesis [15] aswell as the papers [8, 13, 14], and [12]. However, it was not until recentlythat the theorem was discovered. After stating and proving Theorem 2.3,we state a modification of the theorem to accommodate not only a weaklyconvergent sequence of probability measures, but also a corresponding uni-formly convergent sequence of H-valued functions. The second focus of thispaper is the application of Theorem 2.3 to improving the stability theory ofFeynman’s operational calculi. It is possible, however, that Theorem 2.3 maybe of independent interest. Before proceeding any further the reader may finda short discussion of Feynman’s operational calculus useful.

Feynman’s operational calculus originated with the 1951 paper [3] andconcerns itself with the formation of functions of non-commuting operators.Indeed, even functions as simple as f (x, y) = xy are not well-defined if xand y do not commute. Indeed, some possibilities are f (x, y) = yx, f (x, y) =12 (xy + yx), and f (x, y) = 1

3 xy + 23 yx. One then has to decide, usually with a

particular problem in mind, how to form a given function of non-commutingoperators. One method of dealing with this problem is the approach developedby Jefferies and Johnson in the series of papers [4–7] and expanded on in thepapers [8, 11], and others. The Jefferies–Johnson approach to the operationalcalculus uses measures on intervals [0, T] to determine the order of operatorsin products. In the original setting used by Jefferies and Johnson, only contin-uous measures were used. However, Johnson and the current author extendedthe operational calculus to measures with both continuous and discrete partsin [11]. [The reader can see the difference in the operational calculus thatresults when moving from continuous time ordering measures to time orderingmeasures with a non-zero discrete part if they compare (3.19) on page 14 toequations (3.15) and (3.17) on page 12.]

The discussion above, then, begs the question of how measures can be usedto determine the order of operators in products. Feynman’s heuristic rules forthe formation of functions of non-commuting operators give us a starting point.

1) Attach time indices to the operators to specify the order of operators inproducts.

2) With time indices attached, form functions of these operators by treatingthem as though they were commuting.

3) Finally, “disentangle” the resulting expressions; i.e. restore the conven-tional ordering of the operators.

As is well known, the central problem of the operational calculus is thedisentangling process. Indeed in his 1951 paper, [3], Feynman points out that“The process is not always easy to perform and, in fact, is the central problemof this operator calculus.”

We first address rule (1) above. It is in the use of this rule that we will seemeasures used to track the action of operators in products. First, it may bethat the operators involved may come with time indices naturally attached.For example, we might have operators of multiplication by time dependentpotentials. However, it is also commonly the case that the operators used are

Weak convergence and vector-valued functions 273

independent of time. Given such an operator A, we can (as Feynman mostoften did) attach time indices according to Lebesgue measure as follows:

A = 1

t

∫ t

0A(s) ds

where A(s) := A for 0 � s � t. This device does appear a bit artificial butdoes turn out to be extremely useful in many situations. We also note thatmathematical or physical considerations may dictate that one use a measuredifferent from Lebesgue measure. For example, if μ is a probability measureon the interval [0, T], and if A is a linear operator, we can write

A =∫

A(s) μ(ds)

where once again A(s) := A for 0 � s � T. When we write A in this fashion,we are able to use the time variable to keep track of when the operator Aacts. Indeed, if we have two operators A and B, consider the product A(s)B(t)(here, time indices have been attached). If t < s, then we have A(s)B(t) = ABsince here we want B to act first (on the right). If, on the other hand, s < t,then A(s)B(t) = BA since A has the earlier time index. In other words, theoperator with the smaller (or earlier) time index, acts to the right of (or before)an operator with a larger (or later) time index. (It needs to be kept in mind thatthese equalities are heuristic in nature.) For a much more detailed discussionof using measures to attach time indices, see Chapter 14 of the book [9] andthe references contained therein.

Concerning the rules (2) and (3) above, we mention that, once we haveattached time indices to the operators involved, we calculate functions of thenon-commuting operators as if they actually do commute. These calculationsare, of course, heuristic in nature but the idea is that with time indicesattached, one carries out the necessary calculations giving no thought tothe operator ordering problem; the time indices will enable us to restorethe desired ordering of the operators once the calculations are finished; thisis the disentangling process and is typically the most difficult part of anygiven problem.

We now move on to discuss, in general terms, how the operational calculuscan be made mathematically rigorous. Suppose that Ai : [0, T] → L(H), i =1, . . . , n, are given and that we associate to each Ai(·) a Borel probabilitymeasure μi on [0, T]; this is the so-called time ordering measure and, asmentioned above, serves to keep track of when a given operator or operator-valued function acts in products. We construct a commutative Banach algebra(the disentangling algebra) DT

((A1(·), μ1)

∼ , . . . , (An(·), μn)∼) of functions an-

alytic on a certain polydisk. With this commutative Banach algebra in hand, wecan carry out the disentangling calculations called for by Feynman’s “rules” ina mathematically rigorous fashion. Once the disentangling is carried out in thealgebra DT , we map the result to the non-commutative setting of L(X) usingthe so-called disentangling map T T

μ1,...,μn; it is the image under the disentangling

map that is the disentangled operator given by the application of Feynman’s“rules”. We note that changing the n-tuple of time ordering measures will,

274 L. Nielsen

in general, change the operational calculus, as it will usually change the actionof the disentangling map. Of course, a change in the operators will alsogenerally change the operational calculus.

The stability theory for the Jefferies–Johnson formulation of the opera-tional calculus was developed initially in [15] and expanded on in [10, 13, 14],and [12]. In particular, stability with respect to the time ordering measures, onefocus of the current paper, can be described as follows. We select sequences{μik}∞k=1 of Borel probability measures on [0, T] such that μik ⇀ μi as k → ∞.We then have, for each k ∈ N, a particular operational calculus, given by theaction of T T

μ1k,...,μnk, indexed by the n-tuple (μ1k, . . . , μnk) of measures and thus

a sequence of operational calculi. The stability question is then the question ofwhether the sequence of operational calculi has a limiting operational calculusas k → ∞.

The stability theory, in its current form, is not entirely satisfactory, at leastin the time dependent setting. For example, consider the following stabilitytheorem in the time dependent setting (in the setting of the current paper, thespace X will be a separable Hilbert space):

Theorem Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respect tothe usual topology on [0, T] and the norm topology on L(X). Associate toeach Ai(·) a continuous Borel probability measure μi on [0, T]. Let {μik}∞k=1,i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, foreach i = 1, . . . , n, μik ⇀ μi. Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)

where for k = 0 the summand is DT((A1(·), μ1)

∼ , . . . , (An(·), μn)∼). Denote by

‖ · ‖UD the norm on UD.Then

limk→∞∣∣� (T T

μ1k,...,μnk(πk(θ f ))

)− �(T T

μ1,...,μn(π0(θ f ))

)∣∣ = 0

for all � ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ UD.

Remark 1.1 This theorem appears as Theorem 3.1 of [14].

We see that the presence of the sequences of measures gives us a countablefamily of disentangling algebras DT

((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼). Of

course, we also obtain a sequence{T T

μ1k,...,μnk

}∞k=1

of disentangling maps (or,if you like, a sequence of operational calculi). Now, consider the conclusion ofthe theorem. In order to obtain the conclusion we have to choose a functionalfrom the dual of L(H); this is not terribly satisfactory. In fact, it would be muchmore preferable to obtain a conclusion such as

∥∥T Tμ1k,...,μnk

(πk(θ f ))φ − T Tμ1,...,μn

(π0(θ f ))φ∥∥H → 0


for every φ ∈ H; i.e. strong operator convergence. Indeed, in an applied setting,say in non-relativistic quantum mechanics, strong operator convergence isdesirable. It is at this point that the main theorem, Theorem 2.3 below,comes into play. The reason that a functional from L(H)∗ had to be usedin the theorem as stated above is due to the weak convergence hypotheseson the sequence of measures—a bounded and continuous real or complexvalued function is needed in order to apply the standard theorems on weakconvergence of probability measures. With Theorem 2.3 in hand, we can obtainthe preferred strong operator convergence. While the proof of the theorem asstated above has appeared in [14], we will outline the proof below noting whereTheorem 2.3 comes into play.

As a second example of applying Theorem 2.3, we will state and prove atheorem similar in nature to the theorem above but where the time orderingmeasures are allowed to have both continuous and discrete parts. This theoremhas not previously appeared in print though the proof is similar to the proofof the theorem above; however, the presence of the discrete measures changesthe proof in some significant ways.

2 The Weak Convergence Theorem

In this section we will state and prove the weak convergence theorem.Throughout this section H will be a separable Hilbert space. (While thisassumption on the Hilbert space is, strictly speaking, not necessary, it doessimplify the exposition. We could make the assumption that our Hilbert spacevalued functions take their values in a separable subpace of H and proceedaccordingly.) Also, S will be an arbitrary metric space. Before stating thetheorem, we will remind the reader of the definition of weak convergence of asequence of probability measures. This definition can be found in many places,for example [1].

Definition 2.1 Let S be a metric space and let {μk}∞k=1 be a sequence of Borelprobability measures on S. We say that this sequence converges weakly to theBorel probability measure μ on S if

limk→∞

∫S

f (s) μk(ds) =∫

Sf (s) μ(ds)

for every bounded continuous real-valued function f on S. Weak convergenceof μk to μ will be denoted by μk ⇀ μ.

Remark 2.2 Of course, weak convergence in this probabilistic sense is, from afunctional analytic viewpoint, weak-∗ convergence.

We now state the theorem.

276 L. Nielsen

Theorem 2.3 Let H be a separable Hilbert space and let S be a metric space. Let{μk}∞k=1 and μ be Borel probability measures on S such that μk ⇀ μ. Then

limk→∞

∫S

f (s) μk(ds) =∫

Sf (s) μ(ds) (2.1)

in norm on H for every continuous f : S → H such that sups∈S ‖ f (s)‖H < ∞.

Proof Let f : S → H be continuous and such that sups∈S ‖ f (s)‖H < ∞. Notethat, as f is continuous, it is Bochner integrable with respect to each of ourprobability measures. Since H is separable, there is a countable orthonormalbasis {en}∞n=1 for H. For n ∈ N define the orthogonal projection Pn : H →span {e1, . . . , en} by

Pn

⎛⎝ ∞∑

j=1

α je j

⎞⎠ :=

n∑j=1

α je j. (2.2)

Next define the map ψn : span {e1, . . . , en} → Cn by

ψn

⎛⎝ n∑

j=1

α je j

⎞⎠ := (α1, . . . , αn) . (2.3)

Then, clearly, ψn is linear. Moreover, since

∥∥∥∥∥∥ψn

⎛⎝ n∑

j=1

α je j

⎞⎠∥∥∥∥∥∥

2

Cn

= |α1|2 + · · · + |αn|2 =∥∥∥∥∥∥

n∑j=1

α je j

∥∥∥∥∥∥2

H

, (2.4)

it follows that ψn, n ∈ N, is an isometry. We consider the map ψn ◦ Pn ◦ f :S → C

n. Because f is continuous and norm bounded, and since Pn and ψn

are continuous and of norm 1, the map ψn ◦ Pn ◦ f is continuous and boundedfrom S into C

n. For convenience write

ψn ◦ Pn ◦ f (s) := (g1(s), . . . , gn(s)) (2.5)

for bounded and continuous complex-valued functions g1, . . . , gn on the metricspace S. We can write, for j = 1, . . . , n, g j(s) = u j(s) + iv j(s) where u j and v j

are continuous and bounded real-valued functions on S. It follows that

limk→∞

∫S

g j(s) μk(ds) = limk→∞

(∫S

u j(s) μk(ds) + i∫

Sv j(s) μk(ds)

)

=∫

Su j(s) μ(ds) + i

∫Sv j(s) μ(ds) (2.6)

=∫

Sg j(s) μ(ds)


and so, applying the above result component-wise, we have

limk→∞

∫Sψn ◦ Pn ◦ f (s) μk(ds) =

∫Sψn ◦ Pn ◦ f (s) μ(ds). (2.7)

Because of the linearity of ψn, we may write∫

Sψn ◦ Pn ◦ f (s) μk(ds) −

∫Sψn ◦ Pn ◦ f (s) μ(ds)

= ψn

(∫S

Pn ◦ f (s) μk(ds) −∫

SPn ◦ f (s) μ(ds)

)(2.8)

and so, because ψn is an isometry,∥∥∥∥ψn

(∫S

Pn ◦ f (s) μk(ds) −∫

SPn ◦ f (s) μ(ds)

)∥∥∥∥Cn

=∥∥∥∥∫

SPn ◦ f (s) μk(ds) −

∫S

Pn ◦ f (s) μ(ds)

∥∥∥∥H

. (2.9)

We have shown above that the quantity on the left-hand-side of (2.9) vanishesin the limit as k → ∞ and therefore it follows that

limk→∞

∥∥∥∥∫

SPn ◦ f (s) μk(ds) −

∫S

Pn ◦ f (s) μ(ds)

∥∥∥∥H

= 0. (2.10)

We now move on to investigate limn→∞∫

S Pn ◦ f (s) μk(ds) for any k ∈ N. Itis clear that, for each s ∈ S, we have

limn→∞ ‖Pn ◦ f (s) − f (s)‖H = 0. (2.11)

Since

‖Pn ◦ f (s) − f (s)‖H � 2‖ f (s)‖H � 2 sups∈S

‖ f (s)‖H (2.12)

and since sups∈S ‖ f (s)‖H < ∞, we can take h(s) := 2 sups∈S ‖ f (s)‖H for thescalar dominating function that is needed in order to apply the dominatedconvergence theorem for Bochner integrals. This function is integrable withrespect to each of the measures μk and μ and the value of the integral isthe same for each measure since they are all probability measures. Apply-ing the dominated convergence theorem for Bochner integrals leads to thestatement that

limn→∞

∫S

Pn ◦ f (s) μk(ds) =∫

Sf (s) μk(ds) (2.13)

in norm on H. Note that this limit is, in fact, uniform in k ∈ N.

We have now established the following limit statements. First,

limn→∞

∫S


Sf (s) μk(ds) (2.14)

278 L. Nielsen

in norm on H, uniformly in k ∈ N, and second,

limk→∞

∫S


SPn ◦ f (s) μ(ds) (2.15)

in norm on H for each n ∈ N. We now apply a theorem of E. H. Moore oniterated limits (see [2], page 28). Since we have established the limit on nuniformly in k and the limit on k for each fixed n, Moore’s theorem tellsus that

limn→∞k→∞

∫S

Pn ◦ f (s) μk(ds) = limn→∞ lim

k→∞

∫S

Pn ◦ f (s) μk(ds)

= limk→∞

limn→∞

∫S

Pn ◦ f (s) μk(ds)

=∫

Sf (s) μ(ds) (2.16)

in norm on H. The proof of the theorem is finished. �

We can modify the previous theorem somewhat to obtain a result that wewill find useful below. Indeed, we will be concerned below not only with aweakly convergent sequence of probability measures, but also with a corre-sponding uniformly convergent sequence of H-valued functions. (The real-valued version of this theorem appears as Lemma 3.2 of [12].) The followingtheorem addresses this situation.

Theorem 2.4 Let H be a separable Hilbert space. Let μk, μ be, for k ∈ N, Borelprobability measures on the metric space S. Let fk, f , k ∈ N, be continuousnorm bounded H-valued functions on S. If μk ⇀ μ and if fk → f uniformlyin H-norm on S, then

limk→∞

∫E

fk dμk =∫

Ef dμ (2.17)

in norm for any Borel set E⊂ S with μ(∂ E)=0 (that is, E is a μ-continuity set).

Proof Fix a Borel set E in S such that μ(∂ E) = 0. For any k ∈ N, we may write

∥∥∥∥∫

Efk dμk−

∫E

f dμ

∥∥∥∥H

�∫

E‖ fk− f‖Hdμk+

∥∥∥∥∫

Ef dμk−

∫E

f dμ

∥∥∥∥H

. (2.18)

Since fk → f uniformly in H-norm, given ε > 0, there is a k0 ∈ N such that ifk � k0,

sups∈E

‖ fk(s) − f (s)‖H <ε

2(2.19)


and so, for k � k0,∫

E‖ fk − f‖Hdμk <

ε

2μk(E) � ε

2. (2.20)

The remainder of the proof amounts to showing that the second term in (2.18)vanishes as k → ∞. We will proceed much as in the proof of Theorem 2.3.Indeed, taking any orthonormal basis

{e j}

j∈N, we define, for every n ∈ N,

the orthogonal projection Pn : H → span {e1, . . . , en} exactly as in the proofof Theorem 2.3. Next, we define ψn : span {e1, . . . , en} → C

n exactly as in theproof of Theorem 2.3. Then ψn is a linear isometry for every n ∈ N. We nowconsider

limk→∞

∥∥∥∥∫

Eψn ◦ Pn ◦ f (s) dμk(s)−

∫E

ψn ◦ Pn ◦ f (s) dμ(s)

∥∥∥∥Cn

= limk→∞

∥∥∥∥∫

SχE · ψn◦Pn◦ f (s) dμk(s)−

∫SχE · ψn◦Pn◦ f (s) dμ(s)

∥∥∥∥Cn

(2.21)

Since χE · ψn ◦ Pn ◦ f has its discontinuities contained in ∂ E and sinceμ(∂ E) = 0, Theorem 5.2 of [1] (applied component-wise) shows that

limk→∞

∥∥∥∥∫

SχE · ψn◦Pn◦ f (s) dμk(s)−

∫SχE · ψn◦Pn◦ f (s) dμ(s)

∥∥∥∥Cn

=0. (2.22)

Because ψn is a linear isometry, it follows at once that, for each n ∈ N,∥∥∥∥∫

EPn ◦ f dμk −

∫E

Pn ◦ f dμ

∥∥∥∥H

=∥∥∥∥ψn

(∫E

Pn ◦ f dμk −∫

EPn ◦ f dμ

)∥∥∥∥Cn

=∥∥∥∥∫

Eψn ◦ Pn ◦ f dμk −

∫E

ψn ◦ Pn ◦ f dμ

∥∥∥∥Cn

→ 0 (2.23)

as k → ∞.

Next, we use the fact that ‖Pn ◦ f (s) − f (s)‖H → 0 as n → ∞ for each s ∈ S.Since

∥∥∥∥∫

EPn ◦ f dμk

∥∥∥∥H

=∥∥∥∥∫

SχE · Pn ◦ f dμk

∥∥∥∥H

� sups∈S

|χE(s)| ‖ f (s)‖H � sups∈S

‖ f (s)‖H, (2.24)

(uniformly in k ∈ N) we are free to use the dominated convergence theoremfor Bochner integrals to obtain

limn→∞

∫E

Pn ◦ f dμk =∫

Ef dμ (2.25)

280 L. Nielsen

in norm on H. Moreover, this limit is uniform in k. The limit results inequations (2.23) and (2.25) enable us to apply the theorem of E. H. Moore ([2],page 28) on iterated limits exactly as in the proof of Theorem 2.3 to obtain

limk→∞

∥∥∥∥∫

Efk dμk −

∫E

f dμ

∥∥∥∥H

= 0 (2.26)

and therefore we can choose a k1 ∈ N so that∥∥∥∥∫

Efk dμk −

∫E

f dμ

∥∥∥∥H

<ε

2(2.27)

whenever k > k1. Let k2 = max (k0, k1). Then, for k > k2, we have∥∥∥∥∫

Efk dμk −

∫E

f dμ

∥∥∥∥H

�∫

E‖ fk − f‖Hdμk +

+∥∥∥∥∫

Ef dμk −

∫E

f dμ

∥∥∥∥H

<ε

2+ ε

2= ε.

�

3 The Disentangling Map

We now move on to a discussion of the disentangling map. Before definingthe map, however, we need some preliminary definitions and notation (see[4, 8, 11]). (In fact, we follow the paper [11] quite closely here even thoughthat paper was concerned only with the time independent setting and we arehere concerned with the time dependent setting.) We begin by introducing twocommutative Banach algebras AT and DT . These algebras are closely relatedand play an important role in the rigorous development of the operationalcalculus.

Given n ∈ N and n positive real numbers r1, . . . , rn, let AT(r1, . . . , rn) or,more briefly AT , be the space of complex-valued functions (z1, . . . , zn) →f (z1, . . . , zn) of n complex variables that are analytic at the origin and are suchthat their power series expansion

f (z1, . . . , zn) =∞∑

m1,...,mn=0

am1,...,mn zm11 · · · zmn

n (3.1)

converges absolutely at least in the closed polydisk |z1| � r1, . . . , |zn| � rn. Allof these functions are analytic at least in the open polydisk |z1| < r1, . . . , |zn| <

rn. We remark that the entire functions of (z1, . . . , zn) are in AT(r1, . . . , rn) forany n-tuple (r1, . . . , rn) of positive real numbers.

For f ∈ AT given by (3.1) above, we let

‖ f‖ = ‖ f‖AT :=∞∑

m1,...,mn=0

|am1,...,,mn |rm11 · · · rmn

n . (3.2)


This expression is a norm on AT and turns AT into a commutative Banachalgebra (see Proposition 1.1 of [4]; in fact AT is a weighted �1-space).

We now turn to the construction of the Banach algebra DT . To give themost general definition, we will let X be a separable Banach space. Let Ai :[0, T] → L(X), i = 1, . . . , n, be measurable in the sense that A−1

i (E) is a Borelset in [0, T] for every strongly open subset E of L(X). Associate to each Ai(·)a Borel probability measure μi on [0, T] with

μi = λi + ηi (3.3)

for i = 1, . . . , n where λi is a continuous measure for each i and ηi is a finitelysupported discrete measure for each i. Let {τ1, . . . , τh} be the set obtained bytaking the union of the discrete measures η1, . . . , ηn and write

ηi =h∑

j=1

pijδτ j (3.4)

for each i = 1, . . . , n. (We will assume, for convenience, that τ1 <τ2 < · · ·<τh.)With this notation it may well be that many of the pij’s are equal to zero. Wenow define n positive real numbers r1, . . . , rn by

ri :=∫

[0,T]‖Ai(s)‖L(X)μi(ds) (3.5)

for each i = 1, . . . , n. These real numbers will serve as weights and we ig-nore for the present the nature of the Ai(·) as operators and introduce acommutative Banach algebra DT

((A1(·), μ1)

∼ , . . . , (An(·), μn)∼) (the disen-

tangling algebra) of “analytic functions” f((A1(·), μ1)

∼ , . . . , (An(·), μn)∼)

or, more briefly written, f (A1(·)∼, . . . , An(·)∼) where the objects (A1(·),μ1)

∼ , . . . , (An(·), μn)∼ or, more briefly, A1(·)∼, . . . , An(·)∼ replace the inde-

terminants z1, . . . , zn. For brevity, we will usually refer to the disentangling al-gebra as DT . (We write (Ai(·), μi)

∼ for the objects replacing z1, . . . , zn to stressthat these objects depend not only on the operator-valued functions but also onthe measures we associate with them.) It is worth noting here that the operator-valued functions do not have to be distinct though we will still consider theformal objects obtained from them to be distinct in the Banach algebra DT .All of this having been said, we take DT

((A1(·), μ1)

∼ , . . . , (An(·), μn)∼) to be

the collection of all expressions of the form

f (A1(·)∼, . . . , An(·)∼) =∞∑

m1,...,mn=0

am1,...,mn (A1(·)∼)m1 · · · (An(·)∼)

mn (3.6)

with the norm defined by

‖ f‖DT =∞∑

m1,...,mn=0

∣∣am1,...,mn

∣∣ rm11 · · · rmn

n . (3.7)

Via coordinate-wise addition and multiplication of such expressions it easilyfollows that (3.7) is a norm. Similarly, coordinate-wise addition and multi-plication of the expressions seen in (3.6) causes us to observe that DT is a

282 L. Nielsen

commutative Banach algebra (see Proposition 1.2 of [4]). Moreover theBanach algebras AT and DT can be identified (see Proposition 1.3 of [4]; theproof of Proposition 1.3 in [4] is, of course, given in the time independentsetting although it turns out that the proof in the time dependent setting isthe same).

We work here in the commutative setting of the disentangling algebra DT .The definition of the disentangling map will depend on the disentangling of themonomial

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) := (A1(·)∼)m1 · · · (An(·)∼)

mn . (3.8)

Also, it is the disentangling of the monomial that shows best the connectionbetween Feynman’s ideas and this theory.

We now introduce the notation that is necessary for the disentangling map.For m ∈ N, let Sm be the set of all permutations of the integers {1, . . . , m} andgiven π ∈ Sm, we let

�m(π) = {(s1, . . . , sm) ∈ [0, T]m : 0 < sπ(1) < · · · < sπ(m) < T}. (3.9)

When π is the identity permutation it is common to write �m(π) as �m.For j = 1, . . . , n and all s ∈ [0, T], we let

A j(s)∼ = A j(·)∼; (3.10)

that is, we discard the time dependence of the operator-valued functionsthough we will use the time index to keep track of when a given operator acts.Next, given nonnegative integers m1, . . . , mn and letting m = m1 + · · · + mn,we define

Ci(s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

A1(s)∼ if i ∈ {1, . . . , m1}A2(s)∼ if i ∈ {m1 + 1, . . . , m1 + m2}

......

An(s)∼ if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}(3.11)

for i = 1, . . . , m and s ∈ [0, T]. Even though Ci(s)∼ clearly depends on thenonnegative integers m1, . . . , mn, we will suppress this dependence in ournotation to ease the presentation. Next, in order to accommodate the use ofdiscrete measures below, we will need a refined version of the time ordered sets�m(π) given above in (3.9). Let τ1, . . . , τh ∈ [0, T] be such that 0 < τ1 < · · · <

τh < T. (Of course, in our setting, the τi will be the elements of the union ofthe supports of the discrete measures ηi defined above.) Given m ∈ N, π ∈ Sm,and nonnegative integers r1, . . . , rh+1 such that r1 + · · · + rh+1 = m, define

�m;r1,...,rh+1(π) = {(s1, . . . , sm) ∈ [0, 1]m : 0 < sπ(1) < · · · < sπ(r1) < τ1

< sπ(r1+1)< · · ·<sπ(r1+r2) <τ2 <sπ(r1+r2+1) < · · ·<sπ(r1+···+rh) <τh

< sπ(r1+···+rh+1) < · · ·<sπ(m) <1}. (3.12)

Note: Cases where τ1 = 0 and/or τh = T are sometimes of interest. A commenton these cases can be found after the statement of the proposition below.


We may have different operators evaluated at the same τi when we carryout time-ordering calculations. When this happens, Feynman’s “rules” abovedo not specify the order of operation and so a choice needs to be made. (Thissituation arises in certain physical problems where a particular choice may benatural for the problem at hand.) We can make any choice for the orderingand, indeed, can make different choices at each τi. For convenience, we willmake a definite choice, letting A1(·) act first, A2(·) act second, etc.

We are now prepared to time-order the monomial Pm1,...,mn according to thedirections provided by the measures μ1, . . . , μn. We note that the calculationleading to the time-ordered expression below are much more complicated thanthe corresponding calculation found in Proposition 2.2 of [4] for continuoustime ordering measures. The details of the calculation can be found in [11] andwe simply quote the result here.

Proposition Let m1, . . . , mn ∈ N be given. Then the monomialPm1,...,mn

(A1(·)∼, . . . , An(·)∼) is given in time ordered form by

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) ×

=∑

q11+q12=m1

∑q21+q22=m2

· · ·∑

qn1+qn2=mn

(m1! · · · mn!

q11!q12!q21!q22! · · · qn1!qn2!)

×

×∑

π∈Sq11+q21+···+qn1

·∑

r1+···+rh+1=q11+q21+···+qn1

∑j11+···+ j1h=q12

∑j21+···+ j2h=q22

· · ·∑

jn1+···+ jnh=qn2

·(

q12!q22! · · · qn2!j11! · · · j1h! j21! · · · j2h! · · · jn1! · · · jnh!

)×

×∫

�q11+q21+···+qn1;r1 ,··· ,rh+1(π)

· Cπ(q11+q21+···+qn1)(sπ(q11+q21+···+qn1))

· · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) · [pnh An(τh)

∼] jnh · · · [p2h A2(τh)∼] j2h ×

× [p1h A1(τh)∼] j1h · Cπ(r1+···+rh)(sπ(r1+···+rh)) · · · Cπ(r1+1)(sπ(r1+1)) ×

× [pn1 An(τ1)∼] jn1 · · · [p21 A2(τ1)

∼] j21[

p11 A1(τ1)∼] j11 Cπ(r1)(sπ(r1))

· · · Cπ(1)(sπ(1)) · (λq11

1 × · · · × λqn1n

) (ds1, . . . , dsq11+q21+···+qn1

)(3.13)

where λi is the continuous part of μi [see (3.3)].

Note: If τ1 = 0, the first element to act on the right-hand side of (3.13)above would be

[p11 A1(τ1)

∼] j11 . If τh = 1, the last element to act would be[pnh An(τh)

∼] jnh .Now that we have the time ordered monomial in hand, we can define the

disentangling map T Tμ1,...,μn

which will take us from the commutative setting of

284 L. Nielsen

the disentangling algebra DT to the non-commutative setting of L(X). All thatwe need to do is replace the objects Ci(s) by the corresponding operator-valuedfunctions. This amounts to erasing the tildes; to be precise we define

Ci(s) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

A1(s) if i ∈ {1, . . . , m1}A2(s) if i ∈ {m1 + 1, . . . , m1 + m2}

......

An(s) if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}(3.14)

Similarly, in the evaluations at τ1,. . ., τh we replace Aj(·)∼ with Aj(·) for each j.

Definition 3.1 We define the action of the disentangling map T Tμ1,...,μn

on themonomial Pm1,...,mn by

T Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) ×

=∑

q11+q12=m1

∑q21+q22=m2

· · ·∑

qn1+qn2=mn

(m1! · · · mn!

q11!q12!q21!q22! · · · qn1!qn2!)

×

×∑

π∈Sq11+q21+···+qn1

·∑

r1+···+rh+1=q11+q21+···+qn1

∑j11+···+ j1h=q12

∑j21+···+ j2h=q22

· · ·∑

jn1+···+ jnh=qn2

·(

q12!q22! · · · qn2!j11! · · · j1h! j21! · · · j2h! · · · jn1! · · · jnh!

)×

×∫

�q11+q21+···+qn1;r1 ,··· ,rh+1 (π)

· Cπ(q11+q21+···+qn1)(sπ(q11+q21+···+qn1))

· · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) · [pnh An(τh)

] jnh · · · [p2h A2(τh)] j2h ×

× [p1h A1(τh)] j1h · Cπ(r1+···+rh)(sπ(r1+···+rh)) · · · Cπ(r1+1)(sπ(r1+1)) ×

× [pn1 An(τ1)] jn1[

p21 A2(τ1)] j21[

p11 A1(τ1)] j11 Cπ(r1)(sπ(r1))

· · · Cπ(1)(sπ(1)) · (λq11

1 × · · · × λqn1n

) (ds1, . . . , dsq11+q21+···+qn1

).

(3.15)

Then, for f (A1(·)∼, . . . , An(·)∼) ∈ DT (A1(·)∼, . . . , An(·)∼), given by

f (A1(·)∼, . . . , An(·)∼) =∞∑

m1,...,mn=0

am1,...,mn(A1(·)∼)m1 · · · (An(·)∼)mn (3.16)

we set

T Tμ1,...,μn

f (A1(·)∼, . . . , An(·)∼)

=∞∑

m1,...,mn=0

am1,...,mnT Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) . (3.17)


The disentangling map as defined here does indeed give a bounded linearoperator (in fact, a contraction) from DT to L(X). For a proof of this, seeTheorem 2 of [11]. (The proof given there is for the time independent settingalthough the proof for the time dependent setting is essentially identical exceptfor the weights used.)

Remark 3.2 We mention here that we will sometimes use fμ1,...,μn(A1(·), . . . ,An(·)) in place of T T

μ1,...,μnf (A1(·)∼, . . . , An(·)∼) .

The definition of the disentangling map applied to the monomial Pm1,...,mn ,quite complicated when the time ordering measures have discrete parts, sim-plifies significantly when the time ordering measures are all continuous. (Werecall that the measure μ on S is continuous when μ({x}) = 0 for all s ∈ S.)Indeed, Proposition 1 of [11] shows that if μ1, . . . , μn are continuous, then

T Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼)

=∑

π∈Sm1+···+mn

∫

�m1+···+mn (π)

Cπ(m1+···+mn)(sπ(m1+···+mn)) · · · Cπ(1)(sπ(1)) ×

× (μm11 × · · · × μmn

n )(ds1, . . . , dsm1+···+mn), (3.18)

and this is the disentangling of the monomial that is found in [4] and [8].The disentangling of an arbitrary element of DT is found by applying thedisentangling of the monomial term-by-term in the power series. We state theend result below:

fμ1,...,μn (A1(·)∼, . . . , An(·)∼)

=∞∑

m1,...,mn=0

am1,...,mn

∑π∈Sm

∫

�Tm(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1)) ·

· (μm11 × · · · × μmn

n

)(ds1, . . . , dsm). (3.19)

where m = m1 + · · · + mn.

4 Improving the Operational Calculus

Now that we have proved Theorem 2.3 giving a relation between weakconvergence of sequences of probability measures and Hilbert space valuedfunctions in Section 2 and have outlined the operational calculus in the timedependent setting in Section 3, we are ready to discuss how Theorem 2.3improves the operational calculus, at least when we are working in a separableHilbert space.

Remark 4.1 As mentioned above, the restriction to a separable Hilbert space,while not as general as might be hoped, is often not at all a handicap. Indeed,

286 L. Nielsen

when working in the standard nonrelativistic quantum mechanical setting, wework in the separable Hilbert space L2(Rd).

We will first address the theorem stated in the introduction above (from[14]). For the reader’s convenience we will state the theorem again:

Theorem 4.2 Let Ai : [0, T] → L(X), i = 1, . . . , n, be continuous with respectto the usual topology on [0, T] and the norm topology on L(X). Associate toeach Ai(·) a continuous Borel probability measure μi on [0, T]. Let {μik}∞k=1,i = 1, . . . , n, be sequences of Borel probability measures on [0, T] such that, foreach i = 1, . . . , n, μik ⇀ μi. Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)


∼ , . . . , (An(·), μn)∼). Then

limk→∞∣∣� (T T

μ1k,...,μnk(πk(θ f ))

)− �(T T

μ1,...,μn(π0(θ f ))

)∣∣ = 0

for all � ∈ L(X)∗ and all θ f = ( f, f, f, . . .) ∈ UD.

With Theorem 2.3 in hand, the statement of the theorem above can bechanged to the following:

Theorem 4.3 For i = 1, . . . , n let Ai : [0, T] → L(H), H a separable Hilbertspace, be continuous with respect to the usual topology on [0, T] and the normtopology on L(H). Associate to each Ai(·) a continuous Borel probabilitymeasure μi on [0, T]. For each i = 1, . . . , n, let {μik}∞k=1 be a sequence of con-tinuous Borel probability measures on [0, T] such that μik ⇀ μ as k → ∞.Construct the direct sum Banach algebra

UD :=∑

k∈N∪{0}

⊕DT((A1(·), μ1k)

∼ , . . . , (An(·), μnk)∼)


∼ , . . . , (An(·), μn)∼) . Then, for

any θ f := ( f, f, f, . . .) ∈ UD and any φ ∈ H, we have

limk→∞∥∥T T

μ1k,...,μnk(πk(θ f ))φ − T T

μ1,...,μn(π0(θ( f ))φ

∥∥H = 0 (4.1)

where πk is the canonical projection of UD onto the disentangling algebraindexed by the measures μ1k, . . . , μnk. Of course, we can think of the conclusionin (4.1) as

limk→∞∥∥T T

μ1k,...,μnk( f )φ − T T

μ1,...,μn( f )φ∥∥H = 0.

Remark 4.4 Comparing the statement of Theorem 4.3 to the statement ofTheorem 4.2, we see the difference quite clearly. We obtain strong operator


convergence in Theorem 4.3, a much stronger conclusion than in Theorem 4.2.The proofs are quite similar and below we will sketch the proof, going intosome detail concerning the use of Theorem (2.3).

Proof Let m1, . . . , mn ∈ N and let φ ∈ H. We show first that

limk→∞

‖T Tμ1k,...,μnk

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ −

− T Tμ1,...,μn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ‖H = 0. (4.2)

NOTE: It is in proving this assertion that the difference in the proof of thistheorem as compared to Theorem 4.2 arises. We need only choose a vectorfrom H instead of a linear functional on L(H) (or a linear functional λ ∈ H∗and a vector φ ∈ H).

Using the definition of the disentangling map, remembering that we are inthe continuous measure setting of the operational calculus, we can write thenorm difference above as

∥∥∥∥∑π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm) −

−∑π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11 ×· · ·×μmn

n

)(ds1, . . . , dsm)

∥∥∥∥H

.

(4.3)

We now note that, since the operator-valued functions Ai(·) are all continuous,the function fm : [0, T]m → H given by

fm(s1, . . . , sm) = Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ (4.4)

is then a continuous function. Moreover, it is a norm-bounded function intothe Hilbert space, since each of the Ai(·) is a continuous function on a compactsubset of R. Also, since [0, T]m is a separable metric space, we have μ

m11k ×

· · · × μmnnk ⇀ μ

m11 × · · · × μmn

n as k → ∞ (see [1], Theorem 3.2). It follows fromTheorem 2.3 that

limk→∞

∥∥∥∥∑π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm) −

−∑π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11 ×· · ·×μmn

n

)(ds1,. . ., dsm)

∥∥∥∥H

=0.

(4.5)

This establishes our assertion.

288 L. Nielsen

We now sketch the remainder of the proof, reminding the reader that itfollows the proof of Theorem 3.1 of [14] very closely. Let θ f =( f, f, f, . . .)∈UD and write f as in (3.6) above. For φ ∈ H, we can write

‖T Tμ1k,...,μnk

(πk(θ f ))φ − T Tμ1,...,μn

(π0(θ f ))φ‖H

=∥∥∥∥

∞∑m1,...,mn=0

am1,...,mn

∑π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ ×

× (μm11k × · · · × μ

mnnk

)(ds1, . . . , dsm) −

∞∑m1,...,mn=0

am1,...,mn

∑π∈Sm

∫

�m(π)

×

× Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))φ(μ

m11 × · · · × μmn

n

)(ds1, . . . , dsm)

∥∥∥∥H

� ‖φ‖H( ∞∑

m1,...,mn=0

|am1,...,mn |∑π∈Sm

∫

�m(π)

‖Cπ(m)(sπ(m))‖

· · · ‖Cπ(1)(sπ(1))‖(μ

m11k × · · · × μ

mnnk

)(ds1, . . . , dsm) +

+∞∑

m1,...,mn=0

|am1,...,mn |∑π∈Sm

∫

�m(π)

‖Cπ(m)(sπ(m))‖ · · · ‖Cπ(1)(sπ(1))‖ ×

× (μm11 × · · · × μmn

n

)(ds1, . . . , dsm)

)= ‖φ‖H

(‖ f‖Dk + ‖ f‖D0

)(4.6)

where the subscript Dk refers to the kth- disentangling algebra in the direct sumalgebra UD. (The last line is arrived at via the standard Banach algebra inequal-ity ‖xy‖ � ‖x‖‖y‖ which results in a product of real-valued and consequentlycommutative functions. The disentangling is then “unraveled” or reversed toobtain the last line.) Recall that the norm on UD is

‖ {g�}∞�=1 ‖UD = sup�∈N∪{0}

‖g�‖D�.

Let ε > 0 be given. There is a k0 ∈ N such that ‖θ f ‖UD < ‖ f‖k0 + ε. Using (4.6)we therefore have

‖T Tμ1k,...,μnk

(π0(θ f ))φ − T Tμ1,...,μn

(π0(θ f ))φ‖H � ‖φ‖H(‖ f‖k0 + ‖ f‖0 + ε

)(4.7)

We see, then, that a summable scalar-valued dominating function for

∞∑m1,...,mn=0

|am1,...,mn |∥∥Pm1,...,mn

μ1k,...,μnk(A1(·), . . . , An(·)) φ −

− Pm1,...,mnμ1,...,μn

(A1(·), . . . , An(·))φ∥∥H (4.8)


is

(m1, . . . , mn) → ‖φ‖H|am1,...,mn |(

rm11,k0

· · · rmnn,k0

+ rm11 · · · rmn

n

)+ ε‖φ‖H

2m(4.9)

where the weights ri,k0 are

ri,k0 =∫

[0,T]‖Ai(s)‖μik0(ds) (4.10)

and, similarly,

ri :=∫

[0,T]‖Ai(s)‖μi(ds). (4.11)

We can therefore apply the dominated convergence theorem for Bochnerintegrals and pass the limit on the index k through the sum over m1, . . . , mn.Using (4.2) we finish the proof. �

We now move onto the theorem concerning the stability of the disentanglingmap with respect to time-ordering measures that have discrete parts. In thetime dependent setting, this theorem has not appeared in print. A time inde-pendent version of this theorem has, however, appeared in [12]. We remarkhere that the proof in the time independent setting is more straight forwardthan the proof in the time dependent setting. The main difficulty is, besidesthe obvious combinatoric complexity of the combined continuous/discretesetting, that we once again have a countably infinite family of disentanglingalgebras. The strategy of the proof, as one would expect, is the same asthe proof just above. We will show that the limit of the norm differenceof disentangled monomials vanishes and then verify that we can use thedominated convergence theorem for Bochner integrals to pass the limit onthe sequence of measures through the sum over m1, . . . , mn. The presence ofdiscrete parts to the time ordering measures will cause some complications,as one would expect. Before stating the theorem, however, we need a lemmaconcerning weak convergence of the types of discrete measures that we will beusing. (This lemma is Lemma 3.1 of [12]; the proof will not be presented here.)

Lemma 4.5 η :=∑hi=1 piδτi be a purely discrete probability measure on [0, T]

with finite support. Assume that 0 < τ1 < · · · < τh < T. Let

αi := min (τi − τi−1, τi+1 − τi)

for i =1, . . . , h where we take τ0 = 0 and τh+1 =T. In each interval (τi − αi,

τi + αi), i = 1, . . . , h, choose sequences {τik}∞k=1. For each i = 1, . . . , h choosea sequence {pik}∞k=1 such that

ηk =h∑

i=1

pikδτik (4.12)

290 L. Nielsen

is a probability measure for each k. Then ηk ⇀ η if and only if

{pik → pi and τik → τi if pi �= 0pik → pi and {τik}∞k=1 bounded if pi = 0

(4.13)

for i=1,. . . ,h.

We now state the stability theorem.

Theorem 4.6 Let Ai : [0, T] → L(H), i = 1, . . . , n, be continuous with respectto the norm topology on L(H). Associate to Ai(·), i = 1, . . . , � the continuousBorel probability measure μi on [0, T] and associate to Ai(·), i = � + 1, . . . , n,the purely discrete probability measure ηi with finite support. Let {τ1, . . . , τh} bethe union of the supports of the ηi, assume that 0 < τ1 < · · · < τh < T and write

ηi =h∑

j=1

pijδτ j (4.14)

for each i = � + 1, . . . , n. (As observed above, some of the numbers pij may

be zero.) For each i = � + 1, . . . , n choose sequences{τ jk}∞

k=1 and{

pkij

}∞k=1

as in Lemma 4.5. (In particular, we will assume that each sequence{τ jk}∞

k=1converges.) Then

ηik =h∑

j=1

pkijδτ jk (4.15)

converges weakly to ηi for each i=�+1, . . . , n. Also, choose sequences {μik}∞k=1,i=1, . . . , �, of continuous Borel probability measures such that μik ⇀μi.

Define the direct sum Banach algebra UD as in Theorem 4.3 where thedisentangling algebras forming the summands are DT

((A1(·), μ1k)

∼ , . . . ,

(A�(·), μ�k)∼ ,(

A�+1(·), η�+1,k)∼

, . . . , (An(·), ηnk)∼) for k �= 0 and DT ((A1(·),

μ1)∼ , . . . , (A�(·), μ�)

∼ , (A�+1(·), η�+1)∼ , . . . , (An(·), ηn)

∼) for k = 0.We conclude that, for any θ f = ( f, f, , f, . . .) ∈ UD and φ ∈ H,

limk→∞

‖T Tμ1k,...,μ�k,η�+1,k,...,ηnk

(πk(θ f ))φ − T Tμ1,...,μ�,η�+1,...,ηn

(π0(θ f ))φ‖H = 0. (4.16)

Proof Let φ ∈ H. For m1, . . . , mn ∈ N, we first show that

limk→∞

‖T Tμ1k,...,μ�k,η�+1,k,...,ηnk

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ −

− T Tμ1,...,μ�,η�+1,...,ηn

Pm1,...,mn (A1(·)∼, . . . , An(·)∼) φ‖H = 0. (4.17)


As before, Theorem 2.3 will be crucial. We write (4.17) using the definition ofthe disentangling map for the combined continuous/discrete setting. Indeed,(3.15) lets us write

‖Pm1,...,mnμ1k,...,μ�k,η�+1,k,...,ηnk

(A1(·),. . ., An(·))−Pm1,...,mnμ1,...,μ�,η�+1,...,ηn

(A1(·),. . ., An(·)) ‖H

�∑

j�+1,1+···+ j�+1,h=m�+1

· · ·∑

jn1+···+ jnh=mn

m�+1! · · · mn!j�+1,1! · · · j�+1,h! · · · jn1! · · · jnh! ×

×∑

π∈Sm1+···+m�

∑r1+···+rh+1=m1+···+m�

∥∥∥∥∫

�Tm1+···+m�;r1 ,...,rh+1

(π)

Cπ(m1+···+m�)(sπ(m1+···+m�))

· · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1))[pk

nh An(τhk)] jnh · · ·[pk

�+1,h A�+1(τhk)] j�+1,h ×

×Cπ(r1+···+rh)(sπ(r1+···+rh)) · · · Cπ(r1+1)(sπ(r1+1))[

pkn1 An(τ1k)

] jn1

· · ·[pk�+1,1 A�+1(τ1k)

] j�+1,1 Cπ(r1)(sπ(r1)) · · · Cπ(1)(sπ(1))φ(μ

m11k ×· · ·×μ

m�

�k

)××(ds1, . . . , dsm1+···+m�

)−∫

�Tm1+···+m�;r1 ,...,rh+1

(π)

Cπ(m1+···+m�)(sπ(m1+···+m�))×

· · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1))[

pnh An(τh)] jnh · · · [p�+1,h A�+1(τh)

] j�+1,h ××Cπ(r1+···+rh)(sπ(r1+···+rh)) · · · Cπ(r1+1)(sπ(r1+1))

[pn1 An(τ1)

] jn1

· · · [p�+1,1 A�+1(τ1)] j�+1,1 Cπ(r1)(sπ(r1)) · · · Cπ(1)(sπ(1))φ×

×(μm11 × · · · × μ

m�

�

) (ds1, . . . , dsm1+···+m�

) ∥∥∥∥H

(4.18)

Now, for a nonnegative integer k and any nonnegative integers m1, . . . , mn,j�+1,1,. . ., j�+1,h, jn1,. . ., jnh, r1,. . ., rh+1 such that j�+1,1+· · ·+ j�+1,h =m�+1, . . . ,

jn1 + · · · + jnh = mn, r1 + · · · + rh+1 = m1 + · · · + m� and any π ∈ Sm1+···+m�,

define fk : [0, T]m1+···+m� → H by

fk(s1, . . . , sm1+···+m�

):= Cπ(m1+···+m�)(sπ(m1+···+m�)) · · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) ·

·{

n−1∏α=0

[pk

n−α,h An−α(τhk)] jn−α,h

}Cπ(r1+···+rh+1)(sπ(r1+···+rh+1))

· · · Cπ(r1+1)(sπ(r1+1)) ·{

n−1∏α=0

[pk

n−α,1 An−α(τ1k)] jn−α,1

}×

× Cπ(r1)(sπ(r1)) · · · Cπ(1)(sπ(1))φ. (4.19)

(The functions fk are nothing other than the k-dependent integrands in thenorm difference above.) Note that the dependence on the index k is onlypresent in the evaluations at the support points of the discrete measures.

292 L. Nielsen

Since each Ai(·) is continuous, Ai(τ jk) → Ai(τ j) for all i = 1, . . . , n and j =1, . . . , h. It is clear, then, that fk → f uniformly on [0, T]m1+···+m� where

f(s1, . . . , sm1+···+m�

):= Cπ(m1+···+m�)(sπ(m1+···+m�)) · · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) ×

×{

n−1∏α=0

[pn−α,h An−α(τh)

] jn−α,h

}Cπ(r1+···+rh+1)(sπ(r1+···+rh+1))

· · · Cπ(r1+1)(sπ(r1+1)) ·{

n−1∏α=0

[pn−α,1 An−α(τ1)

] jn−α,1

}×

× Cπ(r1)(sπ(r1)) · · · Cπ(1)(sπ(1))φ. (4.20)

(Of course, the function f is the integrand from the norm difference abovethat does not depend on the index k.) Applying Theorem 2.4, we have∥∥∥∥

∫

�Tm1+···+m�;r1 ,...,rh+1

(π)

Cπ(m1+···+m�)(sπ(m1+···+m�)) · · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) ×

× [pknh An(τhk)

] jnh · · · [pk�+1,h A�+1(τhk)

] j�+1,h Cπ(r1+···+rh)(sπ(r1+···+rh))

· · · Cπ(r1+1)(sπ(r1+1))[

pkn1 An(τ1k)

] jn1 · · · [pk�+1,1 A�+1(τ1k)

] j�+1,1 ×× Cπ(r1)(sπ(r1)) · · · Cπ(1)(sπ(1))φ

(μ

m11k × · · · × μ

m�

�k

) (ds1, . . . , dsm1+···+m�

)−−

∫

�Tm1+···+m�;r1 ,...,rh+1

(π)

Cπ(m1+···+m�)(sπ(m1+···+m�))

· · · Cπ(r1+···+rh+1)(sπ(r1+···+rh+1)) ×× [pnh An(τh)

] jnh · · · [p�+1,h A�+1(τh)] j�+1,h Cπ(r1+···+rh)(sπ(r1+···+rh))

· · · Cπ(r1+1)(sπ(r1+1))[

pn1 An(τ1)] jn1 · · · [p�+1,1 A�+1(τ1)

] j�+1,1 ×

× Cπ(r1)(sπ(r1))· · ·Cπ(1)(sπ(1))φ(μ

m11 ×· · ·×μ

m�

�

) (ds1,. . ., dsm1+···+m�

) ∥∥∥∥H

→0

(4.21)

as k → ∞. This in turn shows at once that

limk→∞

∥∥∥Pm1,...,mnμ1k,...,μ�k,η�+1,k,...,ηnk

(A1(·), . . . , An(·)) φ −

− Pm1,...,mnμ1,...,μ�,η�+1,...,ηn

(A1(·), . . . , An(·)) φ∥∥H = 0 (4.22)

We now let θ f = ( f, f, f, . . .) ∈ UD and write f as in (3.16). Even thoughwe have discrete measures here, the rest of the proof goes through in an


almost identical manner as the proof of the corresponding part of the proofof Theorem 4.3. We first write

∥∥ fμ1k,...,μ�k,η�+1,k,...,ηn,k (A1(·), . . . , An(·)) φ −− fμ1,...,μ�,η�+1,...,ηn (An(·), . . . , An(·)) φ

∥∥H

�∞∑

m1,...,mn=0

∣∣am1,...,mn

∣∣ ∥∥∥Pm1,...,mnμ1k,...,μ�k,η�+1,k,...,ηnk

(A1(·), . . . , An(·)) φ −

− Pm1,...,mnμ1,...,μ�,η�+1,...,ηn

(A1(·), . . . , An(·)) φ∥∥H

�∞∑

m1,...,mn=0

∣∣am1,...,mn

∣∣ (∥∥∥Pm1,...,mnμ1k,...,μ�k,η�+1,k,...,ηnk

(A1(·), . . . , An(·)) φ

∥∥∥H

+

+ ∥∥Pm1,...,mnμ1,...,μ�,η�+1,...,ηn

(A1(·), . . . , An(·)) φ∥∥H

). (4.23)

Now note that

∥∥∥Pm1,...,mnμ1k,...,μ�k,η�+1,k,...,ηnk

(A1(·), . . . , An(·)) φ

∥∥∥H

�∑

j�+1,1+···+ j�+1,h=m�+1

· · ·∑

jn1+···+ jnh=mn

m�+1! · · · mn!j�+1,1! · · · j�+1,h! · · · jn1! · · · jnh! ·

·∑

π∈Sm1+···+m�

∑r1+···+rh+1=m1+···+m�

∫

�Tm1 ,...,m�;r1 ,...,rh+1

(π)

· ∥∥Cπ(m1+···+m�)(sπ(m1+···+m�))∥∥

· · · ∥∥Cπ(r1+···+rh+1)(sπ(r1+···+rh+1))∥∥ ·{

n−1∏α=0

[pk

n−α,h ‖An−α(τhk)‖] jn−α,h

}×

× ∥∥Cπ(r1+···+rh)(sπ(r1+···+rh))∥∥ · · · ∥∥Cπ(r1+1)(sπ(r1+1))

∥∥×

×{

n−1∏α=0

[pk

n−α,1 ‖An−α(τ1k)‖] jn−α,1

}∥∥Cπ(r1)(sπ(1))∥∥ · · · ∥∥Cπ(1)(sπ(1))

∥∥ ‖φ‖H ×

× (μm11 × · · · × μ

m�

�

)(ds1, . . . , dsm1+···+m�

). (4.24)

Of course, we obtain essentially the same expression for the other monomialdisentangling. The only difference is that there is no dependence on the indexk. Once we have arrived at the expression above (and the correspondingexpression for the other monomial), we see that with the norms around all of

294 L. Nielsen

our operators, we have products of commutative real-valued functions. Hence,“unraveling” the disentangling we obtain∥∥∥Pm1,...,mn

μ1k,...,μ�k,η�+1,k,...,ηnk(A1(·), . . . , An(·)) φ

∥∥∥H

�(∫

[0,T]‖A1(s)‖μ1k(ds)

)m1

· · ·(∫

[0,T]‖A�(s)‖μ�k(ds)

)m�

×

×(∫

[0,T]‖A�+1(s)‖ η�+1,k(ds)

)m�+1

· · ·(∫

[0,T]‖An(s) ηnk(ds)

)mn

‖φ‖H(4.25)

and∥∥Pm1,...,mn

μ1,...,μ�,η�+1,...,ηn(A1(·), . . . , An(·)) φ

∥∥H

�(∫

[0,T]‖A1(s)‖μ1(ds)

)m1

· · ·(∫

[0,T]‖A�(s)‖μ�(ds)

)m�

×

×(∫

[0,T]‖A�+1(s)‖ η�+1(ds)

)m�+1

· · ·(∫

[0,T]‖An(s) ηn(ds)

)mn

‖φ‖H.

(4.26)

The integrals just above are the weights for the disentangling algebras underconsideration in this theorem and therefore we have∥∥ fμ1k,...,μ�k,η�+1,k,...,ηn,k (A1(·),. . ., An(·)) φ− fμ1,...,μ�,η�+1,...,ηn (An(·),. . ., An(·)) φ

∥∥H

�

⎛⎝ ∞∑

m1,...,mn=0

∣∣am1,...,mn

∣∣ (rm11,k · · · rmn

n,k + rm11 · · · rmn

n

)⎞⎠ ‖φ‖H

= ‖φ‖H (‖ f‖k + ‖ f‖0) � ‖φ‖H(‖θ f ‖UD + ‖ f‖0

)(4.27)

where the notation is as seen in the proof of Theorem 4.3 above. We canuse the equation just above to obtain a scalar-valued summable bound forthe expression seen in the second line of (4.23). Indeed, since the norm onUD is ‖ {gk} ‖UD = supk∈N

‖gk‖k, we can, given ε > 0, choose a k0 ∈ N such that‖θ f ‖UD < ‖ f‖k0 + ε. Then

‖φ‖H(‖θ f ‖UD + ‖ f‖0

)< ‖φ‖H

(‖ f‖k0 + ‖ f‖0 + ε)

(4.28)

and so our summable dominating function is

(m1, . . . , mn) −→ ‖φ‖H∣∣am1,...,mn

∣∣ (rm11,k0

· · · rmnn,k0

+ rm11 · · · rmn

n

)+

+ ε

2m1+···+mn‖φ‖H. (4.29)

We can therefore pass the limit on k through the sum over m1, . . . , mn in thesecond line of (4.23) and the theorem is proved. �


Remark 4.7 The reader has probably noticed that the theorem above con-cerned the stability of the disentangling map in the situation where some of thetime ordering measures are continuous and some are purely discrete. While itwould be desirable to have a stability theorem in this case, such a theorem hasnot been found. The difficulty in obtaining such a theorem seems to be withthe fact that, given a sequence {μk}∞k=1 of probability measures (with finitelysupported discrete parts) converging weakly to a probability measure μ, thelimit measure μ may be continuous, purely discrete, or have both continuousand discrete parts. Moreover, with a general sequence of measures, it may bethe case that the supports of the continuous and discrete parts of the individualmeasures may change. In the current formulation the Feynman’s operationalcalculus, we are unable to accommodate these features. It may be possible,however, to “smooth out" the sequence of probability measures and so obtaina convergence result for general time ordering measures.

References

1. Billingsley, P.: Convergence of probability measures, 2nd edn., Wiley Series in Probability andStatistics, John Wiley and Sons, Inc., New York, (1968)

2. Dunford, N., Schwartz, J.: Linear Operators, Part I, General Theory, Interscience Publishers,Inc, New York (1958)

3. Feynman, R.P.: An operator calculus having applications in quantum electrodynamics. Phys.Rev. 84, 108–128 (1951)

4. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators:definitions and elementary properties. Russ. J. Math. Phys. 8, 153–178 (2001)

5. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators:tensors, ordered support and disentangling an exponential factor. Math. Notes 70, 744–764(2001)

6. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators:spectral theory. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 5, 171–199 (2002)

7. Jefferies, B., Johnson, G.W.: Feynman’s operational calculi for noncommuting operators: themonogenic calculus. Adv. Appl. Clifford Algebras 11, 233–265 (2002)

8. Jefferies, B., Johnson, G.W., Nielsen, L.: Feynman’s operational calcului for time-dependentnoncommuting operators. J. Korean Math. Soc. 38, 193–226 (2001)

9. Johnson, G.W., Lapidus, M.L.: The Feynman Integral and Feynman’s Operational Calculus.Oxford Science Publications, Oxford Mathematical Monographs, Oxford Univ. Press, Oxfordand New York (2000)

10. Johnson, G.W., Nielsen, L.: A stability theorem for Feynman’s operational calculus. Stochasticprocesses, physics and geometry: new interplays, II, pp. 351–365 (Leipzig, 1999)

11. Johnson, G.W., Nielsen, L.: Feynman’s operational calculi: blending instantaneous and con-tinuous phenomena in Feynman’s operational calculi. Stochastic analysis and mathematicalphysics (SAMP/ANESTOC 2002), pp. 229–254. World Sci. Publ., River Edge, NJ (2004)

12. Nielsen, L.: Stability properties for Feynman’s operational calculus in the combined continu-ous/discrete setting. Acta Appl. Math. 88, 47–49 (2005)

13. Nielsen, L.: Stability properties of Feynman’s operational calculus for exponential functionsof noncommuting operators. Acta Appl. Math. 74, 265–292 (2002)

14. Nielsen, L.: Time dependent stability for Feynman’s operational calculus. Rocky Mountain J.Math. 35, 1347–1368 (2005)

15. Nielsen, L.: Stability properties of Feynman’s operational calculus, Ph.D. Dissertation,Mathematics, University of Nebraska Lincoln (1999)

Math Phys Anal Geom (2007) 10:297–312DOI 10.1007/s11040-008-9034-y

Reproducing Kernels and CoherentStates on Julia Sets

K. Thirulogasanthar · A. Krzyzak · G. Honnouvo

Received: 21 June 2007 / Accepted: 23 January 2008 /Published online: 13 March 2008© Springer Science + Business Media B.V. 2008

Abstract We construct classes of coherent states on domains arising fromdynamical systems. An orthonormal family of vectors associated to the gen-erating transformation of a Julia set is found as a family of square integrablevectors, and, thereby, reproducing kernels and reproducing kernel Hilbertspaces are associated to Julia sets. We also present analogous results ondomains arising from iterated function systems.

Keywords Coherent states · Reproducing kernel · Julia sets · IFS · Attractor

Mathematics Subject Classifications (2000) Primary 81R30 · 46E22

1 Introduction

Hilbert spaces are the underlying mathematical structure of many areas ofphysics and engineering such as quantum physics and signal analysis. An

The research of the first two authors was supported by Natural Sciences and EngineeringResearch Council of Canada.

K. Thirulogasanthar (B) · A. KrzyzakDepartment of Computer Science and Software Engineering, Concordia University,1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8e-mail: [email protected]

A. Krzyzake-mail: [email protected]

G. HonnouvoDepartment of Mathematics and Statistics, Concordia University,1455 De Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8e-mail: [email protected]

298 K. Thirulogasanthar et al.

important family of vectors of the Hilbert space of a physical problem isknown as coherent states, CS for short. This family of vectors is very usefulin describing quantum phenomena [1].

In this paper we let {φm}∞m=0 to be an orthonormal basis of an abstractseparable Hilbert space H. The well-known canonical coherent states aredefined by:

| z〉 = e− r2

2

∞∑

m=0

zm

√m!φm ∈ H,

where z ∈ C, the complex plane, z = reiθ .The definition of canonical coherent states has been generalized as follows:

Let D be an open subset of C. For z ∈ D set

| z〉 = N (|z|)− 12

∞∑

m=0

zm

√ρ(m)

φm ∈ H, (1.1)

where {ρ(m)}∞m=0 is a positive sequence of real numbers and N (|z|) is thenormalization factor ensuring that 〈z | z〉 = 1. If in addition {| z〉, z ∈ D} satisfy

∫

D

| z〉〈z | dμ = IH, (1.2)

where dμ is an appropriately chosen measure on D and IH is the identityoperator on H, then {| z〉, z ∈ D} is said to be a set of coherent states on D.

This generalization has produced families of vectors which have beensuccessfully applied to physical problems [1, 6, 11, 12, 14, 16].

In general, CS can be constructed as follows: Let (�, μ) be a measure spaceand H be a closed subspace of L2(�, μ). Let {�m}dim(H)

m=0 , dim(H) denotes thedimension of H, be an orthonormal basis of H satisfying:

dim(H)∑

m=0

|�m(x)|2 < ∞

for all x ∈ �. Let H be another Hilbert space such that dim(H) = dim(H). Let{φm}dim(H)

m=0 be an orthonormal basis of H. Define

K(x, y) =dim(H)∑

m=0

�m(x)�m(y). (1.3)

Then K(x, y) is a reproducing kernel and H is the corresponding reproducingkernel Hilbert space. For x ∈ �, define

| x〉 = K(x, x)−12

dim(H)∑

m=0

�m(x)φm. (1.4)

Reproducing kernels and coherent states on Julia sets 299

Therefore,

〈x | x〉 = K(x, x)−1dim(H)∑

m=0

�m(x)�m(x) = 1,

and

W : H −→ H with Wφ(x) = K(x, x)12 〈x | φ〉

is an isometry. Then, for φ, ψ ∈ H we have

〈φ | ψ〉H = 〈Wφ | Wψ〉H =∫

�

Wφ(x)Wψ(x)dμ(x)

=∫

�

〈φ | x〉〈x | ψ〉K(x, x)dμ(x),

and ∫

�

| x〉〈x | K(x, x)dμ(x) = IH, (1.5)

where K(x, x) is a positive weight function. Thus, the set of states {| x〉 : x ∈ �}forms a set of CS.

In the case where {�m}dim(H)m=0 is an orthogonal basis of H, one can define

ρ(m) = ‖�m‖2; m = 0, ..., dim(H), and obtain an orthonormal basis{

�m√ρ(m)

}dim(H)

m=0

of H. Then, setting

| x〉 = K(x, x)−12

dim(H)∑

m=0

�m(x)√ρ(m)

φm ∈ H, (1.6)

one obtains the desired result which is analogous to (1.1).The above discussion motivates the following definition.

Definition 1.1 Let D be an open subset of C. Let

�m : D −→ C, m = 0, 1, 2, . . . ,

be a sequence of complex functions. Define

| z〉 = N (|z|)− 12

∞∑

m=0

�m(z)√ρ(m)

φm ∈ H; z ∈ D, (1.7)

where N (|z|) is a normalization factor and {ρ(m)}∞m=0 is a sequence of nonzeropositive real numbers. The set of vectors in (1.7) is said to form a set of CS if

(a) 〈z | z〉 = 1 for all z ∈ D;(b) The states {| z〉 : z ∈ D} satisfy a resolution of the identity:

∫

D

| z〉〈z | dμ = I, (1.8)


where dμ is an appropriately chosen measure and I is the identity operatoron H.

For �m(z) = zm, z ∈ C, the states (1.7), for different ρ(m)’s, were studiedextensively and applied successfully in quantum theories. Interesting linkshave been established with group representation, classical polynomials and Liealgebras [1, 5, 7, 12, 16]. Another class of CS was introduced by Gazeau andKlauder for Hamiltonians with discrete and continuous spectrum by setting�m(J, α) = (

√J)meiemα and ρ(m) = e1e2...em, where em’s are the spectrum of

the Hamiltonian arranged in a suitable way [6]. Recently, in [2, 14], usingdefinition (1.7), CS were presented by setting �m(Z ) = Z m, where Z is a n × nmatrix valued function.

In this note we set �m(z) = gm(z)χAm(z) (see Sections 3 and 4 for precisedefinitions), where g is a complex function. Using iteration, we build classes ofCS on domains arising from dynamical systems with definition (1.7), namely,on Julia sets and fractal sets. In particular, an orthonormal family of vectorsassociated to the generating transformation of a Julia set is found as a family ofsquare integrable vectors, and, thereby, reproducing kernels and reproducingkernel Hilbert spaces are associated to Julia sets. We also present analogousresults on domains arising from iterated function systems. To the best of ourknowledge, such a problem has not been addressed in the literature.

In Section 2, we motivate definition (1.7) by building CS on a generaliterated function system. Section 3 is the main section of the note. In thissection, we build CS on Julia sets using iterations. Moreover, it contains theconstruction of reproducing kernels and reproducing kernel Hilbert spaces.In Section 4, we build CS on fractal sets using iterated function systems.Section 5 discusses the case when zero is contained in the Julia set or in theattractor of an iterated function system.

2 CS on Iterated Function System: General Case

In order to motivate definition (1.7) and to set a stage for the construction ofCS on Julia sets, we present a class of CS on a set which is invariant under theiterates of a map.

Let Q : C → C be a mapping such that for some D ⊆ C we have

Qn(D) ⊆ D, ∀n ∈ N.

Let μ be a probability measure on D and {Am}∞m=0 be a partition of D such that

D = ⋃∞m=0 Am a.e μ, An ∩ Am = ∅ for m = n, and

μ(Am) > 0 ∀m � 0 (2.1)

Assume that D ⊆ {z : A � |z|2 � B} for some positive constants A and B.Then by the invariance of D under the iterates of Q, we have

A � |Qm(z)|2 � B. (2.2)


Let

ρ(m) =∫

Am

|Qm(z)|2dμ(z). (2.3)

Then by (2.1) and (2.2) we have 0 < ρ(m) < ∞ and {ρ(m)}∞m=0 is a positivesequence of real numbers.

In the following we present a class of CS on D in the form of (1.7) with�m(z) = Qm(z)χAm(z), where χAm is the characteristic function of Am, i.e.,

χAm(z) ={

1 if z ∈ Am

0 if z /∈ Am

Theorem 2.1 For z ∈ D, the collection of vectors

| z〉 = N (z)−12

∞∑

m=0

Qm(z)χAm(z)√ρ(m)

φm ∈ H (2.4)

forms a set of CS.

Proof The normalization condition 〈z | z〉 = 1 holds trivially by (2.1) and (2.2)with

0 < N (z) =∞∑

m=0

|Qm(z)χAm(z)|2ρ(m)

< ∞.

We now find a resolution of the identity. Let dν(z) = N (z)dμ(z) be a measureon D. Then,

∫

D

| z〉〈z | dν(z) =∞∑

m=0

∞∑

l=0

∫

D

Qm(z)Ql(z)χAm(z)χAl (z)

N (z)ρ(m)dν(z) | φl〉〈φm |

=∞∑

m=0

∫

D

|Qm(z)|2χAm(z)

ρ(m)dμ(z) | φm〉〈φm |

=∞∑

m=0

∫

Am

|Qm(z)|2ρ(m)

dμ(z) | φm〉〈φm |

=∞∑

m=0

| φm〉〈φm |= IH,

where we have used (2.3) together with the fact that χAm(z)χAl (z) = χAmδml,δml is the Kronecker delta function. ��

Since Julia sets and fractals have rich dynamics and greater interest in math-ematics and physics, as an application of the above theory, in the followingsection, we build CS on Julia sets and fractal sets. Even though a reproducingkernel can be associated with the CS in (2.4), since our primary aim is focusedon Julia sets, we shall associate them with the CS of the Julia sets.


3 CS on Julia Sets

In this section, we define the �m’s of definition (1.7) in terms of the iteratesof a rational function of the Riemann sphere. Thus, naturally, the domain ofinterest falls under Julia sets. Julia sets have useful dynamical and topologicalproperties. We exploit these properties in our proofs.

3.1 Holomorphic Dynamics and the Julia Set

In this section, we state some definitions and well know results which will beused in the sequel. All the results can be found in the references [4, 9, 10].

Definition 3.1 A family of analytic functions having a common domain ofdefinition is called normal if every sequence in this family contains a locallyuniformly convergent subsequence.

Definition 3.2 Denote by C the Riemann sphere. Let Q : C → C be a rationalmap of degree greater than one. We denote the nth iterate of Q by Qn; i.e.,

Qn = Q ◦ Q · · · ◦ Q︸︷︷︸n times

.

The Fatou set is defined by:

F(Q) = {z : ∃ a neighborhood Uz s.t. {Qn}∞n=1 is a normal family in Uz}.The Julia set is defined by J = J(Q) = C\F(Q).

The Julia set has the following properties:

1. The Julia set is a compact set.2. The Julia set is a perfect set.3. The Julia set is completely invariant under Q; i.e., for any z ∈ J(Q) we

have Qn(z) ∈ J(Q), n = 0, ±1, ±2, . . . .

Since the Julia set is Q-invariant, we can define Q : J(Q) → J(Q). Let μ bea probability measure defined on the Julia set:

∫

Jdμ(z) = 1. (3.1)

A famous probability measure which is associated with Julia sets is calledconformal measure [13].

We assume:

A � |Qm(z)|2 � B for all m = 0, 1, 2.. and for all z ∈ J. (3.2)

where A and B are positive constants. We partition the Julia set as follows:

J =∞⋃

m=0

Am a.e μ, Am ∩ An = ∅ for m = n, μ(Am) > 0 ∀m � 0.

(3.3)


Let

ρ(m) =∫

Am

|Qm(z)|2dμ(z). (3.4)

Thus, by (3.2) and (3.3), 0 < ρ(m) < ∞ and {ρ(m)} is a positive sequence ofreal numbers.

Now in view of Theorem (2.1) we present a class of CS on Julia sets in theform of (1.7) with �m(z) = Qm(z)χAm(z) and z ∈ J as follows:

| z〉 = N (z)−12

∞∑

m=0


φm ∈ H. (3.5)

Remark 3.3 Our condition (3.2) is satisfied by polynomial Julia sets, for in-stance, by the quadratic family Q(z) = z2 + c (see the examples below andcompare with Section 5 to see how we deal with the case when 0 is an elementof the Julia set).

We now associate a reproducing kernel and a reproducing kernel Hilbertspace with Julia sets. Consider the sequence of functions {�m}∞m=0 with�m(z) = Qm(z)χAm(z) as L2(J, dμ) functions. Since

〈�m | �n〉 =∫

J�m(z)�n(z)dμ(z)

=∫

JQm(z)Qn(z)χAm(z)χAn(z)dμ(z)

= δmn

∫

Am

|Qm(z)|2dμ(z)

= δmnρ(m),

the family {�m}∞m=0 is orthogonal, and consequently { �m√ρ(m)

}∞m=0def= {�m}∞m=0 is

orthonormal in L2(J, dμ). Let dν = N (z)dμ(z) and H = L2(J, dν), then by theresolution of the identity, for φ ∈ H, the functions

� : J −→ C defined by �(z) = 〈z | φ〉are elements of H. Define

W : H −→ H by φ �→ �.

Using the resolution of the identity we have

‖Wφ‖2 = ‖�‖2 =∫

J�(z)�(z)dν(z)

=∫

J〈φ | z〉〈z | φ〉dν(z) = ‖φ‖2.

Thus, W is a partial isometry.


Let HK = WH, then HK is a closed subspace of H. Define

K(z, z′) = 〈z | z′〉 = N (z)−12 N (z′)−

12

∞∑

m=0

�m(z)�m(z′). (3.6)

Now, for any � ∈ HK, using the resolution of the identity, we have:

�(z) =∫

J〈z | z′〉〈z′ | φ〉dν(z′)

=∫

JK(z, z′)�(z′)dν(z′).

Thus, the reproducing property is satisfied by any � ∈ HK. It follows that K isa reproducing kernel and HK is the corresponding reproducing kernel Hilbertspace.

One can easily see that the kernel K satisfies:

(a) Hermiticity, K(z, z′) = K(z′, z);(b) Positivity, K(z, z) > 0;(c) Idempotence,

∫

JK(z, z′′)K(z′′, z′)dν(z′′) = K(z, z′).

From this point to Example (3.4) assume that the Julia set is symmetric withrespect to the real axis of the complex plane (note: Julia sets of rational mapswith real coefficients are symmetric with respect to the real axis [9, 10]). Thefollowing argument is standard in the theory of CS [1], however for the sake ofcompleteness we present it for Julia sets in brief.

Now we realize a resolution of the identity on a closed subset of L2(J, dμ)

which is isomorphic to the reproducing kernel Hilbert space Hk. For φ ∈ H, set

f (z) = N (z)12 〈z | φ〉. (3.7)

Using the resolution of the identity, we have:

〈 f | f 〉 =∫

Jf (z) f (z)dμ(z)

=∫

J〈φ | z〉〈z | φ〉dν(z) = ‖φ‖2,

i.e., f ∈ L2(J, dμ). One can also characterize a reproducing kernel and areproducing kernel Hilbert space in terms of the orthonormal family {�m}∞m=0.


For this we define

W : H −→ L2(J, dμ) with Wφ(z) = N (z)12 〈z | φ〉.

Let WH = H. Then W is an isometry from H to H given by:

Wφm(z) = N (z)12 〈φm | z〉

= N (z)12 〈N (z)−

12

∞∑

m=0


φm | φm〉

= Qm(z)χAm(z)√ρ(m)

= �m(z); ∀z ∈ J.

In fact, H is the closed linear span of the orthonormal family {�m}∞m=0. Further,H is the reproducing kernel Hilbert space corresponding to the reproducingkernel

K(z, z′) =∞∑

m=0

�m(z)�m(z′).

Using (3.6) we also obtain

K(z, z′) = N (z)12 N (z′)

12 K(z, z′).

A reproducing kernel Hilbert space is uniquely determined by its reproducingkernel. Thus, the Hilbert spaces HK and H are isomorphic. Let

ηz = N (z)12 W | z〉. (3.8)

Using the isometry property of W and using (3.7), for any f ∈ H and z ∈ J, wecan write

f (z) = 〈ηz | f 〉H. (3.9)

Therefore, the reproducing kernel is given by:

〈ηz′ | ηz〉H = N (z)12 N (z′)

12 〈z′ | z〉 =

∞∑

m=0

�m(z)�(z′) = K(z, z′). (3.10)

Since K is the reproducing kernel for the reproducing kernel Hilbert space H,for any f ∈ H and z ∈ J, we have

∫

JK(z, z′) f (z′)dμ(z′) = f (z). (3.11)

Using (3.9), (3.10) and (3.11) we obtain:∫

J| ηz〉〈ηz | dμ(z) = IH; (3.12)

i.e., a resolution of the identity for the Hilbert space H.Now, we present examples of families of rational maps which satisfy condi-

tion (3.2). These examples are well studied in the complex dynamics literature[4, 9, 10].


Example 3.4 Let Q(z) = eiα�kj=1

z−a j

1−a jz, where |a j| < 1 and a j ∈ C, j = 1, . . . , k.

Q(z) is known as Blaschke product. The following is well know fact about theJulia sets of Blaschke products [10]:

The Julia set of a Blaschke product is S1 if and only if there exists z0 ∈ D

such that Q(z0) = z0, where D is the open unit disc and S1 is the unit circle.

Thus, if Q(z0) = z0 for some z0 ∈ D, Q(z) clearly satisfies condition (3.2).For the probability measure μ, we take normalized Lebesgue arc measure. Wepartition the unit circle [0, 2π) = ∪∞

m=0 Am as follows:

Am =[

2π

2m+1,

2π

2m

), m = 0, 1, . . .

Then,

μ(Am) = 1

2π

∫ 2π2m

2π

2m+1

dθ = 1

2m+1∀m � 0.

Consequently, one can easily derive the explicit formulas of ρ(m) and N (z).

Example 3.5 The quadratic family, Q(z) = z2 + c, is the most famous familyin holomorphic dynamics. Although it is the simplest non-linear example, ithas very rich and complicated dynamics. Its Julia set is always symmetric withrespect to the origin and contained within the circle |z| = 2. Replacing c byits conjugate has the effect of reflecting Jc through the horizontal axis [4, 9,10]. Now, we give examples of Julia sets of the quadratic family that satisfycondition (3.2). The simplest case is when c = 0, where the Julia set is the unitcircle. The Julia set of Q(z) = z2 + c, for a very small c, is a quasi-circle. Forc = i/4 the Julia set is shown in Fig. 1. It is symmetric with respect to the originwhich does not belong to the Julia set. Hence, we can draw a neighborhoodN0 around the origin and inside the Julia set which is bounded inside the circle|z| = 2. Therefore, Q(z) satisfies (3.2).

Fig. 1 Q(z) = z2 + i/4


Fig. 2 Q(z) = z2 + i

We now present an example of the quadratic family which does not satisfycondition (3.2).

Example 3.6 Let Q(z) = z2 + i. The Julia set of this function, which is calledDendrite, is shown in Fig. 2. Here we cannot have a neighborhood N0 of theorigin such that N0 ∩ J = ∅ since 0 is an element of the Julia set. Thus, Q(z) =z2 + i does not satisfy condition (3.2).

4 CS on Fractal Sets

4.1 Iterated Function System

Let (X, d) be a complete metric space and τ1, τ2, ..., τK be a collection of trans-formations from X into itself. We assume that τ1, τ2, ..., τK are contractions;i.e., for k = 1, . . . , K

maxk

d(τk(x), τk(y)) ≤ α · d(x, y),

where 0 < α < 1. An iterated function system, IFS for short, with state depen-dent probabilities , T = {τ1, . . . , τK; p1(x), . . . , pK(x)} is defined by choosingτk(x) with probability pk(x), pk(x) ≥ 0,

∑Kk=1 pk(x) = 1. The iterates of T are

given by:

Tm(x) = τkm ◦ τkm−1 ◦ · · · ◦ τk1(x) (4.1)

with probability

pkm(τkm−1 ◦ · · · ◦ τk1(x)) · pkm−1(τkm−2 ◦ · · · ◦ τk1(x)) · · · pk1(x).

Let (H(X), h(d)) denote the space of nonempty compact subsets corre-sponding to (X, d) with the Hausdorff metric h(d) [3]. T : H(X) → H(X) isgiven by:

T(B) = ∪Kk=1τk(B)


for all B ∈ H(X). The Banach contraction theorem furnishes an attractor tothe dynamical system (X, T) [3]. In the special case when X = R

2, the attractorof an IFS is called a fractal [3]. The presence of the probabilities allows moreweight on some transformations over the others. Thus, the fractal may haveparts which are more “dense" than the remaining parts.

We consider X = R2. Then, the transformations are two component

objects, i.e.,

Tm(x) = ( f m(x), gm(x)).

We interpret Tm(x) as the complex number f m(x) + igm(x) and denote it byTm(x). We consider a Hilbert space H over complex numbers, thus the objectTm(x)φ ∈ H for all φ ∈ H.

Let A denote the attractor of the IFS and B the Borel subsets of (A, d). Letμ be a probability measure on A:

∫

A

dμ(x) = 1. (4.2)

In particular, one can use the probabilities to construct a measure on theattractor. For example, if the probabilities are constants, the measure canbe constructed in the following way: μ(A) = 1, μ(τk(A)) = pk, μ(τl ◦ τk(A)) =pl · pk, and so on.

4.2 Construction of CS

In this section, we assume that X = R2 and d = | · | is the Euclidean metric

on R2. We also assume that there exists a neighborhood, N0, of the origin

such that

N0 ∩ A = ∅. (4.3)

Remark 4.1 Observe that if T satisfies (4.3), we have:

A � |Tm(x)| � B for all m = 0, 1, 2.. and for all x ∈ A, (4.4)

where A and B are positive constants. The contraction property of the IFS,which guarantees the existence of the attractor, guarantees the existence ofthe upper bound B. This is essential in the proof of our theorems.

We partition the attractor as follows:

A =∞⋃

m=0

Am a.e Am ∩ An = ∅ for m = n, μ(Am) > 0 ∀m � 0. (4.5)

Let

ρ(m) =∫

Am

|Tm(x)|2dμ(x). (4.6)



Now in view of Theorem (2.1) we present a class of CS on fractal sets in theform of (1.7) with �m(x) = Tm(x)χAm(x) and x ∈ A as follows:

| x〉 = N (x)−12

∞∑

m=0

Tm(x)χAm(x)√ρ(m)

φm ∈ H. (4.7)

forms a set of CS.

Remark 4.2 Following the procedure discussed in Section 3, we can associatea reproducing kernel and a reproducing kernel Hilbert space for the attractor.

(a) The set of vectors �m(x) = Tm(x)χAm(x) is an orthogonal family in theHilbert space L2(A, dμ) and the family { �m√

ρ(m)}∞m=0 = {�m}∞m=0 is ortho-

normal in the same Hilbert space.(b) There is a partial isometry,

W : H −→ L2(A, dμ), defined by Wφ(x) = N (x)12 〈φ | x〉.

(b) WH = H ⊂ L2(A, dμ) is a reproducing kernel Hilbert space and theassociated reproducing kernel is

K(x, y) =∞∑

m=0

�m(x)�m(y).

The construction depends on the invariance of the attractor under iterationand on condition (4.3). We present an example which satisfies condition (4.3).

Example 4.3 Let T = {τ1, τ2, τ3}, x = (t1, t2) ∈ R2, where τ1 = (

12 t1, 1

2 t2) +

(12 , 0

), τ2 = (

12 t1, 1

2 t2) + (1, 0) and τ3 = (

12 t1, 1

2 t2) +

(34 ,

√3

4

). Notice that T is a

contraction with a contractivity factor 12 . The attractor of T is shown in Fig. 3

Fig. 3 Sierpinski triangle,the attractor of the IFS inExample 4.3 andExample 4.4


with the origin outside the attractor. For this example, the vertices of the

triangle are (1, 0), (2, 0) and(

32 ,

√3

2

). Thus, T satisfies condition (4.3).

Now, we present an example which does not satisfy condition (4.3).

Example 4.4 Let T = {τ1, τ2, τ3}, x = (t1, t2) ∈ R2, where τ1 = (

12 t1, 1

2 t2), τ2 =

(12 t1, 1

2 t2) + (

12 , 0

)and τ3 = (

12 t1, 1

2 t2) +

(14 ,

√3

4

). Notice that T is a contraction

with a contractivity factor 12 . The attractor of T is shown in Fig. 3. For this

example, the vertices of the triangle are (0, 0), (1, 0) and(

12 ,

√3

2

). Thus, T does

not satisfy condition (4.3).

5 CS on Julia Sets When the Origin is in the Julia Set

This section is motivated by Examples 3.6 and 4.4. We build CS on Julia setswhen the origin is an element of the Julia set. For this purpose we use thedistance between a reference point and the orbit of an element of the Julia set.

Let J be the Julia set of the transformation Q and F be its Fatou set. Weassume that F = ∅. Let A be a positive constant and d be a metric on C. Fixz0 ∈ F such that

infz∈J

m=0,1,2..

d(Qm(z), z0) � A > 0, (5.1)

and assume that there exists another positive constant B such that

A � d(Qm(z), z0) � B for all z ∈ J and m = 0, 1, 2, ... (5.2)

Let {Am}∞m=0 and μ be as in Section 3. Define

ρ(m) =∫

Am

(d(Qm(z), z0)

)2dμ(z). (5.3)


Once again, in view of Theorem (2.1) we present a class of CS on Julia setsin the form of (1.7) with �m(z) = d(Qm(z), z0)χAm(z) and z ∈ J as follows:

| z〉 = N (z)−12

∞∑

m=0

d(Qm(z), z0)χAm(z)√ρ(m)

φm ∈ H. (5.4)

Example 5.1 Consider the Julia set of Example 3.6. Since the Julia set is insidethe circle {z ∈ C : |z| = 2}, one can fix z0 in the Fatou set, for example, with|z0| = 3. Thus, condition (5.2) is satisfied.


Remark 5.2 The above theory can be directly extended to the attractors ofSection 4 when the origin is inside the attractor. For instance, in Example 4.4,one can pick up an x0 in the big hole of the attractor.

6 A Hamiltonian Formalism

We briefly adapt a Hamiltonian formalism described in [1], with the ortho-normal basis of the Hilbert space of CS (3.5). (the same method can also bedirectly adapted to other sets of CS discussed in this article). For this, let

xm = ρ(m)

ρ(m − 1)∀m � 1,

where ρ(m) is given by (3.4), and assume x0 := 0. Then ρ(m) = x1x2 · · · xm =xm!. Define a Hamiltonian by

H =∞∑

m=0

xm|φm〉〈φm|. (6.1)

Then we have H|φm〉 = xm|φm〉 for m � 0. If we calculate the average energyof the CS (3.5) with the Hamiltonian H, we get

E(z) = 〈z|H|z〉 = 1

N (z)

∞∑

m=0

|Qm+1(z)|2χAm+1(z)

xm! < ∞,

i.e., the CS (3.5) are CS of the Hamiltonian H with finite energy.

Remark 6.1 Since, in general, the component Qm(z)χAm(z) appearing in CS(3.5) cannot be written as

Qm(z)χAm(z) = f (z)Qm−1(z)χAm−1(z)

for some function f (z) (independent of m), we cannot define an annihilationoperator, a such that a | z〉 = f (z) | z〉. Thereby, an oscillator algebra cannotbe associated with CS (3.5). For detailed explanation along this line of argu-ment, see [15].

Acknowledgement K. Thirulogasanthar would like to thank Wael Bahsoun for helpful discus-sions about dynamical systems.

References

1. Ali, S.T., Antoine, J-P., Gazeau, J-P.: Coherent States, Wavelets and Their Generalizations.Springer, New York (2000)

2. Ali, S.T., Englis, M., Gaseau, J-P.: Vector coherent states from Plancherel’s theorem, Cliffordalgebras and matrix domains. J. Phys. A 37, 6067–6089 (2004)

3. Barnsley, M.: Fractals Everywhere. Academic Press, London (1988)4. Beardon, A.: Iteration of Rational Functions. Springer-Verlag (1991)


5. Borzov, V.V.: Orthogonal polynomials and generalized oscillator algebras. IntegralTransform. Spec. Funct. 12, 115–138 (2001)

6. Gazeau, J-P., Klauder, J.R.: Coherent states for systems with discrete and continuous spec-trum. J. Phys. A 32, 123–132 (1999)

7. Klauder, J.R., Skagerstam, B.S.: Coherent States, Applications in Physics and MathematicalPhysics. World Scientific, Singapore (1985)

8. Klauder, J.R., Penson, K.A., Sixdeniers, J-M.: Constructing coherent states through solutionsof Steieljes and Hausdorff moment problems. Phys. Rev. A 64, 013817 (2001)

9. McMullen, C.: Complex Dynamics and Renormalization. Princeton University Press (1994)10. Milnor, J.: Dynamics in One Complex Variable: Introductory Lectures. SUNY Stony Brook

(1990)11. Novaes, M., Gazeau, J-P.: Multidimensional generalized coherent states. J. Phys. A 36,

199–212 (2003)12. Pérélomov, A.M.: Generalized Coherent States and Their Applications. Springer-Verlag,

Berlin (1986)13. Przytycki, F., Urbanski, M.: Fractals in the Plane—The Ergodic Theory Methods. Cambridge

Univestity Press (preprint)14. Thirulogasanthar, K., Twareque Ali, S.: A class of vector coherent states defined over matrix

domains. J. Math. Phys. 44, 5070–5083 (2003)15. Thirulogasanthar, K., Honnouvo, G.: Coherent states with complex functions. Internat. J.

Theoret. Phys. 43, 1053–1071 (2004)16. Rowe, D.J., Repca, J.: Vector coherent state theory as a theory of induced representations.

J. Math. Phys. 32, 2614–2634 (1991)

Math Phys Anal Geom (2007) 10:313–358DOI 10.1007/s11040-008-9035-x

Scattering Theory for Open Quantum Systemswith Finite Rank Coupling

Jussi Behrndt · Mark M. Malamud ·Hagen Neidhardt

Received: 25 June 2007 / Accepted: 23 January 2008 /Published online: 6 March 2008© Springer Science + Business Media B.V. 2008

Abstract Quantum systems which interact with their environment are oftenmodeled by maximal dissipative operators or so-called Pseudo-Hamiltonians.In this paper the scattering theory for such open systems is considered. First itis assumed that a single maximal dissipative operator AD in a Hilbert space H isused to describe an open quantum system. In this case the minimal self-adjointdilation ˜K of AD can be regarded as the Hamiltonian of a closed system whichcontains the open system {AD, H}, but since ˜K is necessarily not semiboundedfrom below, this model is difficult to interpret from a physical point of view.In the second part of the paper an open quantum system is modeled with afamily {A(μ)} of maximal dissipative operators depending on energy μ, and itis shown that the open system can be embedded into a closed system where theHamiltonian is semibounded. Surprisingly it turns out that the correspondingscattering matrix can be completely recovered from scattering matrices of

Dedicated to Valentin A. Zagrebnov on the occasion of his 60th birthday.

Jussi Behrndt gratefully acknowledges support by DFG, Grant 3765/1.

Hagen Neidhardt gratefully acknowledges support by DFG, Grant 1480/2.

J. BehrndtTechnische Universität Berlin, Institut für Mathematik,MA 6-4, Straße des 17. Juni 136, 10623 Berlin, Germanye-mail: [email protected]

M. M. MalamudDepartment of Mathematics, Donetsk National University,Universitetskaya 24, 83055 Donetsk, Ukrainee-mail: [email protected]

H. Neidhardt (B)WIAS Berlin, Mohrenstr. 39, 10117 Berlin, Germanye-mail: [email protected]

314 J. Behrndt et al.

single pseudo-Hamiltonians as in the first part of the paper. The general resultsare applied to a class of Sturm–Liouville operators arising in dissipative andquantum transmitting Schrödinger–Poisson systems.

Keywords Scattering theory · Open quantum system ·Maximal dissipative operator · Pseudo-Hamiltonian · Quasi-Hamiltonian ·Lax–Phillips scattering · Scattering matrix · Characteristic function ·Boundary triplet · Weyl function · Sturm–Liouville operator

Mathematics Subject Classifications (2000) 47A40 · 47A55 · 47B25 ·47B44 · 47E05

1 Introduction

Quantum systems which interact with their environment appear naturallyin various physical problems and have been intensively studied in the lastdecades, see e.g. the monographes [22, 25, 41]. Such an open quantum systemis often modeled with the help of a maximal dissipative operator, i.e., a closedlinear operator AD in some Hilbert space H which satisfies

Im (AD f, f ) � 0, f ∈ dom(AD),

and does not admit a proper extension in H with this property. The dynamics inthe open quantum system are described by the contraction semigroup e−itAD ,t � 0. In the physical literature the maximal dissipative operator AD is usuallycalled a pseudo-Hamiltonian. It is well known that AD admits a self-adjointdilation ˜K in a Hilbert space K which contains H as a closed subspace, that is,˜K is a self-adjoint operator in K and

PH

(

˜K − λ)−1 �H= (AD − λ)−1

holds for all λ ∈ C+ := {z ∈ C : Im (z) > 0}, cf. [42]. Since the operator ˜Kis self-adjoint it can be regarded as the Hamiltonian or so-called quasi-Hamiltonian of a closed quantum system which contains the open quantumsystem {AD, H} as a subsystem.

In this paper we first assume that an open quantum system is described bya single pseudo-Hamiltonian AD in H and that AD is an extension of a closeddensely defined symmetric operator A in H with finite equal deficiency indices.Then the self-adjoint dilation ˜K can be realized as a self-adjoint extensionof the symmetric operator A ⊕ G in K = H ⊕ L2(R,HD), where HD is finite-dimensional and G is the symmetric operator in L2(R,HD) given by

Gg := −id

dxg, dom(G) = {g ∈ W1

2(R,HD) : g(0) = 0}

,

see Section 3.1. If A0 is a self-adjoint extension of A in H and G0 denotes theusual self-adjoint momentum operator in L2(R,HD),

G0g := −id

dxg, dom(G) = W1

2(R,HD),

Scattering theory for open quantum systems with finite rank coupling 315

then the dilation ˜K can be regarded as a singular perturbation (or, moreprecisely, a finite rank perturbation in resolvent sense) of the ‘unperturbedoperator’ K0 := A0 ⊕ G0, cf. [8, 49]. From a physical point of view K0

describes a situation where both subsystems {A0, H} and {G0, L2(R,HD)} donot interact while ˜K takes into account an interaction of the subsystems. Sincethe spectrum σ(G0) of the momentum operator is the whole real axis, standardperturbation results yield σ

(

˜K) = σ(K0) = R and, in particular, K0 and ˜K are

necessarily not semibounded from below. For this reason K0 and ˜K are oftencalled quasi-Hamiltonians rather than Hamiltonians.

The pair{

˜K, K0}

is a complete scattering system in K = H ⊕ L2(R,HD),that is, the wave operators

W±(

˜K, K0) := s- lim

t→±∞ eit˜Ke−itK0 Pac(K0)

exist and are complete, cf. [9, 17, 72, 73]. Here Pac(K0) denotes the orthogonalprojection in K onto the absolutely continuous subspace Kac(K0) of K0. Thescattering operator

S(

˜K, K0) := W+

(

˜K, K0)∗

W−(

˜K, K0)

of the scattering system{

˜K, K0}

regarded as an operator in Kac(K0) is unitary,commutes with the absolutely continuous part Kac

0 of K0 and is unitarilyequivalent to a multiplication operator induced by a (matrix-valued) func-tion

{

˜S(λ)}

λ∈Rin a spectral representation L2(R, dλ,Kλ) of Kac

0 = Aac0 ⊕ G0,

cf. [17]. The family{

˜S(λ)}

is called the scattering matrix of the scatteringsystem

{

˜K, K0}

and is one of the most important quantities in the analysisof scattering processes.

In our setting the scattering matrix{

˜S(λ)}

decomposes into a 2 × 2 blockmatrix function in L2(R, dλ,Kλ) and it is one of our main goals in Section 3to show that the left upper corner in this decomposition coincides withthe scattering matrix {SD(λ)} of the dissipative scattering system {AD, A0},cf. [63, 65, 66]. The right lower corner of

{

˜S(λ)}

can be interpreted as theLax–Phillips scattering matrix {SLP(λ)} corresponding to the Lax–Phillips scat-tering system

{

˜K,D−,D+}

. Here D± := L2(R±,HD) are so-called incomingand outgoing subspaces for the dilation ˜K; we refer to [17, 57] for details onLax–Phillips scattering theory. The scattering matrices

{

˜S(λ)}

, {SD(λ)} and{SLP(λ)} are all explicitely expressed in terms of an ‘abstract’ Titchmarsh–Weyl function M(·) and a dissipative matrix D which corresponds to themaximal dissipative operator AD in H and plays the role of an ‘abstract’boundary condition. With the help of this representation of {SLP(λ)} we easilyrecover the famous relation

SLP(λ) = WAD(λ − i0)∗

found by Adamyan and Arov in [2–5] between the Lax–Phillips scatteringmatrix and the characteristic function WAD(·) of the maximal dissipative op-erator AD, cf. Corollary 3.11. We point out that M(·) and D are completelydetermined by the operators A ⊂ A0 and AD from the inner system. This is


interesting also from the viewpoint of inverse problems, namely, the scatteringmatrix

{

˜S(λ)}

of{

˜K, K0}

, in particular, the Lax–Phillips scattering matrix{SLP(λ)} can be recovered having to disposal only the dissipative scatteringsystem {AD, A0}, see Theorem 3.6 and Remark 3.7.

We emphasize that this simple and somehow straightforward embeddingmethod of an open quantum system into a closed quantum system by choosinga self-adjoint dilation ˜K of the pseudo-Hamiltonian AD is very convenientfor mathematical scattering theory, but difficult to legitimate from a physicalpoint of view, since the quasi-Hamiltonians ˜K and K0 are necessarily notsemibounded from below.

In the second part of the paper we investigate open quantum systems whichare described by an appropriate chosen family of maximal dissipative opera-tors {A(μ)}, μ ∈ C+, instead of a single pseudo-Hamiltonian AD. Similarly tothe first part of the paper we assume that the maximal dissipative operatorsA(μ) are extensions of a fixed symmetric operator A in H with equal finitedeficiency indices. Under suitable assumptions on the family {A(μ)} thereexists a symmetric operator T in a Hilbert space G and a self-adjoint extension˜L of L = A ⊕ T in L = H ⊕ G such that

PH

(

˜L − μ)−1 �H= (A(μ) − μ

)−1, μ ∈ C+, (1.1)

holds, see Section 4.2. For example, in one-dimensional models for carriertransport in semiconductors the operators A(μ) are regular Sturm–Liouvilledifferential operators in L2((a, b)) with μ-dependent dissipative boundaryconditions and the ‘linearization’ ˜L is a singular Sturm–Liouville operator inL2(R), cf. [14, 38, 43, 53] and Section 4.4. We remark that one can regard andinterpret relation (1.1) also from an opposite point of view. Namely, if a self-adjoint operator ˜L in a Hilbert space L is given, then the compression of theresolvent of ˜L onto any closed subspace H of L defines a family of maximaldissipative operators {A(μ)} via (1.1), so that each closed quantum system{

˜L, L}

naturally contains open quantum subsystems {{A(μ)}, H} of the typewe investigate here. Nevertheless, since from a purely mathematical point ofview both approaches are equivalent we will not explicitely discuss this secondinterpretation.

If A0 and T0 are self-adjoint extensions of A and T in H and G, respectively,then again ˜L can be regarded as a singular perturbation of the self-adjointoperator L0 := A0 ⊕ T0 in L. As above L0 describes a situation where thesubsystems {A0, H} and {T0, G} do not interact while ˜L takes into account acertain interaction. We note that if A and T have finite deficiency indices,then the operator ˜L is semibounded from below if and only if A and T aresemibounded from below. Well-known results imply that the pair

{

˜L, L0}

is a complete scattering system in the closed quantum system and again thescattering matrix

{

˜S(λ)}

decomposes into a 2 × 2 block matrix function whichcan be calculated in terms of abstract Titchmarsh–Weyl functions. In thisframework the problem is quite similar to the problem of zero-range potentialswith internal structure investigated by Pavlov and his group in the eighties,see for example [55, 69, 70]. However, using boundary triplets and abstract


Titchmarsh–Weyl functions we present here a general framework in whichsingular perturbation problems can be embedded and solved.

On the other hand it can be shown that the family {A(μ)}, μ ∈ C+, admitsa continuation to R, that is, the limit A(μ + i0) exists for a.e. μ ∈ R in thestrong resolvent sense and defines a maximal dissipative operator. The familyA(μ + i0), μ ∈ R, can be regarded as a family of energy dependent pseudo-Hamiltonians in H and, in particular, each pseudo-Hamiltonian A(μ + i0)

gives rise to a quasi-Hamiltonian ˜Kμ in H ⊕ L2(R,Hμ), a complete scatteringsystem

{

˜Kμ, A0 ⊕ −i(d/dx)}

and a corresponding scattering matrix{

˜Sμ(λ)}

asillustrated in the first part of the introduction.

One of our main observations in Section 4 is that the scattering matrix{

˜S(λ)}


˜L, L0}

in H⊕G is related to the scattering ma-trices

{

˜Sμ(λ)}

of the systems{

˜Kμ, A0 ⊕−i(d/dx)}

, μ∈R, in H⊕L2(R,Hμ) via

˜S(μ) = ˜Sμ(μ) for a.e. μ ∈ R. (1.2)

In other words, the scattering matrix{

˜S(λ)}


˜L, L0}

can be completely recovered from scattering matrices of scattering systems forsingle quasi-Hamiltonians. Furthermore, under certain continuity propertiesof the abstract Titchmarsh–Weyl functions this implies ˜S(λ) ≈ ˜Sμ(λ) for all λ

in a sufficiently small neighborhood of the fixed energy μ ∈ R, which justifiesthe concept of single quasi-Hamiltonians for small energy ranges.

Similarly to the case of a single pseudo-Hamiltonian the diagonal entries of{

˜S(μ)}

or{

˜Sμ(μ)}

can be interpreted as scattering matrices corresponding toenergy dependent dissipative scattering systems and energy-dependent Lax–Phillips scattering systems. Moreover, if

{

SLPμ (λ)

}

is the scattering matrix ofthe Lax–Phillips scattering system

{

˜Kμ, L2(R±,Hμ)}

and WA(μ)(·) denote thecharacteristic functions of the maximal dissipative operators A(μ), then anenergy-dependent modification

SLPμ (μ) = WA(μ)(μ − i0)∗

of the classical Adamyan-Arov result holds for a.e. μ ∈ R, cf. Section 4.3.

The paper is organized as follows. In Section 2 we give a brief introductioninto extension and spectral theory of symmetric and self-adjoint operators withthe help of boundary triplets and associated Weyl functions. These conceptswill play an important role throughout the paper. Furthermore, we recall arecent result on the representation of the scattering matrix of a scatteringsystem consisting of two self-adjoint extensions of a symmetric operator from[18], see also [6]. Section 3 is devoted to open quantum systems describedby a single pseudo-Hamiltonian AD in H. In Theorem 3.2 a minimal self-adjoint dilation ˜K in H ⊕ L2(R,HD) of the maximal dissipative operator AD

is explicitely constructed. Section 3.2 and Section 3.3 deal with the scatteringmatrix of

{

˜K, K0}

and the interpretation of the diagonal entries as scatteringmatrices of the dissipative scattering system {AD, A0} and the Lax–Phillipsscattering system

{

˜K, L2(R±,HD)}

. In Section 3.4 we give an example of apseudo-Hamiltonian which arises in the theory of dissipative Schrödinger–


Poisson systems, cf. [15, 16, 50]. In Section 4 the family {A(μ)} of maximaldissipative operators in H is introduced and, following ideas of [29], weconstruct a self-adjoint operator ˜L in a Hilbert space L, H ⊂ L, such that(1.1) holds. After some preparatory work the relation (1.2) between thescattering matrices of

{

˜L, L0}

and the scattering systems consisting of quasi-Hamiltonians is verified in Section 4.3. Finally, in Section 4.4 we consider a so-called quantum transmitting Schrödinger–Poisson system as an example for anopen quantum system which consists of a family of energy-dependent pseudo-Hamiltonians, cf. [14, 20, 23, 38, 43, 53].

We note that the scattering theory for open quantum systems developed inthis paper is applicable to various quantenmechanic problems, e.g. quantumpumps, see [10–12, 46].

Notations Throughout this paper (H, (·, ·)) and (G, (·, ·)) denote separableHilbert spaces. The linear space of bounded linear operators defined on H withvalues in G will be denoted by [H, G]. If H = G we simply write [H]. The setof closed operators in H is denoted by C(H). The resolvent set ρ(S) of a closedlinear operator S ∈ C(H) is the set of all λ ∈ C such that (S − λ)−1 ∈ [H], thespectrum σ(S) of S is the complement of ρ(S) in C. The notations σp(S), σc(S),σac(S) and σr(S) stand for the point, continuous, absolutely continuous andresidual spectrum of S, respectively. The domain, kernel and range of a linearoperator are denoted by dom(·), ker(·) and ran (·), respectively.

2 Self-adjoint Extensions and Scattering Systems

In this section we briefly review the notion of abstract boundary triplets andassociated Weyl functions in the extension theory of symmetric operators, seee.g. [31, 32, 34, 45]. For scattering systems consisting of a pair of self-adjointextensions of a symmetric operator with finite deficiency indices we recall aresult on the representation of the scattering matrix in terms of a Weyl functionproved in [18].

2.1 Boundary Triplets and Closed Extensions

Let A be a densely defined closed symmetric operator in the separable Hilbertspace H with equal deficiency indices n±(A) = dim ker(A∗ ∓ i) � ∞. We usethe concept of boundary triplets for the description of the closed extensionsA� ⊆ A∗ of A in H.

Definition 2.1 A triplet � = {H, �0, �1} is called a boundary triplet for theadjoint operator A∗ if H is a Hilbert space and �0, �1 : dom(A∗) → H arelinear mappings such that the ‘abstract Green identity’

(A∗ f, g) − ( f, A∗g) = (�1 f, �0g) − (�0 f, �1g)

holds for all f, g ∈ dom(A∗) and the map � := (�0, �1)� : dom(A∗) → H × H

is surjective.


We refer to [32] and [34] for a detailed study of boundary triplets and recallonly some important facts. First of all a boundary triplet � = {H, �0, �1} forA∗ exists since the deficiency indices n±(A) of A are assumed to be equal.Then n±(A) = dimH and A = A∗ � ker(�0) ∩ ker(�1) holds. We note that aboundary triplet for A∗ is not unique.

In order to describe the closed extensions A� ⊆ A∗ of A with the help ofa boundary triplet � = {H, �0, �1} for A∗ we have to consider the set ˜C(H)

of closed linear relations in H, that is, the set of closed linear subspaces ofH × H. We usually use a column vector notation for the elements in a linearrelation �. A closed linear operator in H is identified with its graph, so thatthe set C(H) of closed linear operators in H is viewed as a subset of ˜C(H),in particular, a linear relation � is an operator if and only if the multivaluedpart mul(�) = { f ′ : ( 0

f ′) ∈ �

}

is trivial. For the usual definitions of the linearoperations with linear relations, the inverse, the resolvent set and the spectrumwe refer to [36]. Recall that the adjoint relation �∗ ∈ ˜C(H) of a linear relation� in H is defined as

�∗ ={(

kk′

)

: (h′, k) = (h, k′) for all(

hh′

)

∈ �

}

,

and � is said to be symmetric (self-adjoint) if � ⊂ �∗ (� = �∗, respectively).Note that this definition extends the definition of the adjoint operator. For aself-adjoint relation � = �∗ in H the multivalued part mul(�) is the orthogonalcomplement of dom(�) in H. Setting Hop := dom(�) and H∞ = mul(�) oneverifies that � can be written as the direct orthogonal sum of a self-adjointoperator �op in the Hilbert space Hop and the ‘pure’ relation �∞ = {( 0

f ′) :

f ′ ∈ mul(�)}

in the Hilbert space H∞.A linear relation � in H is called dissipative if Im (h′, h) � 0 holds for all

(h, h′)� ∈ � and � is called maximal dissipative if it is dissipative and doesnot admit proper dissipative extensions in H; then � is necessarily closed,� ∈ ˜C(H). We remark that a linear relation � is maximal dissipative if andonly if � is dissipative and some λ ∈ C+ (and hence every λ ∈ C+) belongs toρ(�).

A description of all closed (symmetric, self-adjoint, (maximal) dissipative)extensions of A is given in the next proposition.

Proposition 2.2 Let A be a densely defined closed symmetric operator in H withequal deficiency indices and let � = {H, �0, �1} be a boundary triplet for A∗.Then the mapping

� �→ A� := A∗ �{

f ∈ dom(A∗) : (�0 f, �1 f )� ∈ �}

(2.1)

establishes a bijective correspondence between the set ˜C(H) and the set of closedextensions A� ⊆ A∗ of A where (·, ·)� is the transposed vector. Furthermore,

(A�)∗ = A�∗

holds for any � ∈ ˜C(H). The extension A� in (2.1) is symmetric (self-adjoint,dissipative, maximal dissipative) if and only if � is symmetric (self-adjoint,dissipative, maximal dissipative, respectively).


It follows immediately from this proposition that if � = {H, �0, �1} is aboundary triplet for A∗, then the extensions

A0 := A∗ � ker(�0) and A1 := A∗ � ker(�1)

are self-adjoint. In the sequel usually the extension A0 corresponding to theboundary mapping �0 is regarded as a ‘fixed’ self-adjoint extension. We notethat the closed extension A� in (2.1) is disjoint with A0, that is dom(A�) ∩dom(A0) = dom(A), if and only if � ∈ C(H). In this case (2.1) takes the form

A� = A∗ � ker(

�1 − ��0)

. (2.2)

Without loss of generality we will often restrict ourselves to simple symmet-ric operators. Recall that a symmetric operator is said to be simple if there isno nontrivial subspace which reduces it to a self-adjoint operator. By [54] eachsymmetric operator A in H can be written as the direct orthogonal sum A ⊕ As

of a simple symmetric operator A in the Hilbert space

H = clospan{ker(A∗ − λ) : λ ∈ C\R}and a self-adjoint operator As in H �H. Here clospan{·} denotes the closedlinear span. Obviously A is simple if and only if H coincides with H. Noticethat if � = {H, �0, �1} is a boundary triplet for the adjoint A∗ of a non-simplesymmetric operator A = A ⊕ As, then � = {H,�0,�1

}

, where

�0 := �0 � dom((

A)∗)

and �1 := �1 � dom((

A)∗)

is a boundary triplet for the simple part(

A)∗ ∈ C

(

H)

such that the extensionA� = A∗ � �(−1)�, � ∈ ˜C(H), in H is given by A� ⊕ As, A� := ( A )∗ � �(−1)� ∈C(

H)

, and the Weyl functions and γ -fields of � = {H, �0, �1} and � ={

H,�0,�1}

coincide.We say that a maximal dissipative operator is completely non-self-adjoint

if there is no nontrivial reducing subspace in which it is self-adjoint. Note thateach maximal dissipative operator can be decomposed orthogonally into a self-adjoint part and a completely non-self-adjoint part, see e.g. [42].

2.2 Weyl Functions, γ -fields and Resolvents of Extensions

Let, as in Section 2.1, A be a densely defined closed symmetric operator inH with equal deficiency indices. If λ ∈ C is a point of regular type of A, i.e.(A−λ)−1 is bounded, we denote the defect subspace of A by Nλ =ker(A∗−λ).The following definition can be found in [31, 32, 34].

Definition 2.3 Let � = {H, �0, �1} be a boundary triplet for A∗. The operatorvalued functions γ (·) : ρ(A0) → [H, H] and M(·) : ρ(A0) → [H] defined by

γ (λ) := (�0 � Nλ

)−1 and M(λ) := �1γ (λ), λ ∈ ρ(A0), (2.3)

are called the γ -field and the Weyl function, respectively, corresponding to theboundary triplet �.


It follows from the identity dom(A∗) = ker(�0)+Nλ, λ ∈ ρ(A0), where asabove A0 = A∗ � ker(�0), that the γ -field γ (·) and the Weyl function M(·) in(2.3) are well defined. Moreover, both γ (·) and M(·) are holomorphic on ρ(A0)

and the relations

γ (λ) = (I + (λ − μ)(A0 − λ)−1)

γ (μ), λ, μ ∈ ρ(A0),

and

M(λ) − M(μ)∗ = (λ − μ)γ (μ)∗γ (λ), λ, μ ∈ ρ(A0), (2.4)

are valid (see [32]). The identity (2.4) yields that M(·) is a Nevanlinna function,that is, M(·) is a ([H]-valued) holomorphic function on C\R and

M(λ) = M(λ)∗ andIm (M(λ))

Im (λ)� 0 (2.5)

hold for all λ ∈ C\R. The union of C\R and the set of all points λ ∈ R suchthat M can be analytically continued to λ and the continuations from C+ andC− coincide is denoted by h(M). Besides (2.5) it follows also from (2.4) thatthe Weyl function M(·) satisfies 0 ∈ ρ(Im (M(λ))) for all λ ∈ C\R; Nevanlinnafunctions with this additional property are sometimes called uniformly strict,cf. [30]. Conversely, each [H]-valued Nevanlinna function τ with the additionalproperty 0 ∈ ρ(Im (τ (λ))) for some (and hence for all) λ ∈ C\R can be realizedas a Weyl function corresponding to some boundary triplet, we refer to[32, 56, 58] for further details.

Let again � = {H, �0, �1} be a boundary triplet for A∗ with correspondingγ -field γ (·) and Weyl function M(·). The spectrum and the resolvent set ofthe closed (not necessarily self-adjoint) extensions of A can be described withthe help of the function M(·). More precisely, if A� ⊆ A∗ is the extensioncorresponding to � ∈ ˜C(H) via (2.1), then a point λ ∈ ρ(A0) belongs to ρ(A�)

(σi(A�), i = p, c, r) if and only if 0 ∈ ρ(� − M(λ)) (resp. 0 ∈ σi(� − M(λ)), i =p, c, r). Moreover, for λ ∈ ρ(A0) ∩ ρ(A�) the well-known resolvent formula

(A� − λ)−1 = (A0 − λ)−1 + γ (λ)(

� − M(λ))−1

γ(

λ)∗

(2.6)

holds, cf. [31, 32, 34]. Formula (2.6) is a generalization of the known Kreinformula for canonical resolvents. We emphasize that it is valid for any closedextension A� ⊆ A∗ of A with a nonempty resolvent set.

2.3 Self-adjoint Extensions and Scattering

Let A be a densely defined closed symmetric operator with equal finitedeficiency indices, i.e., n+(A) = n−(A) < ∞. Let � = {H, �0, �1}, A0 :=A∗ � ker(�0), be a boundary triplet for A∗ and let A� be a self-adjointextension of A which corresponds to a self-adjoint � ∈ ˜C(H). Since here dimHis finite by (2.6)

(A� − λ)−1 − (A0 − λ)−1, λ ∈ ρ(A�) ∩ ρ(A0)


is a finite rank operator and therefore the pair {A�, A0} performs a so-calledcomplete scattering system, that is, the wave operators

W±(A�, A0) := s- limt→±∞ eitA�e−itA0 Pac(A0)

exist and their ranges coincide with the absolutely continuous subspaceHac(A�) of A�, cf. [17, 52, 72, 73]. Pac(A0) denotes the orthogonal projectiononto the absolutely continuous subspace Hac(A0) of A0. The scattering operatorS(A�, A0) of the scattering system {A�, A0} is then defined by

S(A�, A0) := W+(A�, A0)∗W−(A�, A0).

If we regard the scattering operator as an operator in Hac(A0), then S(A�, A0)

is unitary, commutes with the absolutely continuous part

Aac0 := A0 � dom(A0) ∩ Hac(A0)

of A0 and it follows that S(A�, A0) is unitarily equivalent to a multiplica-tion operator induced by a family {S�(λ)} of unitary operators in a spectralrepresentation of Aac

0 , see e.g. [17, Proposition 9.57]. This family is called thescattering matrix of the scattering system {A�, A0}.

We note that if the symmetric operator A is not simple, then the Hilbertspace H can be decomposed as H = H ⊕ (H )⊥ (cf. the end of Section 2.1) suchthat the scattering operator is given by the orthogonal sum S

(

A�, A0)⊕ I,

where A� = A� ⊕ As and A0 = A0 ⊕ As, and hence it is sufficient to considersimple symmetric operators A in the following.

Since the deficiency indices of A are finite the Weyl function M(·) corre-sponding to the boundary triplet � = {H, �0, �1} is a matrix-valued Nevan-linna function. By Fatous theorem (see [37, 44]) then the limit

M(λ + i0) := limε→+0

M(λ + iε) (2.7)

from the upper half-plane exists for a.e. λ ∈ R. We denote the set of real pointswhere the limit in (2.7) exits by �M and we agree to use a similar notation forarbitrary scalar and matrix-valued Nevanlinna functions. Furthermore we willmake use of the notation

HM(λ) := ran(

Im (M(λ)))

, λ ∈ �M, (2.8)

and we will usually regard HM(λ) as a subspace of H. The orthogonal projectionand restriction onto HM(λ) will be denoted by PM(λ) and �HM(λ)

, respectively.Note that for λ ∈ ρ(A0) ∩ R the Hilbert space HM(λ) is trivial by (2.4). Againwe agree to use a notation analogous to (2.8) for arbitrary Nevanlinna func-tions. The family {PM(λ)}λ∈�M of orthogonal projections in H onto HM(λ),λ ∈ �M, is measurable and defines an orthogonal projection in the Hilbertspace L2(R, dλ,H); sometimes we write L2(R,H) instead of L2(R, dλ,H). Therange of this projection is denoted by L2(R, dλ,HM(λ)).

Besides the Weyl function M(·) we will also make use of the function

λ �→ N�(λ) := (� − M(λ))−1

, λ ∈ C\R, (2.9)


where � ∈ ˜C(H) is the self-adjoint relation corresponding to the extensionA� via (2.1). Since λ ∈ ρ(A0) ∩ ρ(A�) if and only if 0 ∈ ρ(� − M(λ)) thefunction N�(·) is well defined. It is not difficult to see that N�(·) is an[H]-valued Nevanlinna function and hence N�(λ + i0) = limε→0 N�(λ + iε)exists for almost every λ ∈ R, we denote this set by �N� . We claim that

N�(λ + i0) = (� − M(λ + i0))−1

, λ ∈ �M ∩ �N�, (2.10)

holds. In fact, if � is a self-adjoint matrix then (2.10) follows immediatelyfrom N�(λ)(� − M(λ)) = (� − M(λ))N�(λ) = IH, λ ∈ C+. If � ∈ ˜C(H) has anontrivial multivalued part we decompose � as � = �op ⊕ �∞, where �op isa self-adjoint matrix in Hop = dom(�op) and �∞ is a pure relation in H∞ =H � Hop, cf. Section 2.1, and denote the orthogonal projection and restrictionin H onto Hop by Pop and �Hop , respectively. Then we have

λ �→ N�(λ) = (�op − Pop M(λ)�Hop

)−1Pop, λ ∈ C\R,

(see e.g. [56, page 137]) and from

N�(λ + i0) = (�op − Pop M(λ + i0)�Hop

)−1Pop

for all λ ∈ �M ∩ �N� we conclude (2.10). Observe that R\(�M ∩ �N�) hasLebesgue measure zero.

The next representation theorem of the scattering matrix is essential in thefollowing, cf. [18, Theorem 3.8]. Since the scattering matrix is only determinedup to a set of Lebesgue measure zero we choose the representative of theequivalence class defined on �M ∩ �N� .

Theorem 2.4 Let A be a densely defined closed simple symmetric operator withfinite deficiency indices in the separable Hilbert space H, let � = {H, �0, �1} bea boundary triplet for A∗ with corresponding Weyl function M(·) and definethe spaces HM(λ), λ ∈ �M, as in (2.8). Furthermore, let A0 = A∗ � ker(�0) andlet A� = A∗ � �(−1)�, � ∈ ˜C(H), be a self-adjoint extension of A. Then thefollowing holds.

1. Aac0 is unitarily equivalent to the multiplication operator with the free

variable in L2(R, dλ,HM(λ)).2. In L2(R, dλ,HM(λ)) the scattering matrix {S�(λ)} of the complete scattering

system {A�, A0} is given by

S�(λ) = IHM(λ)+ 2iPM(λ)

√

Im (M(λ))(

� − M(λ))−1√

Im (M(λ)) �HM(λ)

for all λ ∈ �M ∩ �N� , where M(λ) := M(λ + i0).

A similar representation of the S-matrix for point interactions was obtainedin [6], see also [8]. In order to show the usefulness of Theorem 2.4 and to makethe reader more familiar with the notion of boundary triplets and associatedWeyl functions we calculate the scattering matrix of the scattering system{−d2/dx2 + δ, −d2/dx2} in the following simple example.


Example 2.5 Let us consider the densely defined closed simple symmetricoperator

(Af )(x) := − f ′′(x), dom(A) = { f ∈ W22(R) : f (0) = 0

}

,

in L2(R), see e.g. [7]. Clearly A has deficiency indices n+(A) = n−(A) = 1 andit is well-known that the adjoint operator A∗ is given by

(A∗ f )(x) = − f ′′(x),

dom(A∗) = { f ∈ W22(R\{0}) : f (0+) = f (0−), f ′′ ∈ L2(R)

}

.

It is not difficult to verify that � = {C, �0, �1}, where

�0 f := f ′(0+) − f ′(0−) and �1 f := − f (0+), f ∈ dom(A∗),

is a boundary triplet for A∗ and A0 = A∗ � ker(�0) coincides with the usualself-adjoint second order differential operator defined on W2

2(R). Moreoverthe defect space ker(A∗ − λ), λ �∈ [0, ∞), is spanned by the function

x �→ ei√

λxχR+(x) + e−i√

λxχR−(x), λ �∈ [0, ∞),

where the square root is defined on C with a cut along [0, ∞) and fixed byIm(√

λ)

> 0 for λ �∈ [0, ∞) and by√

λ � 0 for λ ∈ [0, ∞). Therefore we findthat the Weyl function M(·) corresponding to � = {C, �0, �1} is given by

M(λ) = �1 fλ�0 fλ

= i

2√

λ, fλ ∈ ker(A∗ − λ), λ �∈ [0, ∞).

Let α ∈ R\{0} and consider the self-adjoint extension A−α−1 corresponding tothe parameter −α−1, A−α−1 = A∗ � ker(�1 + α−1�0), i.e.

(A−α−1 f )(x) = − f ′′(x)

dom(A−α−1) = { f ∈ dom(A∗) : α f (0±) = f ′(0+) − f ′(0−)}

.

This self-adjoint operator is often denoted by −d2/dx2 + αδ, see [7]. It followsimmediately from Theorem 2.4 that the scattering matrix {S(λ)} of the scatter-ing system {A−α−1 , A0} is given by

S(λ) = 2√

λ − iα

2√

λ + iα, λ > 0.

We note that scattering systems of the form{−d2/dx2+αδ′, −d2/dx2

}

, α ∈ R,can be investigated in a similar way as above. Other examples can be foundin [18].

3 Dissipative and Lax–Phillips Scattering Systems

In this section we regard scattering systems {AD, A0} consisting of a maximaldissipative and a self-adjoint extension of a symmetric operator A with finitedeficiency indices. In the theory of open quantum system the maximal dissi-pative operator AD is often called a pseudo-Hamiltonian. We shall explicitely


construct a dilation (or so-called quasi-Hamiltonian) ˜K of AD and calculatethe scattering matrix of the scattering system

{

˜K, A0 ⊕ G0}

, where G0 is a self-adjoint first order differential operator. The diagonal entries of the scatteringmatrix then turn out to be the scattering matrix of the dissipative scatteringsystem {AD, A0} and of a so-called Lax–Phillips scattering system, respectively.

We emphasize that this efficient and somehow straightforward method forthe analysis of scattering processes for open quantum systems has the essentialdisadvantage that the quasi-Hamiltonians ˜K and A0 ⊕ G0 are necessarily notsemibounded from below.

3.1 Self-adjoint Dilations of Maximal Dissipative Operators

Let in the following A be a densely defined closed simple symmetric operatorin the separable Hilbert space H with equal finite deficiency indices n±(A) =n < ∞, let � = {H, �0, �1}, A0 = A∗ � ker(�0), be a boundary triplet for A∗and let D ∈ [H] be a dissipative n × n-matrix. Then the closed extension

AD = A∗ � ker(�1 − D�0)

of A corresponding to � = D via (2.1)–(2.2) is maximal dissipative and C+belongs to ρ(AD). Note that here we restrict ourselves to maximal dissipativeextensions AD corresponding to dissipative matrices D instead of maximaldissipative relations in the finite dimensional space H. This is no essentialrestriction, see Remark 3.3 at the end of this subsection. For λ ∈ ρ(AD) ∩ρ(A0) the resolvent of the extension AD is given by

(AD − λ)−1 = (A0 − λ)−1 + γ (λ)(D − M(λ))−1γ(

λ)∗

, (3.1)

cf. (2.6). Write the dissipative matrix D ∈ [H] as

D = Re (D) + iIm (D),

decompose H as the direct orthogonal sum of the finite dimensional subspacesker(Im (D)) and HD := ran (Im (D)),

H = ker(Im (D)) ⊕ HD, (3.2)

and denote by PD and �HD the orthogonal projection and restriction in H ontoHD. Since Im (D) � 0 the self-adjoint matrix −PDIm (D)�HD∈ [HD] is strictlypositive and the next lemma shows how −iPDIm (D)�HD (and iPDIm (D)�HD )can be realized as a Weyl function of a differential operator.

Lemma 3.1 Let G be the symmetric first order differential operator in theHilbert space L2(R,HD) defined by

(Gg)(x) = −ig′(x), dom(G) = {g ∈ W12(R,HD) : g(0) = 0

}

.


Then G is simple, n±(G) = dimHD and the adjoint operator G∗g = −ig′is defined on dom(G∗) = W1

2(R−,HD) ⊕ W12(R+,HD). Moreover, the triplet

�G = {HD, ϒ0, ϒ1}, where

ϒ0g := 1√2

(−PDIm (D)�HD

)− 12(

g(0+) − g(0−))

,

ϒ1g := i√2

(−PDIm (D)�HD

) 12(

g(0+) + g(0−))

,

g ∈ dom(G∗), is a boundary triplet for G∗ and G0 := G∗ � ker(ϒ0) is theusual self-adjoint first order differential operator in L2(R,HD) with domaindom(G0) = W1

2(R,HD) and σ(G0) = R. The Weyl function τ(·) correspondingto the boundary triplet �G = {HD, ϒ0, ϒ1} is given by

τ(λ) ={

−iPDIm (D)�HD, λ ∈ C+,

iPDIm (D)�HD, λ ∈ C−.(3.3)

Proof Besides the assertion that �G = {HD, ϒ0, ϒ1} is a boundary triplet forG∗ with Weyl function τ(·) given by (3.3) the statements of the lemma arewell-known. We note only that the simplicity of G follows from [1, VIII.104]and the fact that G can be written as a finite direct orthogonal sum of first orderdifferential operators on R− and R+.

A straightforward calculation shows that the identity

(G∗g, k) − (g, G∗k) = i(g(0+), k(0+)) − i(g(0−), k(0−))

= (ϒ1g, ϒ0k) − (ϒ0g, ϒ1k)

holds for all g, k ∈ dom(G∗). Moreover, the mapping (ϒ0, ϒ1)� is surjective.

Indeed, for an element (h, h′)� ∈ HD × HD we choose g ∈ dom(G∗) such that

g(0+) = 1√2

{

(−PDIm (D)�HD

) 12 h − i

(−PDIm (D)�HD

)− 12 h′}

and

g(0−) = 1√2

{

−(−PDIm (D)�HD

) 12 h − i

(−PDIm (D)�HD

)− 12 h′}

holds. Then a simple calculation shows ϒ0g = h, ϒ1g = h′ and therefore �G ={HD, ϒ0, ϒ1} is a boundary triplet for G∗. It is not difficult to check that thedefect subspace Nλ = ker(G∗ − λ) is

Nλ ={

span{

x �→ eiλxχR+(x)ξ : ξ ∈ HD}

, λ ∈ C+,

span{

x �→ eiλxχR−(x)ξ : ξ ∈ HD}

, λ ∈ C−,

and hence we conclude that the Weyl function of �G = {HD, ϒ0, ϒ1} is givenby (3.3). ��

Let AD be the maximal dissipative extension of A in H from above and let Gbe the first order differential operator from Lemma 3.1. Clearly K := A ⊕ G


is a densely defined closed simple symmetric operator in the separable Hilbertspace

K := H ⊕ L2(R,HD),

with equal finite deficiency indices n±(K) = n±(A) + n±(G) < ∞ and theadjoint is K∗ = A∗ ⊕ G∗. The elements in dom(K∗) = dom(A∗) ⊕ dom(G∗)will be written in the form f ⊕ g, f ∈ dom(A∗), g ∈ dom(G∗). In the nexttheorem we construct a self-adjoint extension ˜K of K in K which is a minimalself-adjoint dilation of the dissipative operator AD in H. The construction isbased on the idea of the coupling method from [29]. It is worth to mentionthat in the case of a (scalar) Sturm–Liouville operator with real potentialand dissipative boundary condition our construction coincides with the oneproposed by B.S. Pavlov in [68, 71], cf. Example 3.5 below.

Theorem 3.2 Let A, � = {H, �0, �1} and AD be as in the beginning of thissection, let G and �G = {HD, ϒ0, ϒ1} be as in Lemma 3.1 and K = A ⊕ G.Then

˜K = K∗ �

⎧

⎨

⎩

f ⊕ g ∈ dom(K∗) :PD�0 f − ϒ0g = 0,

(1 − PD)(�1 − Re (D)�0) f = 0,

PD(�1 − Re (D)�0) f + ϒ1g = 0

⎫

⎬

⎭

, (3.4)

is a minimal self-adjoint dilation of the maximal dissipative operator AD, that is,for all λ ∈ C+

PH

(

˜K − λ)−1 �H= (AD − λ)−1

holds and the minimality condition K = clospan{ (

˜K − λ)−1

H : λ ∈ C\R}

issatisfied. Moreover, σ

(

˜K) = R.

Proof Let γ (·), ν(·) and M(·), τ (·) be the γ -fields and Weyl functions of theboundary triplets � = {H, �0, �1} and �G = {HD, ϒ0, ϒ1}, respectively. Thenit is straightforward to check that ˜� = {˜H,˜�0,˜�1

}

, where

˜H := H ⊕ HD, ˜�0 :=(

�0

ϒ0

)

and ˜�1 :=(

�1 − Re (D)�0

ϒ1

)

, (3.5)

is a boundary triplet for K∗ = A∗ ⊕ G∗ and the corresponding Weyl function˜M(·) and γ -field γ (·) are given by

˜M(λ) =(

M(λ) − Re (D) 00 τ(λ)

)

, λ ∈ C\R, (3.6)

and

γ (λ) =(

γ (λ) 00 ν(λ)

)

, λ ∈ C\R, (3.7)

respectively. Note also that K0 := K∗ � ker(

˜�0) = A0 ⊕ G0 holds.


With respect to the decomposition ˜H = ker(Im (D)) ⊕ HD ⊕ HD of ˜H(cf. (3.2)) we define the linear relation ˜� by

˜� :={(

(u, v, v)�(0, −w, w)�

)

: u ∈ ker(Im (D), v, w ∈ HD

}

∈ ˜C (˜H) . (3.8)

We leave it to the reader to check that ˜� is self-adjoint. Hence it follows fromProposition 2.2 that the operator K

˜� = K∗ � ˜�(−1)˜� is a self-adjoint extension

of the symmetric operator K = A ⊕ G in K = H ⊕ L2(R,HD) and one verifieswithout difficulty that this extension coincides with ˜K from (3.4), ˜K = K

˜�.In order to calculate

(

˜K − λ)−1

, λ ∈ C\R, we use the block matrixdecomposition

M(λ) − Re (D) =(

MD11(λ) MD

12(λ)

MD21(λ) MD

22(λ)

)

∈ [ker(Im (D)) ⊕ HD]

(3.9)

of M(λ) − Re (D) ∈ [H]. Then the definition of ˜� in (3.8) and (3.6) imply

(

˜� − ˜M(λ))−1 =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

⎛

⎜

⎜

⎜

⎝

⎛

⎜

⎝

−MD11(λ)u − MD

12(λ)v

−w − MD21(λ)u − MD

22(λ)v

w − τ(λ)v

⎞

⎟

⎠

(u, v, v)�

⎞

⎟

⎟

⎟

⎠

: u ∈ ker(Im (D))

v, w ∈ HD

⎫

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎭

and since every λ ∈ C\R belongs to ρ(

˜K) ∩ ρ(K0), K0 = A0 ⊕ G0, it follows

that(

˜� − ˜M(λ))−1

, λ ∈ C\R, is the graph of a bounded everywhere defined

operator. In order to calculate(

˜� − ˜M(λ))−1

in a more explicit form we set

x := −MD11(λ)u − MD

12(λ)v,

y := −w − MD21(λ)u − MD

22(λ)v,

z := w − τ(λ)v.

(3.10)

This yields(

xy + z

)

= −(

MD11(λ) MD

12(λ)

MD21(λ) MD

22(λ) + τ(λ)

)

(

uv

)

and by (3.3) and (3.9) we have

−(

MD11(λ) MD

12(λ)

MD21(λ) MD

22(λ) + τ(λ)

)

={

D − M(λ), λ ∈ C+,

D∗ − M(λ), λ ∈ C−. (3.11)

Hence for λ ∈ C+ we find(

uv

)

= (D − M(λ))−1(

xy + z

)

which implies(

uv

)

= (D − M(λ))−1(

xy

)

+ (D − M(λ))−1 �HD z (3.12)


and

v = PD(

D − M(λ))−1(

xy

)

+ PD(

D − M(λ))−1 �HD z. (3.13)

Therefore, by inserting (3.10), (3.12) and (3.13) into the above expression for(

˜� − ˜M(λ))−1

we obtain

(

˜� − ˜M(λ))−1 =

(

(D − M(λ))−1 (D − M(λ))−1 �HD

PD(D − M(λ))−1 PD(D − M(λ))−1 �HD

)

(3.14)

for all λ ∈ C+ and by (2.6) the resolvent of the self-adjoint extension ˜K admitsthe representation

(

˜K − λ)−1 = (K0 − λ)−1 + γ (λ)

(

˜� − ˜M(λ))−1

γ(

λ)∗

, (3.15)

λ ∈ C\R. It follows from K0 = A0 ⊕ G0, (3.7) and (3.14) that for λ ∈ C+ thecompressed resolvent of ˜K onto H is given by

PH

(

˜K − λ)−1 � H = (A0 − λ)−1 + γ (λ)

(

D − M(λ))−1

γ(

λ)∗

,

where PH denotes the orthogonal projection in K onto H. Taking into account(3.1) we get

PH

(

˜K − λ)−1 � H = (AD − λ)−1, λ ∈ C+,

and hence ˜K is a self-adjoint dilation of AD. Since σ(G0) = R it follows fromwell-known perturbation results and (3.15) that σ

(

˜K) = R holds.

It remains to show that ˜K satisfies the minimality condition

K = H ⊕ L2(R,HD) = clospan{ (

˜K − λ)−1

H : λ ∈ C\R}

. (3.16)

First of all s-limt→+∞(−it)(

˜K − it)−1 = IK implies that H is a subset of the right

hand side of (3.16). The orthogonal projection in K onto L2(R,HD) is denotedby PL2 . Then we conclude from (3.7), (3.14) and (3.15) that for λ ∈ C+

PL2

(

˜K − λ)−1 �H= ν(λ)PD

(

D − M(λ))−1

γ(

λ)∗

(3.17)

holds and this gives

ran(

PL2

(

˜K − λ)−1 �H

)

= ker(G∗ − λ), λ ∈ C+.

From (3.11) it follows that similar to the matrix representation (3.14) theleft lower corner of

(

˜� − ˜M(λ))−1

is given by PD(D∗ − M(λ))−1 for λ ∈ C−.Hence, the analogon of (3.17) for λ ∈ C− implies that

ran(

PL2

(

˜K − λ)−1 �H

) = ker(G∗ − λ)

is true for λ ∈ C−. Since by Lemma 3.1 the symmetric operator G is simple itfollows that

L2(R,HD) = clospan{

ker(G∗ − λ) : λ ∈ C\R}

holds, cf. Section 2.1, and therefore the minimality condition (3.16) holds. ��


Remark 3.3 We note that also in the case when the parameter D is not adissipative matrix but a maximal dissipative relation in H a minimal self-adjointdilation of AD can be constructed in a similar way as in Theorem 3.2.

Indeed, let A and � = {H, �0, �1} be as in the beginning of this section andlet ˜D ∈ ˜C(H) be a maximal dissipative relation in H. Then ˜D can be written asthe direct orthogonal sum of a dissipative matrix ˜Dop in Hop := H � mul(˜D)

and an undetermined part or ‘pure relation’ ˜D∞ := {( 0y

) : y ∈ mul(˜D)}

. Itfollows that

B := A∗ � �(−1){(

0y

) : y ∈ mul(˜D)} = A∗ � �(−1)

˜D∞

is a closed symmetric extension of A and{

Hop, �0 �dom(B∗), Pop�1 �dom(B∗)}

is a boundary triplet for

B∗ = A∗ �{

f ∈ dom(A∗) : (1 − Pop)�0 f = 0}

with A∗ � ker(�0) = B∗ � ker(�0 �dom(B∗)). In terms of this boundary triplet themaximal dissipative extension A

˜D = �(−1)˜D coincides with the extension

B˜Dop

= B∗ � ker(

Pop�1 �dom(B∗) −˜Dop�0 �dom(B∗))

,

corresponding to the operator part ˜Dop ∈ [Hop] of ˜D.

Remark 3.4 In the special case ker(Im D) = {0} the relations (3.4) take theform

�0 f − ϒ0g = 0 and (�1 − Re (D)�0) f + ϒ1g = 0,

so that ˜K is a coupling of the self-adjoint operators A0 and G0 correspondingto the coupling of the boundary triplets �A = {H, �0, �1 − Re (D)�0} and�G = {H, ϒ0, ϒ1} in the sense of [29]. In the case ker(Im D) �= {0} anotherconstruction of ˜K is based on the concept of boundary relations (see [30]).

A minimal self-adjoint dilation ˜K for a scalar Sturm–Liouville operator witha complex (dissipative) boundary condition has originally been constructed byB.S. Pavlov in [68]. For the scalar case (n = 1) the operator in (3.20) in thefollowing example coincides with the one in [68].

Example 3.5 Let Q+ ∈ L1loc(R+, [Cn]) be a matrix valued function such that

Q+(·) = Q+(·)∗, and let A be the usual minimal operator in H = L2(R+, Cn)

associated with the Sturm–Liouville differential expression −d2/dx2 + Q+,

A = − d2

dx2+ Q+, dom(A) = { f ∈ Dmax,+ : f (0) = f ′(0) = 0

}

,

where Dmax,+ is the maximal domain defined by

Dmax,+ ={

f ∈ L2(R+, Cn) : f, f ′ ∈ AC(R+, C

n), − f ′′ + Q+ f ∈ L2(R+, Cn)}

.


It is well known that the adjoint operator A∗ is given by

A∗ = − d2

dx2+ Q+, dom(A∗) = Dmax,+.

In the following we assume that the limit point case prevails at +∞, so thatthe deficiency indices n±(A) of A are both equal to n. In this case a boundarytriplet � = {Cn, �0, �1} for A∗ is

�0 f := f (0), �1 f := f ′(0), f ∈ dom(A∗) = Dmax,+. (3.18)

For any dissipative matrix D ∈ [Cn] we consider the (maximal) dissipativeextension AD of A determined by

AD = A∗ � ker(�1 − D�0), Im D � 0. (3.19)

(a) First suppose 0 ∈ ρ(Im D). Then HD = Cn and by Theorem 3.2 and

Remark 3.4 the (minimal) self-adjoint dilation ˜K of the operator AD isa self-adjoint operator in K = L2(R+, C

n) ⊕ L2(R, Cn) defined by

˜K( f ⊕ g) = (− f ′′ + Q+ f)⊕ −ig′,

dom(

˜K) =

⎧

⎪

⎨

⎪

⎩

f ∈ Dmax,+, g ∈ W12(R−, C

n) ⊕ W12(R+, C

n)

f ′(0) − Df (0) = −i(−2Im D)1/2g(0−),

f ′(0) − D∗ f (0) = −i(−2Im D)1/2g(0+)

⎫

⎪

⎬

⎪

⎭

.(3.20)

(b) Let now ker(Im D) �= {0}, so that HD = ran (Im D) = Ck �= C

n. Accordingto Theorem 3.2 the (minimal) self-adjoint dilation ˜K of the operator AD

in K = L2(R+, Cn) ⊕ L2(R, C

k) is defined by

˜K( f ⊕ g) = (− f ′′ + Q+ f)⊕ −ig′,

dom(

˜K) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

f ∈ Dmax,+, g ∈ W12(R−, C

k) ⊕ W12(R+, C

k)

PD[ f ′(0) − Df (0)] = −i(−2PDIm (D)�HD)1/2g(0−),

PD[ f ′(0) − D∗ f (0)] = −i(−2PDIm (D)�HD)1/2g(0+),

f ′(0) − Re(D) f (0) ∈ HD

⎫

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎭

.

3.2 Dilations and Dissipative Scattering Systems

Let, as in the previous section, A be a densely defined closed simple symmetricoperator in H with equal finite deficiency indices and let � = {H, �0, �1} be aboundary triplet for A∗, A0 = A∗ � ker �0, with corresponding Weyl functionM(·). Let D ∈ [H] be a dissipative matrix and let AD = A∗ � ker(�1 − D�0)

be the corresponding maximal dissipative extension in H. Since the functionC+ � λ �→ M(λ) − D is a Nevanlinna function the limits

M(λ + i0) − D = limε→+0

M(λ + iε) − D

and

ND(λ + i0) = limε→+0

ND(λ + iε) = limε→+0

(

D − M(λ + iε))−1


exist for a.e. λ ∈ R. We denote these sets of real points λ by �M and �ND . Thenwe have

ND(λ + i0) = (D − M(λ + i0))−1

, λ ∈ �M ∩ �ND, (3.21)

cf. Section 2.3. Let G be the symmetric first order differential opera-tor in L2(R,HD) and let �G = {HD, ϒ0, ϒ1} be the boundary triplet fromLemma 3.1. Then G0 = G∗ � ker(ϒ0) is the usual self-adjoint differentiation op-erator in L2(R,HD) and K0 = A0 ⊕ G0 is self-adjoint in K = H ⊕ L2(R,HD).In the next theorem we consider the complete scattering system

{

˜K, K0}

,where ˜K is the minimal self-adjoint dilation of AD in K from Theorem 3.2.

Theorem 3.6 Let A, � = {H, �0, �1}, M(·) and AD be as above and defineHM(λ), λ ∈ �M, as in (2.8). Let K0 = A0 ⊕ G0 and let ˜K be the minimal self-adjoint dilation of AD from Theorem 3.2. Then the following holds.

1. Kac0 = Aac

0 ⊕ G0 is unitarily equivalent to the multiplication operator withthe free variable in L2(R, dλ,HM(λ) ⊕ HD).

2. In L2(R, dλ,HM(λ) ⊕ HD) the scattering matrix{

˜S(λ)}

of the complete

scattering system{

˜K, K0}

is given by

˜S(λ) =(

IHM(λ)0

0 IHD

)

+ 2i

(

˜T11(λ) ˜T12(λ)

˜T21(λ) ˜T22(λ)

)

∈ [HM(λ) ⊕ HD]

for all λ ∈ �M ∩ �ND , where

˜T11(λ) = PM(λ)

√

Im (M(λ))(

D − M(λ))−1√

Im (M(λ)) �HM(λ),

˜T12(λ) = PM(λ)

√

Im (M(λ))(

D − M(λ))−1√−Im (D) �HD,

˜T21(λ) = PD

√−Im (D)(

D − M(λ))−1√


˜T22(λ) = PD

√−Im (D)(

D − M(λ))−1√−Im (D) �HD

and M(λ) = M(λ + i0).

Proof Let K = A ⊕ G and let ˜� = {H ⊕ HD,˜�0,˜�1}

be the boundary tripletfor K∗ from (3.5). Note that since A and G are densely defined closed simplesymmetric operators also K is a densely defined closed simple symmetricoperator. Recall that for λ ∈ C+ the Weyl function of ˜� = {H ⊕ HD,˜�0,˜�1

}

isgiven by

˜M(λ) =(

M(λ) − Re (D) 00 −iPDIm (D) �HD

)

. (3.22)

Then Theorem 2.4 implies that

L2(

R, dλ,H˜M(λ)

)

, H˜M(λ) = HM(λ) ⊕ HD, λ ∈ �M,


performs a spectral representation of the absolutely continuous part

Kac0 = K0 � dom(K0) ∩ Kac(K0)

= A0 ⊕ G0 �(

dom(A0) ∩ Hac(A0))⊕ L2(R,HD) = Aac

0 ⊕ G0

of K0 such that the scattering matrix{

˜S(λ)}


˜K, K0}

isgiven by

˜S(λ) = IH˜M(λ)

+ 2iP˜M(λ)

√

Im ( ˜M(λ))(

˜� − ˜M(λ))−1√

Im ( ˜M(λ)) �H˜M(λ)

(3.23)

for all λ ∈ �˜M ∩ �N

˜� , where P˜M(λ) and �H

˜M(λ)are the projection and restriction

in ˜H = H ⊕ HD onto H˜M(λ). Here ˜� is the self-adjoint relation from (3.8), the

function N˜� is defined analogously to (2.9) and

N˜�(λ + i0) = (˜� − ˜M(λ + i0)

)−1

holds for all λ ∈ �˜M ∩ �N

˜� , cf. (2.10).By (3.22) we have

√

Im(

˜M(λ + i0)) =

(√Im (M(λ + i0)) 0

0 PD√−Im (D) �HD

)

for all λ ∈ �˜M = �M and (3.14) yields

(

˜� − ˜M(λ + i0))−1 =

(

(D − M(λ + i0))−1 (D − M(λ + i0))−1 �HD

PD(D − M(λ + i0))−1 PD(D − M(λ + i0))−1 �HD

)

for λ ∈ �M ∩ �N˜� . It follows that the sets �M ∩ �N

˜� and �M ∩ �ND , see(3.21), coincide and by inserting the above expressions into (3.23) we concludethat for each λ ∈ �M ∩ �ND the scattering matrix

{

˜S(λ)}

is a two-by-two blockoperator matrix with respect to the decomposition

H˜M(λ) = HM(λ) ⊕ HD, λ ∈ �M ∩ �ND,

with the entries from assertion (2). ��

Remark 3.7 It is worth to note that the scattering matrix{

˜S(λ)}

of the scatter-ing system

{

˜K, K0}

in Theorem 3.6 depends only on the dissipative matrix Dand the Weyl function M(·) of the boundary triplet � = {H, �0, �1} for A∗. Inother words, the scattering matrix

{

˜S(λ)}

is completely determined by objectscorresponding to the operators A, A0 and AD in H.

Let AD and A0 be as in the beginning of this section. In the following wewill focus on the so-called dissipative scattering system {AD, A0} and we referthe reader to [26, 27, 59–65] for a detailed investigation of such scatteringsystems. We recall only that the wave operators W±(AD, A0) of the dissipativescattering system {AD, A0} are defined by

W+(AD, A0) = s- limt→+∞ eitA∗

D e−itA0 Pac(A0)


and

W−(AD, A0) = s- limt→+∞ e−itAD eitA0 Pac(A0),

where e−itAD := s-limn→∞(1 + (it/n)AD)−n, see e.g. [52, Section IX]. The scat-tering operator

SD := W+(AD, A0)∗W−(AD, A0)

of the dissipative scattering system {AD, A0} will be regarded as an operatorin Hac(A0). Then SD is a contraction which in general is not unitary. Since SD

and Aac0 commute it follows that SD is unitarily equivalent to a multiplication

operator induced by a family {SD(λ)} of contractive operators in a spectralrepresentation of Aac

0 .With the help of Theorem 3.6 we obtain a representation of the scattering

matrix of the dissipative scattering system {AD, A0} in terms of the Weylfunction M(·) of � = {H, �0, �1} in the following corollary, cf. Theorem 2.4.

Corollary 3.8 Let A, � = {H, �0, �1}, A0 = A∗ � ker(�0), M(·) and AD be asabove and define HM(λ), λ ∈ �M, as in (2.8). Then the following holds.

1. Aac0 is unitarily equivalent to the multiplication operator with the free

variable in L2(R, dλ,HM(λ)).2. In L2(R, dλ,HM(λ)) the scattering matrix {SD(λ)} of the dissipative scatter-

ing system {AD, A0} is given by

SD(λ) = IHM(λ)+ 2iPM(λ)

√

Im (M(λ))(

D − M(λ))−1√Im (M(λ)) �HM(λ)

for all λ ∈ �M ∩ �ND , where M(λ) = M(λ + i0).

Proof Let ˜K be the minimal self-adjoint dilation of AD from Theorem 3.2.Since for t � 0 we have

PHe−it˜K � H = s- limn→∞ PH

(

1 + itn˜K)−n

�H= s- limn→∞

(

1 + itn

AD

)−n

= e−itAD

it follows that the wave operators W+(AD, A0) and W−(AD, A0) coincide with

PHW+(

˜K, K0)

�H = s- limt→+∞ PHeit˜Ke−itK0 Pac(K0) �H

= s- limt→+∞ PHeit˜K �H e−itA0 Pac(A0)

and

PHW−(

˜K, K0)

�H = s- limt→−∞ PHeit˜Ke−itK0 Pac(K0) �H

= s- limt→+∞ PHe−it˜K �H eitA0 Pac(A0),

respectively. This implies that the scattering operator SD coincides with thecompression PHac(A0)S

(

˜K, K0)

�Hac(A0) of the scattering operator S(

˜K, K0)


onto Hac(A0). Therefore the scattering matrix SD(λ) of the dissipative scat-tering system is given by the upper left corner

{

IHM(λ)+ 2i˜T11(λ)

}

, λ ∈ �M ∩ �ND,

of the scattering matrix{

˜S(λ)}


˜K, K0}

, seeTheorem 3.6. ��

3.3 Lax–Phillips Scattering Systems

Let again A, � = {H, �0, �1}, {AD, A0} and G, G0, �G = {HD, ϒ0, ϒ1} be asin the previous subsections. In Corollary 3.8 we have shown that the scatteringmatrix of the dissipative scattering system {AD, A0} is the left upper corner inthe block operator matrix representation of the scattering matrix

{

˜S(λ)}

of thescattering system

{

˜K, K0}

, where ˜K is a minimal self-adjoint dilation of AD inK = H ⊕ L2(R,HD) and K0 = A0 ⊕ G0, cf. Theorem 3.6.

In the following we are going to interpret the right lower corner of{

˜S(λ)}

as the scattering matrix corresponding to a Lax–Phillips scattering system, seee.g. [17, 57] for further details. To this end we decompose the space L2(R,HD)

into the orthogonal sum of the subspaces

D− := L2(R−,HD) and D+ := L2(R+,HD). (3.24)

Then clearly K = H ⊕ D− ⊕ D+ and we agree to denote the elements in K inthe form f ⊕ g− ⊕ g+, f ∈ H, g± ∈ D± and g = g− ⊕ g+ ∈ L2(R,HD). By J+and J− we denote the operators

J+ : L2(R,HD) → K, g �→ 0 ⊕ 0 ⊕ g+,

and

J− : L2(R,HD) → K, g �→ 0 ⊕ g− ⊕ 0,

respectively. Note that J+ + J− is the embedding of L2(R,HD) into K. In thenext lemma we show that D+ and D− are so-called outgoing and incomingsubspaces for the self-adjoint dilation ˜K in K.

Lemma 3.9 Let ˜K be the self-adjoint operator from Theorem 3.2, let D± be asin (3.24) and let A0 = A∗ � ker(�0) be as above. Then

e−it˜KD± ⊆ D±, t ∈ R±, and⋂

t∈R

e−it˜KD± = {0}

and, if in addition σ(A0) is singular, then⋃

t∈R

e−it˜KD+ =⋃

t∈R

e−it˜KD− = Kac (˜K)

. (3.25)

Proof Let us first show that

e−it˜K � D± = J±e−itG0 � D±, t ∈ R±, (3.26)


holds. In fact, since e−itG0 is the right shift group we have

e−itG0(dom(G) ∩ D±) ⊆ dom(G) ∩ D±, t ∈ R±,

where dom(G) ∩ D± = {W1,2(R,HD) : f (x) = 0, x ∈ R±}. Let us fix somet ∈ R± and denote the symmetric operator A ⊕ G by K. Since

J±(

dom(G) ∩ D±) ⊂ dom(K) ⊂ dom

(

˜K)

the function

ft,±(s) := ei(s−t)˜K J±e−isG0 �D± f±, s ∈ R±, f± ∈ dom(G) ∩ D±,

is differentiable and

dds

ft,±(s) = iei(s−t)˜K(˜K − 0H ⊕ G0

)

J±e−isG0 �D± f± = 0, t ∈ R±,

holds. Hence we have ft,±(0) = ft,±(t) and together with the observation thatthe set dom(G) ∩ D± is dense in D± this immediately implies (3.26). Then weobtain e−it˜KD± ⊆ D±, t ∈ R±, and

⋂

t∈R

e−it˜KD± ⊆⋂

t∈R±

e−it˜KD± =⋂

t∈R±

J±e−itG0D± = {0}.

Let us show (3.25). Since A has finite deficiency indices the wave operatorsW±

(

˜K, A0 ⊕ G0)

exist and are complete, i.e.,

ran(

W±(

˜K, A0 ⊕ G0)) = Kac (

˜K)

holds. Since A0 is singular we have

W±(

˜K, A0 ⊕ G0) = s- lim

t→±∞ eit˜K(J+ + J−)e−itG0 �L2

and it follows from (3.26) that W±(

˜K, A0 ⊕ G0)

f± = f± for f± ∈ D±, sothat in particular D± and e−itG0D± ∈ Kac

(

˜K)

for t ∈ R±. Assume now thatg ∈ L2(R,HD) vanishes identically on some open interval (−∞, α). Then forr > 0 sufficiently large e−irG0 g ∈ D+ and by (3.26) for t > r

eit˜K(J+ + J−)e−i(t−r)G0 e−irG0 g = eir˜K J+e−irG0 g.

Since the elements g ∈ L2(R,HD) which vanish on intervals (−∞, α) form adense set in L2(R,HD) and the wave operator W+

(

˜K, A0 ⊕ G0)

is completewe conclude that

⋃

r∈R+

eir˜KD+ (3.27)

is a dense set in Kac(

˜K)

. A similar argument shows that the set (3.27) withR+ and D+ replaced by R− and D−, respectively, is also dense in Kac

(

˜K)

. Thisimplies (3.25). ��


According to Lemma 3.9 the system{

˜K,D−,D+}

is a Lax–Phillips scatteringsystem and in particular the Lax–Phillips wave operators

�± := s- limt→±∞ eit˜K J±e−itG0 : L2(R,HD) → K

exist, cf. [17]. We note that s-limt→±∞ J∓e−itG0 = 0 and therefore the restric-tions of the wave operators W±

(

˜K, K0)


˜K, K0}

,K0 = A0 ⊕ G0, onto L2(R,HD),

W±(

˜K, K0)

�L2= s- limt→±∞ eit˜K(J+ + J−)e−itG0 ,

coincide with the Lax–Phillips wave operators �±. Hence the Lax–Phillipsscattering operator SLP := �∗+�− admits the representation

SLP = PL2 S(

˜K, K0)

�L2 ,

where S(

˜K, K0) = W+

(

˜K, K0)∗

W−(

˜K, K0)

is the scattering operator of thescattering system

{

˜K, K0}

. The Lax–Phillips scattering operator SLP is a con-traction in L2(R,HD) and commutes with the self-adjoint differential operatorG0. Hence SLP is unitarily equivalent to a multiplication operator induced bya family {SLP(λ)} of contractive operators in L2(R,HD), this family is calledthe Lax–Phillips scattering matrix.

The above considerations together with Theorem 3.6 immediately imply thefollowing corollary on the representation of the Lax–Phillips scattering matrix.

Corollary 3.10 Let{

˜K,D−,D+}

be the Lax–Phillips scattering system consid-ered in Lemma 3.9 and let A, � = {H, �0, �1}, AD, M(·) and G0 be as in theprevious subsections. Then G0 = Gac

0 is unitarily equivalent to the multiplicationoperator with the free variable in L2(R,HD) = L2(R, dλ,HD) and the Lax–Phillips scattering matrix {SLP(λ)} admits the representation

SLP(λ) = IHD + 2iPD

√

Im (−D)(

D − M(λ))−1√

Im (−D) �HD (3.28)

for λ ∈ �M ∩ �ND , where M(λ) = M(λ + i0).

Let again AD be the maximal dissipative extension of A corresponding tothe maximal dissipative matrix D ∈ [H] and let HD = ran (Im (D)). By [33] thecharacteristic function WAD of the completely non-self-adjoint part of AD isgiven by

WAD : C− → [HD]μ �→ IHD − 2iPD

√−Im (D)(

D∗ − M(μ))−1√−Im (D) �HD .

(3.29)

Comparing (3.28) and (3.29) we obtain the famous relation betweenthe Lax–Phillips scattering matrix and the characteristic function found byAdamyan and Arov in [2–5].


Corollary 3.11 Let the assumption be as in Corollary 3.10. Then the Lax–Phillips scattering matrix {SLP(λ)} and the characteristic function WAD of themaximal dissipative operator AD are related by

SLP(λ) = WAD(λ − i0)∗, λ ∈ �M ∩ �ND .

Next we consider the special case that the spectrum σ(A0) of the self-adjoint extension A0 = A∗ � ker(�0) is purely singular, Hac(A0) = {0}. As usuallet M(·) be the Weyl function corresponding to � = {H, �0, �1}. Then wehave HM(λ) = ran (Im (M(λ + i0))) = {0} for a.e. λ ∈ �M, cf. [21], and if evenσ(A0) = σp(A0) then HM(λ) = {0} for all λ ∈ �M. Therefore Theorem 3.6 andCorollaries 3.10 and 3.11 imply the following statement.

Corollary 3.12 Let the assumption be as in Corollary 3.10, let K0 = A0 ⊕ G0

and assume in addition that σ(A0) is purely singular. Then the scatteringmatrix

{

˜S(λ)}

of the complete scattering system{

˜K, K0}

coincides with theLax–Phillips scattering matrix {SLP(λ)} of the Lax–Phillips scattering system{

˜K,D−,D+}

, that is,

˜S(λ) = SLP(λ) = WAD(λ − i0)∗ (3.30)

for a.e. λ ∈ R. If even σ(A0) = σp(A0), then (3.30) holds for all λ ∈ �M ∩ �ND .

3.4 1D-Schrödinger Operators with Dissipative Boundary Conditions

In this subsection we consider an open quantum system consisting of aself-adjoint and a maximal dissipative extension of a symmetric regularSturm–Liouville differential operator. Such maximal dissipative operators orpseudo-Hamiltonians are used in the description of carrier transport in semi-conductors, see e.g. [13, 15, 38, 43, 50, 51, 53].

Assume that −∞ < xl < xr < ∞ and let V ∈ L∞((xl, xr)) be a real valuedfunction. Moreover, let m ∈ L∞((xl, xr)) be a real function such that m > 0 andm−1 ∈ L∞((xl, xr)). It is well-known that

(Af )(x) := −1

2

ddx

1

m(x)

ddx

f (x) + V(x) f (x),

dom(A) :=⎧

⎨

⎩

f ∈ L2((xl, xr)) :f, 1

m f ′ ∈ W12((xl, xr))

f (xl) = f (xr) = 0(

1m f ′) (xl) = ( 1

m f ′) (xr) = 0

⎫

⎬

⎭

,

is a densely defined closed simple symmetric operator in the Hilbert spaceH := L2((xl, xr)). The deficiency indices of A are n+(A) = n−(A) = 2 and theadjoint operator A∗ is given by

(A∗ f )(x) = −1

2

ddx

1

m(x)

ddx

f (x) + V(x) f (x),

dom(A∗) ={

f ∈ H : f,1

mf ′ ∈ W1

2((xl, xr))

}

.


It is straightforward to verify that � = {C2, �0, �1}, where

�0 f :=(

f (xl)

f (xr)

)

and �1 f :=((

12m f ′) (xl)

− ( 12m f ′) (xr)

)

, (3.31)

f ∈ dom(A∗), is a boundary triplet for A∗. Note that the self-adjoint extensionA0 = A∗ � ker(�0) corresponds to Dirichlet boundary conditions, that is,

dom(A0) ={

f ∈ H : f,1

mf ′ ∈ W1

2((xl, xr)), f (xl) = f (xr) = 0

}

.

It is well known that A0 is semibounded from below and that σ(A0) consistsof eigenvalues accumulating to +∞. As usual we denote the Weyl functioncorresponding to � = {C2, �0, �1} by M(·). Here M(·) is a two-by-two matrix-valued function which has poles at the eigenvalues of A0 and in particularwe have

HM(λ) = ran(

Im (M(λ))) = {0} for all λ ∈ �M. (3.32)

If ϕλ, ψλ ∈ L2((xl, xr)) are fundamental solutions of

−1

2

(

1

mf ′)′

+ V f = λ f

satisfying the boundary conditions

ϕλ(xl) = 1,

(

1

mϕ′

λ

)

(xl) = 0, ψλ(xl) = 0,

(

1

mψ ′

λ

)

(xl) = 1, (3.33)

then M can be written as

M(λ) = 1

2ψλ(xr)

(−ϕλ(xr) 11 −( 1

mψ ′λ

)

(xr)

)

, λ ∈ ρ(A0). (3.34)

We are interested in maximal dissipative extensions

AD = A∗ � ker(�1 − D�0)

of A where D ∈ [C2] has the special form

D =(−κl 0

0 −κr

)

, Im (κl) � 0, Im (κr) � 0. (3.35)

Of course, if both κl and κr are real constants then HD = ran (Im (D)) = {0}and AD is self-adjoint. In this case AD can be identified with the self-adjointdilation ˜K acting in H ⊕ L2(R, {0})=H, cf. Theorem 3.2.

Let us first consider the situation where both κl and κr have positive imagi-nary parts. Then HD = C

2 and the self-adjoint dilation ˜K from Theorem 3.2 isgiven by

˜K( f ⊕ g− ⊕ g+) =(

− 1

2

(

1

mf ′)′

+ V f)

⊕ −ig′− ⊕ −ig′

+,

dom˜K ={

f, 1m f ′ ∈ W1

2((xl, xr)),

g± ∈ W12(R±, C

2): �0 f − ϒ0g = 0,

(�1 − Re (D)�0) f + ϒ1g = 0

}

.


Here �G = {C2, ϒ0, ϒ1} is the boundary triplet for first order differentialoperator G ⊂ G∗ in L2(R, C

2) from Lemma 3.1 and we have decomposed theelements f ⊕ g in H ⊕ L2(R, C

2) as agreed in the beginning of Section 3.3. Letus set

g−(0−) =(

gl(0−)

gr(0−)

)

and g+(0+) =(

gl(0+)

gr(0+)

)

.

Then a straightforward calculation using the definitions of � = {C2, �0, �1}and �G = {C2, ϒ0, ϒ1} in (3.31) and Lemma 3.1, respectively, shows that anelement f ⊕ g− ⊕ g+ belongs to dom

(

˜K)

if and only if

(

1

2mf ′)

(xl) + κl f (xl) = −i√

2Im (κl)gl(0−),

(

1

2mf ′)

(xl) + κ l f (xl) = −i√

2Im (κl)gl(0+),

(

1

2mf ′)

(xr) − κr f (xr) = i√

2Im (κr)gr(0−),

(

1

2mf ′)

(xr) − κr f (xr) = i√

2Im (κr)gr(0+)

hold. We note that this dilation ˜K is isomorphic in the sense of [42, Section I.4]to those used in [15, 16, 50, 51].

Theorem 3.6 and the fact that σ(A0) is singular (cf. (3.32)) imply thatthe scattering matrix

{

˜S(λ)}


˜K, K0}

, K0 = A0 ⊕ G0,coincides with

SLP(λ) = IC2 + 2i√−Im (D)

(

D − M(λ))−1√−Im (D) ∈ [C2]

for all λ �∈ σp(A0) ∩ R, where M(λ) = M(λ + i0) (cf. Corollary 3.12). By (3.35)here

√−Im (D) is a diagonal matrix with entries√

Im (κl) and√

Im (κr). Weleave it to the reader to compute SLP(λ) explicitely in terms of the fundamentalsolutions ϕλ and ψλ in (3.33). According to Corollary 3.11 the continuationof the characteristic function WAD of the completely non-self-adjoint pseudo-Hamiltonian AD from C− to R\{σp(A0)} coincides with SLP(λ)∗,

WAD(λ − i0) = IC2 − 2i√−Im (D)

(

D∗ − M(

λ))−1√−Im (D)

= SLP(λ)∗.

Next we consider briefly the case where one of the entries of D in (3.35) isreal. Assume e.g. κl ∈ R. In this case HD = C={0} ⊕ C, PD is the orthogonal


projection onto the second component in C2 and G is a first order differential

operator in L2(R, C). The self-adjoint dilation ˜K is

˜K( f ⊕ g− ⊕ g+) =(

− 1

2

(

1

mf ′)′

+ V f)

⊕ −ig′− ⊕ −ig′

+,

dom(˜K) =⎧

⎨

⎩

f, 1m f ′ ∈ W1

2((xl, xr)),

g± ∈ W12(R±, C

2):

PD�0 f − ϒ0g = 0,

(1 − PD)(�1 − Re (D)�0) f = 0,

PD(�1 − Re (D)�0) f + ϒ1g = 0

⎫

⎬

⎭

,

and explicitely this means that an element f ⊕ g− ⊕ g+ belongs to dom(

˜K)

ifand only if

(

1

2mf)′

(xr) − κr f (xr) = i√

2Im (κr)g+(0+),

(

1

2mf)′

(xr) − κr f (xr) = i√

2Im (κr)g−(0−),

(

1

2mf)′

(xl) + κl f (xl) = 0

hold. The scattering matrix of{

˜K, K0}

is given by

SLP(λ) = IHD + 2iIm (κr)PD(

D − M(λ))−1 �HD, λ ∈ �M,

which is now a scalar function, and is related to the characteristic function ofthe maximal dissipative operator AD by SLP(λ) = WAD(λ − i0)∗.

4 Energy Dependent Scattering Systems

In this section we consider families {A−τ(λ), A0} of scattering systems, whereτ(·) is a matrix Nevanlinna function and {A−τ(λ)} is a family of maximaldissipative extensions of a symmetric operator A with finite deficiency indices.Such scattering systems arise naturally in the description of open quantumsystems, see e.g. Section 4.4 where a problem arising in model the theory ofa so-called quantum transmitting Schrödinger–Poisson system is described.Following ideas in [29] (see also [19, 24, 35, 47, 48]) the family {A−τ(λ)} is‘linearized’ in an abstract way, that is, we construct a self-adjoint extension˜L of A which acts in a larger Hilbert space H ⊕ G and satisfies

PH

(

˜L − λ)−1 �H= (A−τ(λ) − λ

)−1,

so that, roughly speaking, the open quantum system is embedded into a closedsystem. The corresponding Hamiltonian ˜L is semibounded if and only if A0 issemibounded and τ(·) is holomorphic on some interval (−∞, η). The essentialobservation here is that the scattering matrix of

{

˜L, L0}

, where L0 is thedirect orthogonal sum of A0 and a self-adjoint operator connected with τ(·),pointwise coincides with the scattering matrix of a scattering system

{

˜K, K0}

as


investigated in the previous section. From a physical point of view this undersuitable justifies continuity assumptions the use of quasi-Hamiltonians ˜K forthe analysis of scattering processes in suitable small energy ranges.

4.1 The Štraus Family and its Characteristic Functions

Let A be a densely defined closed simple symmetric operator in the separableHilbert space H with equal finite deficiency indices n±(A) = n < ∞ and let� = {H, �0, �1} be a boundary triplet for A∗. Assume that τ(·) is an [H]-valuedNevanlinna function and consider the family {A−τ(λ)},

A−τ(λ) := A∗ � ker(

�1 + τ(λ)�0)

, λ ∈ C+,

of closed extension of A. Sometimes it is convenient to consider A−τ(λ) for allλ ∈ h(τ ), that is, for all λ ∈ C\R and all real points λ where τ is holomorphic,cf. Section 2.2. Since Im τ(λ) � 0 for λ ∈ C+ it follows that each A−τ(λ), λ ∈ C+,is a maximal dissipative extension of A in H. The family {A−τ(λ)}λ∈C+ is calledthe Štraus family of A associated with τ (cf. [67] and e.g. [28, Section 3.3]) andfor brevity we shall often call {A−τ(λ)} simply Štraus family.

Since H is finite dimensional Fatous theorem (see [37, 44]) implies thatthe limit τ(λ + i0) = limε→+0 τ(λ + iε) from the upper half-plane exists fora.e. λ ∈ R. As in Section 2.3 we denote set of real points λ where this limitexists by �τ . If there is no danger of confusion we will usually write τ(λ)

instead of τ(λ + i0) for λ ∈ �τ . Obviously, the Lebesgue measure of R \ �τ

is zero. Hence the Štraus family {A−τ(λ)}λ∈C+ admits a continuation to C+ ∪ �τ

which is also denoted by {A−τ(λ)}, λ ∈ C+ ∪ �τ . We remark that in the caseIm (τ (λ)) = 0 for some λ ∈ C+ ∪ �τ the maximal dissipative operator A−τ(λ) isself-adjoint.

Let M(·) be the Weyl function corresponding to the boundary triplet � ={H, �0, �1}. Then M(·) is an [H]-valued Nevanlinna function and Im (M(λ)) isstrictly positive for λ ∈ C+. Therefore

N−τ(λ)(λ) := −(τ(λ) + M(λ))−1

, λ ∈ C+,

is a well-defined Nevanlinna function, see also (2.9). The set of all real λ wherethe limit

N−τ(λ+i0)(λ + i0) = limε→+0

−(τ(λ + iε) + M(λ + iε))−1

exists will for brevity be denoted by �N .Furthermore, for fixed λ ∈ �τ we define an [H]-valued Nevanlinna function

Q−τ(λ)(·) by

Q−τ(λ)(μ) := −(τ(λ) + M(μ))−1

, μ ∈ C+, (4.1)

and denote by �Qλ the set of all real points μ where the limit

Q−τ(λ)(μ + i0) = limε→+0

Q−τ(λ)(μ + iε) (4.2)


exists. Note that the complements R \ �N and R \ �Qλ are of Lebesguemeasure zero. The next lemma will be used in Section 4.3.

Lemma 4.1 Let A, � = {H, �0, �1}, M(·) and τ(·) be as above. Then thefollowing assertions (1)–(3) are true.

1. If λ ∈ �τ and μ ∈ �M ∩ �Qλ , then the operator τ(λ) + M(μ) is invertibleand

(

τ(λ) + M(μ))−1 = lim

ε→+0

(

τ(λ) + M(μ + iε))−1

. (4.3)

2. If λ ∈ �τ ∩ �M ∩ �N, then the operator τ(λ) + M(λ) is invertible and

(

τ(λ) + M(λ))−1 = lim

ε→+0

(

τ(λ + iε) + M(λ + iε))−1

. (4.4)

3. If λ ∈ �τ ∩ �M ∩ �N, then λ ∈ �Qλ and

(

τ(λ) + M(λ))−1 = lim

ε→+0

(

τ(λ) + M(λ + iε))−1

. (4.5)

Proof

1. If λ ∈ �τ , μ ∈ �M, then

limε→+0

(

τ(λ) + M(μ + iε)) = τ(λ) + M(μ).

Since(

τ(λ) + M(μ + iε))

Q−τ(λ)(μ + iε) =Q−τ(λ)(μ + iε)

(

τ(λ) + M(μ + iε)) = −IH

for all ε > 0, we get

−IH = (τ(λ) + M(μ))

Q−τ(λ)(μ) = Q−τ(λ)(μ)(

τ(λ) + M(μ))

for λ ∈ �τ and μ ∈ �M ∩ �Qλ which proves (4.3).2. For λ ∈ �τ ∩ �M clearly

limε→+0

(

τ(λ + iε) + M(λ + iε)) = τ(λ) + M(λ)

exists. Since (τ (λ) + M(λ))N−τ(λ)(λ) = N−τ(λ)(λ)(τ (λ) + M(λ)) = −IH forall λ ∈ C+ we have

−IH = (τ(λ) + M(λ))

N−τ(λ)(λ) = N−τ(λ)(λ)(

τ(λ) + M(λ))

for λ ∈ �τ ∩ �M ∩ �N which verifies (4.4).3. Let λ ∈ �τ ∩ �M ∩ �N . Let us show that λ ∈ �Qλ , i.e., we have to show

that limε→+0(τ (λ) + M(λ + iε))−1 exists. Since τ(λ) + M(λ) is boundedlyinvertible and τ(λ) + M(λ + iε), ε > 0, converges in the operator norm to


τ(λ) + M(λ) the family {(τ (λ) + M(λ + iε))−1}ε>0 is uniformly bounded.Using

(

τ(λ) + M(λ + iε))−1 − (τ(λ) + M(λ)

)−1

= −(τ(λ) + M(λ + iε))−1(

M(λ + iε) − M(λ))(

τ(λ) + M(λ))−1

,

ε > 0, one obtains the existence of limε→+0(τ (λ) + M(λ + iε))−1 and (4.5).��

Let A, � = {H, �0, �1} and M(·) be as in the beginning of this section andlet as above τ(·) be a matrix Nevanlinna function with values in [H]. Foreach maximal dissipative operator from the Štraus family {A−τ(λ)}λ∈C+ thecharacteristic function WA−τ (λ)

is given by

WA−τ (λ): C− → [Hτ(λ)] (4.6)

μ �→ IHτ (λ)+ 2iPτ(λ)

√

Im (τ (λ))(

τ(λ)∗ + M(μ))−1√

Im (τ (λ)) �Hτ (λ),

(see [33] and (3.29)), where we have used Hτ(λ) = ran (Im (τ (λ))), λ ∈ �τ ,and denoted the projection and restriction onto Hτ(λ) by Pτ(λ) and �Hτ (λ)

,respectively.

If we regard the Štraus family {A−τ(λ)} on the larger set C+ ∪ �τ , then forλ ∈ �τ the characteristic function WA−τ (λ)

(·) is defined as in (4.6). Note that inthe case Im (τ (λ)) = 0 for λ ∈ �τ the characteristic function of the self-adjointextension A−τ(λ) of A is the identity operator on the trivial space Hτ(λ) ={0}. Since the characteristic functions WA−τ (λ)

(·), λ ∈ C+ ∪ �τ , are contractive[Hτ(λ)]-valued functions in the lower half-plane, the limits

WA−τ (λ)(μ − i0) = lim

ε→+0WA−τ (λ)

(μ − iε)

exist for a.e. μ ∈ R, cf. [42]. The next proposition is a simple consequence ofLemma 4.1.

Proposition 4.2 Let A, � = {H, �0, �1} and M(·) be as above and let τ(·) bean [H]-valued Nevanlinna function. Let {A−τ(λ)}λ∈C+∪�τ be the Štraus familyof maximal dissipative extensions of A and let WA−τ (λ)

(·) be the correspondingcharacteristic functions. Then the following holds.

1. If λ ∈ �τ and μ ∈ �M ∩ �Qλ , then the limit WA−τ (λ)(μ − i0) exists and

WA−τ (λ)(μ − i0)

= IHτ (λ)+ 2iPτ(λ)

√

Im (τ (λ))(τ (λ)∗ + M(μ)∗)−1√

Im (τ (λ)) �Hτ (λ).

2. If λ ∈ �τ ∩ �M ∩ �N, then the limit WA−τ (λ)(λ − i0) exists and

WA−τ (λ)(λ − i0)

= IHτ (λ)+ 2iPτ(λ)

√

Im (τ (λ))(τ (λ)∗ + M(λ)∗)−1√

Im (τ (λ)) �Hτ (λ).


4.2 Coupling of Symmetric Operators and Coupled Scattering Systems

Let, as in the previous subsection, A be a densely defined closed simplesymmetric operator in H with equal finite deficiency indices n ± (A) = n andlet � = {H, �0, �1} be a boundary triplet for A∗ with corresponding Weylfunction M(·). Let τ(·) be an [H]-valued Nevanlinna function and assume inaddition that τ can be realized as the Weyl function corresponding to a denselydefined closed simple symmetric operator T in some separable Hilbert spaceG and a suitable boundary triplet �T = {H, ϒ0, ϒ1} for T∗. It is worth to notethat the Nevanlinna function τ(·) has this property if and only if Im (τ (λ)) isinvertible for some (and hence for all) λ ∈ C+ and

limy→∞

1

y

(

τ(iy)h, h) = 0 and lim

y→∞ y Im(

τ(iy)h, h) = ∞ (4.7)

hold for all h ∈ H, h �= 0, (see e.g. [56, Corollary 2.5 and Corollary 2.6] and[32, 58]).

In the following the function −τ(·) and the Štraus family

A−τ(λ) = A∗ � ker(

�1 + τ(λ)�0)

(4.8)

are in a certain sense the counterparts of the dissipative matrix D ∈ [H] and thecorresponding maximal dissipative extension AD from Section 3.1. Similarlyto Theorem 3.2 we construct an ‘energy dependent dilation’ in Theorem 4.3below, that is, we find a self-adjoint operator ˜L such that

PH

(

˜L − λ)−1 �H= (A−τ(λ) − λ

)−1

holds.First we fix a separable Hilbert space G, a densely defined closed simple

symmetric operator T ∈ C(G) and a boundary triplet �T = {H, ϒ0, ϒ1} forT∗ such that τ(·) is the corresponding Weyl function. We note that T andG are unique up to unitary equivalence and the resolvent set ρ(T0) of theself-adjoint operator T0 := T∗ � ker(ϒ0) coincides with the set h(τ ) of pointsof holomorphy of τ , cf. Section 2.2. Since the deficiency indices of T aren+(T) = n−(T) = n it follows that

L := A ⊕ T, dom(L) = dom(A) ⊕ dom(T),

is a densely defined closed simple symmetric operator in the separable Hilbertspace L := H ⊕ G with deficiency indices n±(L) = n±(A) + n±(T) = 2n.

The following theorem has originally been proved in [29, Section 5]. For thesake of completeness we present a direct proof here, cf. [19].

Theorem 4.3 Let A, � = {H, �0, �1}, M(·) and τ(·) be as above, let T be adensely defined closed simple symmetric operator in G and �T = {H, ϒ0, ϒ1}be a boundary triplet for T∗ with Weyl function τ(·). Then

˜L = L∗ �{

f ⊕ g ∈ dom(L∗) : �0 f − ϒ0g = 0�1 f + ϒ1g = 0

}

(4.9)


is a self-adjoint operator in L such that

PH

(

˜L − λ)−1 �H = (A−τ(λ) − λ

)−1

holds for all λ ∈ ρ(A0) ∩ h(τ ) ∩ h(−(M + τ)−1) and the minimality condition

L = clospan{(

˜L − λ)−1

H : λ ∈ C\R}

is satisfied. Moreover, ˜L is semibounded from below if and only if A0 issemibounded from below and (−∞, η) ⊂ h(τ ) for some η ∈ R.

Proof It is easy to see that ˜� = {H ⊕ H,˜�0,˜�1}

, where ˜�0 := (�0, ϒ0)� and

˜�1 := (�1, ϒ1)�, is a boundary triplet for L∗ = A∗ ⊕ T∗. If γ (·) and ν(·) denote

the γ -fields of � = {H, �0, �1} and �T = {H, ϒ0, ϒ1}, respectively, then the γ -field γ and Weyl function ˜M of ˜� = {H ⊕ H,˜�0,˜�1

}

are given by

λ �→ γ (λ) =(

γ (λ) 00 ν(λ)

)

and λ �→ ˜M(λ) =(

M(λ) 00 τ(λ)

)

,

λ ∈ ρ(A0) ∩ ρ(T0), A0 = A∗ � ker(�0), T0 = T∗ � ker(ϒ0). A simple calculationshows that the relation

� :={(

(v, v)�(w, −w)�

)

: v, w ∈ H}

∈ ˜C(H ⊕ H) (4.10)

is self-adjoint in H ⊕ H, hence the operator L� = L∗ � ˜�(−1)� is a self-adjointextension of L in L = H ⊕ G and L� coincides with ˜L in (4.9). Hence, withL0 = L∗ � ker

(

˜�0) = A0 ⊕ T0 we have

(

˜L − λ)−1 = (L0 − λ)−1 + γ (λ)

(

� − ˜M(λ))−1

γ(

λ)∗

(4.11)

for all λ ∈ ρ(

˜L) ∩ ρ(L0) by (2.6). Note that the difference of the resolvents of

˜L and L0 is a finite rank operator and therefore by well-known perturbationresults ˜L is semibounded if and only if L0 is semibounded, that is, A0 and T0

are both semibounded. From ρ(T0) = h(τ ) we conclude that the last assertionof the theorem holds.

Similar considerations as in the proof of Theorem 3.2 show that

(

� − ˜M(λ))−1 = −

(

(M(λ) + τ(λ))−1 (M(λ) + τ(λ))−1

(M(λ) + τ(λ))−1 (M(λ) + τ(λ))−1

)

(4.12)

holds for all λ ∈ ρ(

˜L) ∩ ρ(L0). Therefore the compressed resolvent of ˜L has

the form

PH

(

L − λ)−1 � H = (A0 − λ)−1 − γ (λ)

(

M(λ) + τ(λ))−1

γ(

λ)∗

and coincides with (A−τ(λ) − λ)−1 for all λ belonging to

ρ(L0) ∩ ρ(

˜L) = ρ(A0) ∩ h(τ ) ∩ h

(−(M + τ)−1)

,

see Section 2.2. The minimality condition follows from the fact that T is simple,clospan{ker(T∗ − λ) : λ ∈ C\R} and (4.11) in a similar way as in the proof ofTheorem 3.2 ��


Example 4.4 Let A be the symmetric Sturm–Liouville differential operatorfrom Example 3.5 and let � = {Cn, �0, �1} be the boundary triplet for A∗defined by (3.18). Besides the operator A we consider the minimal operator Tin G = L2(R−, C

n) associated with the Sturm–Liouville differential expression−d2/dx2 + Q−,

T = − d2

dx2+ Q−, dom(T) = {g ∈ Dmax,− : g(0) = g′(0) = 0

}

.

Analogously to Example 3.5 it is assumed that Q− ∈ L1loc(R−, [Cn]) satisfies

Q−(·) = Q−(·)∗, that the limit point case prevails at −∞ and the maximaldomain Dmax,− is defined in the same way as Dmax,+ in Example 3.5 with R+and Q+ replaced by R− and Q−, respectively.

It is easy to see that �T = {Cn, ϒ0, ϒ1}, where

ϒ0g := g(0), ϒ1g := −g′(0), g ∈ dom(T∗) = Dmax,−, (4.13)

is a boundary triplet for T∗. For f ∈ dom(A∗) and g ∈ dom(T∗) the conditions�0 f − ϒ0g = 0 and �1 f + ϒ1g = 0 in (4.9) stand for

f (0+) = g(0−) and f ′(0+) = g′(0−),

so that the operator ˜L in Theorem 4.3 is the self-adjoint Sturm–Liouvilleoperator

˜L = − d2

dx2+ Q, Q(x) =

{

Q+(x), x ∈ R+,

Q−(x), x ∈ R−,

in L2(R, Cn).

Let A, � = {H, �0, �1}, M(·) and T, �T = {H, ϒ0, ϒ1}, τ(·) be as in the be-ginning of this subsection. We define the families {HM(λ)}λ∈�M and {Hτ(λ)}λ∈�τ

of Hilbert spaces HM(λ) and Hτ(λ) by

HM(λ) = ran(

Im (M(λ + i0)))

and Hτ(λ) = ran(

Im (τ (λ + i0)))

(4.14)

for all real points λ belonging to �M and �τ , respectively, cf. Section 2.3. Asusual the projections and restrictions in H onto HM(λ) and Hτ(λ) are denotedby PM(λ), �HM(λ)

and Pτ(λ), �Hτ (λ), respectively.

The next theorem is the counterpart of Theorem 3.6 in the present frame-work. We consider the complete scattering system

{

˜L, L0}

consisting of theself-adjoint operators ˜L from Theorem 4.3 and

L0 := A0 ⊕ T0, A0 = A∗ � ker(�0), T0 = T∗ � ker(ϒ0),

and express the scattering matrix{

˜S(λ)}

in terms of the function M(·) and τ(·).

Theorem 4.5 Let A, � = {H, �0, �1}, M(·) and T, �T = {H, ϒ0, ϒ1}, τ(·) be asabove. Define HM(λ), Hτ(λ) as in (4.14) and let L0 = A0 ⊕ T0 and ˜L be as inTheorem 4.3. Then the following holds.

1. Lac0 = Aac

0 ⊕ Tac0 is unitarily equivalent to the multiplication operator with

the free variable in L2(R, dλ,HM(λ) ⊕ Hτ(λ)).


2. In L2(R, dλ,HM(λ) ⊕ Hτ(λ)) the scattering matrix{

˜S(λ)}

of the completescattering system

{

˜L, L0}

is given by

˜S(λ) = IHM(λ)⊕Hτ (λ)− 2i

(

˜T11(λ) ˜T12(λ)˜T21(λ) ˜T22(λ)

)

∈ [HM(λ) ⊕ Hτ(λ)] (4.15)

for all λ ∈ �M ∩ �τ ∩ �N, where

˜T11(λ) = PM(λ)

√

Im (M(λ))(

M(λ) + τ(λ))−1√


˜T12(λ) = PM(λ)

√

Im (M(λ))(

M(λ) + τ(λ))−1√

Im (τ (λ)) �Hτ (λ),

˜T21(λ) = Pτ(λ)

√

Im (τ (λ))(

M(λ) + τ(λ))−1√


˜T22(λ) = Pτ(λ)

√

Im (τ (λ))(

M(λ) + τ(λ))−1√

Im (τ (λ)) �Hτ (λ)

and M(λ) = M(λ + i0), τ(λ) = τ(λ + i0).

Proof Let L = A ⊕ T and let ˜� = {H ⊕ H,˜�0,˜�1}

be the boundary triplet for

L∗ from the proof of Theorem 4.3. The corresponding Weyl function ˜M is

λ �→ ˜M(λ) =(

M(λ) 00 τ(λ)

)

, λ ∈ ρ(A0) ∩ ρ(T0), (4.16)

and since L is a densely defined closed simple symmetric operator in theseparable Hilbert space L = H ⊕ G we can apply Theorem 2.4. First of all weimmediately conclude from

H˜M(λ) = HM(λ) ⊕ Hτ(λ), λ ∈ �

˜M = �M ∩ �τ ,

that the absolutely continuous part Lac0 = Aac

0 ⊕ Tac0 of L0 is unitarily equiva-

lent to the multiplication operator with the free variable in the direct integralL2(R, dλ,HM(λ) ⊕ Hτ(λ)). Moreover

˜S(λ) = I˜Hλ

+ 2iP˜M(λ)

√

Im ( ˜M(λ))(

� − ˜M(λ))−1√

Im ( ˜M(λ)) �H˜M(λ)

(4.17)

holds for λ ∈ �˜M ∩ �N� , where � is the self-adjoint relation from (4.10), the

set �N� is defined as in Section 2.3 and P˜M(λ) and �H

˜M(λ)denote the projection

and restriction in H ⊕ H onto H˜M(λ), respectively.

For λ ∈ �˜M ∩ �N� we have

limε→+0

(

� − ˜M(λ + iε))−1 = (� − ˜M(λ + i0)

)−1,

and(

� − ˜M(λ))−1 = −

(

(M(λ) + τ(λ))−1 (M(λ) + τ(λ))−1

(M(λ) + τ(λ))−1 (M(λ) + τ(λ))−1

)

cf. (4.12). This implies that the sets �˜M ∩ �N� and �M ∩ �τ ∩ �N coincide.

Moreover, by inserting the above expression for(

� − ˜M(λ))−1

, λ ∈ �M ∩ �τ ∩�N , into (4.17) and taking into account (4.16) we find that the scattering matrix{

˜S(λ)}


˜L, L0}

has the form asserted in (2). ��


The following corollary, which is of similar type as Corollary 3.12, is a simpleconsequence of Theorem 4.5 and Proposition 4.2.

Corollary 4.6 Let the assumptions be as in Theorem 4.5, let WA−τ (λ)(·) be the

characteristic function of the extension A−τ(λ) in (4.8) and assume in additionthat σ(A0) is purely singular. Then Lac

0 is unitarily equivalent to the multiplica-tion operator with the free variable in L2(R, dλ,Hτ(λ)) and the scattering matrix{

˜S(λ)}

of the complete scattering system{

˜L, L0}

is given by

˜S(λ) = WA−τ (λ)(λ − i0)∗

= IHτ (λ)− 2iPτ(λ)

√Im (τ (λ))

(

M(λ) + τ(λ))−1√

Im (τ (λ)) �Hτ (λ)

for a.e. λ ∈ R. In the special case σ(A0) = σp(A0) this relation holds for allpoints λ ∈ �M ∩ �τ ∩ �N.

Corollary 4.7 Let the assumptions be as in Corollary 4.6 and suppose that thedefect of A is one, n±(A) = 1. Then

˜S(λ) = WA−τ (λ)(λ − i0)∗ = M(λ) + τ(λ)

M(λ) + τ(λ)

holds for a.e. λ ∈ R with Im τ(λ + i0) �= 0.

4.3 Scattering Matrices of Energy Dependent and Fixed DissipativeScattering Systems

Let A, � = {H, �0, �1}, A0 = A∗ � ker(�0) and τ(·) be as in the previous sub-sections and let {A−τ(λ)} be the Štraus family associated with τ from (4.8). Inthe following we first fix some μ ∈ C+ ∪ �τ and consider the fixed dissipativescattering system {A−τ(μ), A0}. Notice that if μ ∈ �τ it may happen that A−τ(μ)

is self-adjoint. Let us denote by ˜Kμ the minimal self-adjoint dilation of themaximal dissipative extension A−τ(μ) in H ⊕ L2(R, dλ,Hτ(μ)) constructed inTheorem 3.2. Here the fixed Hilbert space Hτ(μ) = ran (Im (τ (μ))) coincideswith H if μ ∈ C+ or Hτ(μ) is a (possibly trivial) subspace of H if μ ∈ �τ .Furthermore, if K0 = A0 ⊕ G0, where G0 is the first order differential op-erator in L2(R, dλ,Hτ(μ)) from Lemma 3.1, then according to Theorem 3.6the absolutely continuous part Kac

0 = Aac0 ⊕ G0 of K0 is unitarily equivalent

to the multiplication operator with the free variable in the direct integralL2(R, dλ,HM(λ) ⊕ Hτ(μ)) and the scattering matrix

{

˜Sμ(λ)}

of the scatteringsystem

{

˜Kμ, K0}

is given by

˜Sμ(λ) = IHM(λ)⊕Hτ (μ)− 2i

(

˜T11,μ(λ) ˜T12,μ(λ)˜T21,μ(λ) ˜T22,μ(λ)

)

∈ [HM(λ) ⊕ Hτ(μ)

]

(4.18)


for all λ ∈ �M ∩ �Qμ , where

˜T11,μ(λ) = PM(λ)

√

Im (M(λ))(

τ(μ) + M(λ))−1√


˜T12,μ(λ) = PM(λ)

√

Im (M(λ))(

τ(μ) + M(λ))−1√

Im (τ (μ)) �Hτ (μ),

˜T21,μ(λ) = Pτ(μ)

√

Im (τ (μ))(

τ(μ) + M(λ))−1√


˜T22,μ(λ) = Pτ(λ)

√

Im (τ (μ))(

τ(μ) + M(λ))−1√

Im (τ (μ)) �Hτ (μ)

and M(λ) = M(λ + i0). Here the set �Qμ and the corresponding functionλ �→ Q−τ(μ)(λ) defined in (4.1)–(4.2) replace �ND and λ �→ (D − M(λ))−1 inTheorem 3.6, respectively.

The following theorem is one of the main results of this paper. Roughlyspeaking it says that the scattering matrix of the scattering system

{

˜L, L0}

fromTheorem 4.5 pointwise coincides with scattering matrices of scattering systems{

˜Kμ, K0}

of the above form.

Theorem 4.8 Let A, � = {H, �0, �1}, M(·) and T, �T = {H, ϒ0, ϒ1}, τ(·) beas in the beginning of Section 4.2 and let L0 = A0 ⊕ T0 and ˜L be as inTheorem 4.3. For μ ∈ �τ denote the minimal self-adjoint dilation of A−τ(μ) inH ⊕ L2(R,Hτ(μ)) by ˜Kμ and let K0 = A0 ⊕ G0, where G0 is the self-adjoint firstorder differential operator in L2(R,Hτ(μ)).

Then for each μ ∈ �M ∩ �τ ∩ �N the value of the scattering matrix{

˜Sμ(λ)}


˜Kμ, K0}

at energy λ=μ coincides with the value of thescattering matrix

{

˜S(λ)}


˜L, L0}

at energy λ=μ, that is,

˜S(μ) = ˜Sμ(μ) for all μ ∈ �M ∩ �τ ∩ �N. (4.19)

Proof According to Lemma 4.1 (3) each real μ ∈ �M ∩ �τ ∩ �N belongs alsoto the set �Qμ . Therefore, by comparing Theorem 4.5 with the scatteringmatrix

{

˜Sμ(λ)}

of{

˜Kμ, K0}

at energy λ = μ in (4.18) we conclude (4.19). ��

Remark 4.9 The statements of Theorem 4.5 and Theorem 4.8 are also inter-esting from the viewpoint of inverse problems. Namely, if τ(·) is a matrixNevanlinna function, satisfying ker(Im (τ (λ))) = 0, λ ∈ C+, and the conditions(4.7), and if {A−τ(λ), A0} is a family of energy dependent dissipative scatteringsystems as considered above, then in general the Hilbert space G and theoperators T ⊂ T0 are not explicitely known, and hence also the scatteringsystem

{

˜L, L0}

is not explicitely known. However, according to Theorem 4.5the scattering matrix

{

˜S(λ)}

can be expressed in terms of τ(·) and the Weylfunction M(·), and by Theorem 4.8

{

˜S(λ)}

can be obtained with the help of thescattering matrices

{

˜Sμ(λ)}

of the scattering systems{

˜Kμ, K0}

.

In the following corollary the scattering matrices {S−τ(μ)(λ)} of the energydependent dissipative scattering systems {A−τ(μ), A0}, μ ∈ �τ , are evaluatedat energy λ = μ.


Corollary 4.10 Let the assumptions be as in Theorem 4.8 and let μ ∈ �M ∩�τ ∩ �N. Then the scattering matrix {S−τ(μ)(λ)} of the dissipative scatteringsystem {A−τ(μ), A0} at energy λ = μ coincides with the upper left corner of thescattering matrix

{

˜S(λ)}


˜L, L0}

at energy λ = μ.

Let again ˜Kμ be the minimal self-adjoint dilation of the maximal dissipativeoperator A−τ(μ) in H ⊕ L2(R, dλ,Hτ(μ)). In the next corollary we focus onthe Lax–Phillips scattering matrices

{

SLPμ (λ)

}

of the Lax–Phillips scatteringsystems

{

˜Kμ,D−,μ,D+,μ

}

, where

D−,μ := L2(

R−,Hτ(μ)

)

and D+,μ := L2(

R+,Hτ(μ)

)

are incoming and outgoing subspaces for ˜Kμ, cf. Lemma 3.9. If WA−τ (μ)(·) is the

characteristic function of A−τ(μ), cf. (4.6), then according to Corollaries 3.10and 3.11 we have

SLPμ (λ) = WA−τ (μ)

(λ − i0)∗

= IHτ (λ)− 2iPτ(λ)

√Im (τ (λ))

(

τ(μ) + M(λ))−1√

Im (τ (λ)) �Hτ (λ)

for all λ ∈ �M ∩ �Qμ , cf. Proposition 4.2 and Corollary 4.6. Statements (2) and(3) of the following corollary can be regarded as generalizations of the classicalAdamyan-Arov result, cf. [2–5], and Corollary 3.11.

Corollary 4.11 Let the assumptions be as in Theorem 4.8 and let μ ∈ �M ∩�τ ∩ �N.

1. The scattering matrix{

SLPμ (λ)

}

of the Lax–Phillips scattering system{

˜Kμ,D−,μ,D+,μ

}

at energy λ = μ coincides with the lower right corner ofthe scattering matrix

{

˜S(λ)}


˜L, L0}

at λ = μ.2. The characteristic function WA−τ (μ)

(·) of A−τ(μ) satisfies

SLPμ (μ) = WA−τ (μ)

(μ − i0)∗

= IHτ (μ)− 2iPτ(μ)

√

Im (τ (μ))(

τ(μ) + M(μ))−1√

Im (τ (μ)) �Hτ (μ).

3. If σ(A0) is purely singular, then

˜S(μ) = SLPμ (μ) = WA−τ (μ)

(μ − i0)∗

holds for a.e. μ ∈ R. In the special case σ(A0) = σp(A0) this is true for allμ ∈ �M ∩ �τ ∩ �N.

4.4 1D-Schrödinger Operators with Transparent Boundary Conditions

As an example we consider an open quantum system of similar type as inSection 3.4. Instead of a single pseudo-Hamiltonian AD here the open quantumsystem is described by a family of energy dependent pseudo-Hamiltonians{A−τ(λ)} which is sometimes called a quantum transmitting family.


Let, as in Section 3.4, (xl, xr) ⊂ R be a bounded interval and let A be thesymmetric Sturm–Liouville operator in H = L2((xl, xr)) given by

(Af )(x) = −1

2

ddx

1

m(x)

ddx

f (x) + V(x) f (x),

dom(A) =

⎧

⎪

⎨

⎪

⎩

f ∈ H :f, 1

m f ′ ∈ W12((xl, xr))

f (xl) = f (xr) = 0(

1m f ′) (xl) = ( 1

m f ′) (xr) = 0

⎫

⎪

⎬

⎪

⎭

,

where V, m, m−1 ∈ L∞((xl, xr)) are real functions and m > 0. Let vl, vr be realconstants, let ml, mr > 0 and define ˜V, m ∈ L∞(R) by

˜V(x) :=

⎧

⎪

⎨

⎪

⎩

vl x ∈ (−∞, xl],V(x) x ∈ (xl, xr),

vr x ∈ [xr, ∞),

(4.20)

and

m(x) :=

⎧

⎪

⎨

⎪

⎩

ml x ∈ (−∞, xl],m(x) x ∈ (xl, xr),

mr x ∈ [xr, ∞),

(4.21)

respectively. We choose the boundary triplet � = {C2, �0, �1},

�0 f =(

f (xl)

f (xr)

)

, �1 f =((

12m f ′) (xl)

− ( 12m f ′) (xr)

)

, f ∈ dom(A∗),

from (3.31) for A∗.In the following we consider the Štraus family

A−τ(λ) = A∗ � ker(

�1 + τ(λ)�0)

, λ ∈ C+ ∪ �τ ,

associated with the 2 × 2-matrix Nevanlinna function

λ �→ τ(λ) =⎛

⎝

i√

λ−vl2ml

0

0 i√

λ−vr2mr

⎞

⎠ ; (4.22)

here the square root is defined on C with a cut along [0, ∞) and fixed byIm(√

λ)

> 0 for λ �∈ [0, ∞) and by√

λ � 0 for λ ∈ [0, ∞), cf. Example 2.5, sothat indeed Im (τ (λ)) > 0 for λ ∈ C+ and τ(λ) = τ(λ), λ ∈ C\R. Moreover it isnot difficult to see that τ(·) is holomorphic on C\[min{vl, vr}, ∞) and �τ = R.The Štraus family {A−τ(λ)}, λ ∈ C+ ∪ �τ , has the explicit form

(

A−τ(λ) f)

(x) := −1

2

ddx

1

md

dxf (x) + V(x) f (x),

dom(

A−τ(λ)

) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

f ∈ H :f, 1

m f ′ ∈ W12((xl, xr)),

(

12m f ′) (xl) = −i

√

λ−vl2ml

f (xl),

(

12m f ′) (xr) = i

√

λ−vr2mr

f (xr)

⎫

⎪

⎪

⎪

⎬

⎪

⎪

⎪

⎭

.

(4.23)


The operator A−τ(λ) is self-adjoint if λ ∈ (−∞, min{vl, vr}] and maximal dis-sipative if λ ∈ (min{vl, vr}, ∞). We note that the Štraus family in (4.23) playsan important role for the quantum transmitting Schrödinger–Poisson systemin [14] where it was called the quantum transmitting family. For this openquantum system the boundary conditions in (4.23) are often called transparentor absorbing boundary conditions, see e.g. [39, 40].

We leave it to the reader to verify that the Nevanlinna function τ(·) in (4.22)satisfies the conditions (4.7). Hence by [32, 56, 58] there exists a separableHilbert space G, a densely defined closed simple symmetric operator T in G

and a boundary triplet �T = {C2, ϒ0, ϒ1} for T∗ such that τ(·) is the corre-sponding Weyl function. Here G, T and �T = {C2, ϒ0, ϒ1} can be explicitlydescribed. Indeed, as Hilbert space G we choose L2((−∞, xl) ∪ (xr, ∞)) andfrequently we identify this space with L2((−∞, xl)) ⊕ L2((xr, ∞)). An elementg ∈ G will be written in the form g = gl ⊕ gr, where gl ∈ L2((−∞, xl)) andgr ∈ L2((xr, ∞)). The operator T in G is defined by

(Tg)(x) :=(− 1

2d

dx1

ml

ddx gl(x) + vlgl(x) 0

0 − 12

ddx

1mr

ddx gr(x) + vrgr(x)

)

,

dom(T) :={

g = gl ⊕ gr ∈ G : g ∈ W22((−∞, xl)) ⊕ W2

2((xr, ∞))

gl(xl) = gr(xr) = g′l(xl) = g′

r(xl) = 0

}

,

and it is well-known that T is a densely defined closed simple symmetricoperator in G with deficiency indices n+(T) = n−(T) = 2. The adjoint operatorT∗ is given by

(T∗g)(x) =(− 1

2d

dx1

ml

ddx gl(x) + vlgl(x) 0

0 − 12

ddx

1mr

ddx gr(x) + vrgr(x)

)

,

dom(T∗) = {g = gl ⊕ gr ∈ G : W22((−∞, xl)) ⊕ W2

2((xr, ∞))}

.

We leave it to the reader to check that �T = {C2, ϒ0, ϒ1}, where

ϒ0g :=(

gl(xl)

gr(xr)

)

and ϒ1g :=(− 1

2mlg′

l(xl)

12mr

g′r(xr)

)

,

g = gl ⊕ gr ∈ dom(T∗), is a boundary triplet for T∗. Note that T0 = T∗ �ker(ϒ0) is the restriction of T∗ to the domain

dom(T0) = {g ∈ dom(T∗) : gl(xl) = gr(xr) = 0}

,

that is, T0 corresponds to Dirichlet boundary conditions. It is not difficultto see that σ(T0) = [min{vl, vr}, ∞) and hence the Weyl function of �T ={C2, ϒ0, ϒ1} is holomorphic on C\[min{vl, vr}, ∞).

Lemma 4.12 Let T ⊂ T∗ and �T = {C2, ϒ0, ϒ1} be as above. Then the corre-sponding Weyl function coincides with τ(·) in (4.22).


Proof A straightforward calculation shows that

hl,λ(x) := i√2ml(λ − vl)

exp{

−i√

2ml(λ − vl)(x − xl)}

belongs to L2((−∞, xl)) for λ ∈ C\[vl, ∞) and satisfies

−1

2

ddx

1

ml

ddx

hl,λ(x) + vlhl,λ(x) = λhl,λ(x).

Analogously the function

kr,λ(x) := i√2ml(λ − vr)

exp{

i√

2mr(λ − vr)(x − xr)}

belongs to L2((xr, ∞)) for λ ∈ C\[vr, ∞) and satisfies

−1

2

ddx

1

mr

ddx

kr,λ(x) + vrkr,λ(x) = λkr,λ(x).

Therefore the functions

hλ := hl,λ ⊕ 0 and kλ := 0 ⊕ kr,λ

belong to G and we have ker(T∗ − λ) = sp{hλ, kλ}.As the Weyl function τ (·) associated with T and �T = {C2, ϒ0, ϒ1} is

defined by

ϒ1gλ = τ (λ)ϒ0gλ for all gλ ∈ ker(T∗ − λ),

λ ∈ C\[min{vl, vr}, ∞), we conclude from

ϒ1hλ = 1

2

(− 1ml

0

)

and ϒ0hλ =( i√

2ml(λ−vl)

0

)

and

ϒ1kλ = 1

2

(

0− 1

mr

)

and ϒ0kλ =(

0i√

2mr(λ−vr)

)

that τ has the form (4.22), τ (·) = τ(·). ��

Let A, � = {C2, �0, �1} and T, �T = {C2, ϒ0, ϒ1} be as above. Then accord-ing to Theorem 4.3 the operator

˜L := A∗ ⊕ T∗ �{

f ⊕ g ∈ dom(A∗ ⊕ T∗) : �0 f − ϒ0g = 0�1 f + ϒ1g = 0

}

(4.24)

is a self-adjoint extension of A ⊕ T in H ⊕ G. We can identify H ⊕ G withL2((−∞, xl)) ⊕ L2((xl, xr)) ⊕ L2((xr, ∞)) and L2(R). The elements f ⊕ g inH ⊕ G, f ∈ H, g = gl ⊕ gr ∈ G will be written in the form gl ⊕ f ⊕ gr. Theconditions �0 f = ϒ0g and �1 f = −ϒ1g, f ∈ dom(A∗), g ∈ dom(T∗), have theform

(

f (xl)

f (xr)

)

=(

gl(xl)

gr(xr)

)

and

( (

12m f ′

l

)

(xl)

− ( 12m f ′

r

)

(xr)

)

=( 1

2mlg′(xl)

− 12mr

g′(xr)

)

.


Therefore an element gl ⊕ f ⊕ gr in the domain of (4.24) has the properties

gl(xl) = f (xl) and f (xr) = gr(xr)

as well as

1

mlg′

l(xl) =(

1

mf ′)

(xl) and(

1

mf ′)

(xr) = 1

mrg′

r(xr)

and the self-adjoint operator ˜L in (4.24) becomes

˜L(gl ⊕ f ⊕ gr) =⎛

⎜

⎜

⎝

− 12

ddx

1ml

ddx gl + vlgl 0 0

0 − 12

ddx

1m

ddx f + V f 0

0 0 − 12

ddx

1mr

ddx gr + vrgr

⎞

⎟

⎟

⎠

.

With the help of (4.20) and (4.21) we see that (4.24) can be regarded as theusual self-adjoint second order differential operator

˜L = −1

2

ddx

1

md

dx+ ˜V

on the maximal domain in L2(R), that is, (4.24) coincides with the so-calledBuslaev–Fomin operator from [14].

Denote by M(·) the Weyl function corresponding to A and the boundarytriplet � = {C2, �0, �1}, cf. (3.33)–(3.34). Since σ(A0) consists of eigenvaluesCorollary 4.6 implies that the scattering matrix

{

˜S(λ)}


˜L, L0}

, L0 = A0 ⊕ T0, is given by

˜S(λ) = IHτ (λ)− 2iPτ(λ)

√

Im (τ (λ))(

M(λ) + τ(λ))−1√

Im (τ (λ)) �Hτ (λ)

for all λ ∈ ρ(A0) ∩ �N , where

Hτ(λ) = ran (Im (τ (λ))) =

⎧

⎪

⎨

⎪

⎩

{0} λ ∈ (−∞, min{vl, vr}],C λ ∈ (min{vl, vr}, max{vl, vr}],C

2 λ ∈ (max{vl, vr}, ∞).

The scattering system{

˜L, L0}

was already investigated in [13, 14]. There itwas in particular shown that the scattering matrix

{

˜S(λ)}

and the characteristicfunction WA−τ (λ)

(·) of the maximal dissipative extension A−τ(λ) from (4.23) areconnected via

˜S(λ) = WA−τ (λ)(λ − i0)∗,

which we here immediately obtain from Corollary 4.6.

Acknowledgements The authors are grateful to Professor Peter Lax for helpful comments andfruitful discussions. Moreover, we would like to thank one of the referees for drawing our attentionto further physical applications.


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The Absolutely Continuous Spectrumof One-dimensional Schrödinger Operators

Christian Remling

Received: 22 October 2007 / Accepted: 5 February 2008 /Published online: 12 March 2008© Springer Science + Business Media B.V. 2008

Abstract This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum.The basic result says that the ω limit points of the potential under the shiftmap are reflectionless on the support of the absolutely continuous part ofthe spectral measure. This implies an Oracle Theorem for such potentials andDenisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators,these issues were discussed in my recent paper (Remling, The absolutelycontinuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007).The treatment of the continuous case in the present paper depends on the samebasic ideas.

Keywords Absolutely continuous spectrum · Schrödinger operator ·Reflectionless potential

Mathematics Subject Classifications (2000) Primary 34L40 · 81Q10

1 Introduction

This note discusses basic properties of one-dimensional Schrödinger operators

H = − d2

dx2+ V(x)

with some absolutely continuous spectrum. It is a supplement to my recentpaper [19]. In [19], I dealt with the discrete case exclusively. As one would

C. Remling (B)Mathematics Department, University of Oklahoma, Norman, OK 73019, USAe-mail: [email protected]: www.math.ou.edu/∼cremling

http://arxiv.org/abs/0706.1101

360 C. Remling

expect, the basic ideas that were presented in [19] can also be used to analyzethe continuous case. It is the purpose of this paper to give such a treatment;basically, this will be a matter of making the appropriate definitions.

Therefore, my general philosophy will be to keep this note brief. I willassume that the reader is familiar with at least the general outline of thediscussion of [19] and only focus on those aspects where the extension to thecontinuous case is perhaps not entirely obvious. By the same token, I will notsay much about related work here; please see again [19] for a fuller discussion.

Given a potential V, we will consider limit points W under the shift(SxV)(t) = V(x + t), as x → ∞. We will thus need a suitable topology on asuitable space of potentials. This will naturally lead us to consider generalizedSchrödinger operators, with measures as potentials.

The basic result, from which everything else will follow, is Theorem 3 below.It says that the limits W are necessarily reflectionless (this notion will be de-fined later) on the support of the absolutely continuous part of the spectralmeasure. This is a very strong condition; it severely restricts the structure ofpotentials with some absolutely continuous spectrum. As in [19], this resultcrucially depends on earlier work of Breimesser and Pearson [7, 8].

We will present two applications of Theorem 3 here; both are analogs ofresults from [19]. The first application gives an easy and transparent proof of acontinuous Denisov–Rakhmanov [11, 12, 17] type theorem.

We denote by �ac the essential support of the absolutely continuous part ofthe spectral measure; this is determined up to sets of (Lebesgue) measure zero.If we write ρ for the spectral measure, we can define (a representative of) �ac

as the set where dρ/dt > 0. The absolutely continuous spectrum, σac, may beobtained from �ac by taking the essential closure. The essential spectrum, σess,can be defined as the set of accumulation points of the spectrum.

Theorem 1 Let V be a uniformly locally integrable (half line) potential (that is,we assume that supn

∫ n+1n |V(x)| dx < ∞). Suppose that

σess = �ac = [0, ∞).

Then

limx→∞

∫V(x + t)ϕ(t) dt = 0

for every continuous ϕ of compact support.

Denisov proved this earlier [11, Theorem 2], under the somewhat strongerassumption that V is bounded.

The conclusion of Theorem 1 says that V(x) tends to zero as x → ∞ inweak ∗ sense (more precisely, it is the sequence of measures V(x + t) dt thatconverges). It will become clear later that this mode of convergence is naturalhere. Also, examples of the type V = U2 + U ′ with a rapidly decaying, butoscillating U show that stronger modes of convergence of V can not beexpected.

Absolutely continuous spectrum of one-dimensional Schrödinger operators 361

Theorem 1 will be proved in Section 4. As in [19, Theorem 1.8], it should bepossible to use the same technique to establish an analogous result for finitegap potentials and spectra (and beyond), but we will not pursue this themehere.

Let us now discuss a second structural consequence of Theorem 3; in [19],I introduced the designation Oracle Theorem for statements of this type. TheOracle Theorem says that for operators with absolutely continuous spectrum,it is possible to approximately predict future values of the potential, witharbitrarily high accuracy, based on information about past values.

The precise formulation will involve measures μ as potentials and someadditional technical devices; these will of course be explained in more detaillater. To get a preliminary impression of what the Oracle Theorem is saying,it is possible to replace μ by a (uniformly locally integrable) potential V inTheorem 2 below.

We will work with spaces VCJ of signed Borel measures μ on intervals J.

For now, we can pretend that a measure μ is in VCJ if |μ|(J) � C|J|, but, for

inessential technical reasons, the actual definition will be slightly different. Ifendowed with the weak ∗ topology, these spaces VC

J are compact and in factmetrizable. The metric d that is used below arises in this way. We will also usea similarly defined space VC of measures on R. See Section 2 for the precisedefinitions.

Finally, Sxμ will denote the shift by x of the measure μ, that is,

∫f (t) d(Sxμ)(t) =

∫f (t − x) dμ(t). (1.1)

If dμ = V dt is a locally integrable potential V, then this reduces to the shiftmap (SxV)(t) = V(x + t) that was introduced above.

Theorem 2 (The Oracle Theorem) Let A ⊂ R be a Borel set of positive(Lebesgue) measure, and let ε > 0, a, b ∈ R (a < b), C > 0. Then there existL > 0 and a continuous function (the oracle)

� : VC(−L,0) → VC

(a,b)

so that the following holds. If μ ∈ VC and the half line operator associated withμ satisfies �ac ⊃ A, then there exists an x0 > 0 so that for all x � x0, we have that

d(�

(χ(−L,0)Sxμ

), χ(a,b)Sxμ

)< ε.

In other words, for large enough x, we can approximately determine thepotential on (x + a, x + b) from its values on (x − L, x), and the function(oracle) that does this prediction is in fact independent of the potential.Moreover, by adjusting a, b , we can also specify in advance how far the oracleshould look into the future.

362 C. Remling

2 Topologies on Spaces of Potentials

We need a topology on a suitable set of potentials that makes this spacecompact and also interacts well with other basic objects such as m functions.This is easy to do if we are satisfied with working with potentials that obey alocal Lp condition with p > 1. Indeed, for every p > 1 (and C > 0), we candefine

VCp =

{

V : R → R :∫ n+1

n|V(x)|p dx � Cp for all n ∈ Z

}

.

Closed balls in Lp are compact in the weak ∗ topology if p > 1; in fact,these compact topological spaces are metrizable. Pick such metrics dn; inother words, if Wj, W ∈ Lp(n, n + 1), ‖Wj‖p, ‖W‖p ≤ C, then dn(Wj, W) → 0precisely if Wj → W in the weak ∗ topology, that is, precisely if

∫ n+1

nWj(x)g(x) dx →

∫ n+1

nW(x)g(x) dx ( j → ∞)

for all g ∈ Lq(n, n + 1), where 1/p + 1/q = 1. Then, using these metrics, de-fine, for V, W ∈ VC

p

d(V, W) =∞∑

n=−∞2−|n| dn(Vn, Wn)

1 + dn(Vn, Wn);

here Vn, Wn denote the restrictions of V, W to (n, n + 1).This metric generates the product topology on VC

p , where this space isnow viewed as the product of the closed balls of radius C in Lp(n, n + 1). Inparticular, (VC

p , d) is a compact metric space.This simple device allows us to establish continuous analogs of the results

of [19] without much difficulty at all, but it is unsatisfactory because the mostnatural and general local condition on the potentials is an L1 condition. SinceL1 is not a dual space, we will then need to consider measures to make ananalogous approach work. Thus we define

VC = {μ ∈ M(R) : |μ|(I) � C max{|I|, 1} for all intervals I ⊂ R} .

Here, M(R) denotes the set of (signed) Borel measures on R. We can nowproceed as above to define a metric on VC: Pick a countable dense (with respectto ‖ · ‖∞) subset { fn : n ∈ N} ⊂ Cc(R), the continuous functions of compactsupport, and put

ρn(μ, ν) =∣∣∣∣

∫fn(x) d(μ − ν)(x)

∣∣∣∣ .

Then define the metric d as

d(μ, ν) =∞∑

n=1

2−n ρn(μ, ν)

1 + ρn(μ, ν). (2.1)


Clearly, d(μ j, μ) → 0 if and only if∫

f (x) dμ j(x) →∫

f (x) dμ(x) ( j → ∞)

for all f ∈ Cc(R). Moreover, (VC, d) is a compact space. To prove this, letμn ∈ VC. By the Banach-Alaoglu Theorem, closed balls in M([−R, R]) arecompact. Use this and a diagonal process to find a subsequence μn j with theproperty that

∫f dμn j → ∫

f dμ for all f ∈ Cc(R), for some μ ∈ M(R). Theproof can now be completed by noting that a measure ν ∈ M(R) is in VC if andonly if

∣∣∣∣

∫f (x) dν(x)

∣∣∣∣ � C max{diam(supp f ), 1}‖ f‖∞

for all f ∈ Cc(R).The same construction can be run if R is replaced by an interval J, and these

spaces, which we will denote by VCJ , will also play an important role later on.

3 Schrödinger Operators with Measures

We are thus led to consider Schrödinger operators with measures as potentials;therefore, we must now clarify what the precise meaning of this object is. Thereis, of course, a considerable amount of previous work on these issues; see, forexample, [1, 3, 5, 6] and the references cited therein. Here, we will follow theapproach of [3]. Actually, Schrödinger operators will not play a central rolein this paper, at least not explicitly. Therefore, we will only indicate how tomake sense out of the Schrödinger equations − f ′′ + μf = zf . We can then usethese to define Titchmarsh-Weyl m functions, spectral measures etc., and werefer the reader to [3] for the (straightforward) definition of domains that yieldself-adjoint operators.

There are two obvious attempts, and these conveniently lead to the sameresult: If I ⊂ R is an open interval and f ∈ C(I), we can call f a solution to theSchrödinger equation

− f ′′ + fμ = zf (3.1)

if (3.1) holds in the sense of distributions on I. Alternatively, one can workwith the quasi-derivative

(Af )(x) = f ′(x) −∫

[0,x]f (t) dμ(t);

if x < 0, then∫[0,x] needs to be replaced with − ∫

(x,0)here. We now say that

f solves (3.1) on I if both f and Af are (locally) absolutely continuous and−(Af )′ = zf on I. This new definition is motivated by the observation that, atleast formally, (Af )′ = f ′′ − fμ.

A slight modification of the argument from the proof of [3, Theorem 2.4]then shows that this latter interpretation of (3.1) is equivalent to the equation

364 C. Remling

holding in D′(I). The basic observation here is that if f is continuous, then(Af )′ = f ′′ − fμ in D′, not only formally.

Note that if f solves (3.1), then f ′ is of bounded variation and the jumps canonly occur at the atoms of μ.

If μ ∈ VC, we have limit point case at both endpoints. This means thatfor z ∈ C

+ (the upper half plane in C), there exist unique (up to a factor)solutions f±(x, z) of (3.1) on R satisfying f− ∈ L2(−∞, 0), f+ ∈ L2(0, ∞). TheTitchmarsh-Weyl m functions of the problems on (−∞, x) and (x, ∞), withDirichlet boundary conditions at x (u(x) = 0), are now defined as follows:

m±(x, z) = ± f ′±(x, z)

f±(x, z)(3.2)

We will use this formula only for points x with μ({x}) = 0 so that the possiblediscontinuities of f ′ cannot cause any problems here.

Definition 1 Let A ⊂ R be a Borel set. We call a potential μ ∈ VC reflectionlesson A if

m+(x, t) = −m−(x, t) for almost every t ∈ A (3.3)

for some x ∈ R with μ({x}) = 0.The set of reflectionless potentials μ ∈ ⋃

C>0 VC on A is denoted by R(A).

This is a key notion for everything that follows. If we have (3.3) for somex, then we automatically get this equation at all points of continuity of μ.Moreover, the exceptional set implicit in (3.3) can be taken to be independentof x. To prove these remarks, observe that if m±(x, t) ≡ limy→0+ m±(x, t + iy)

exists for some x, t ∈ R, then this limit exists for all x (and the same t).Moreover, as a function of x, the m functions are of bounded variation and(using distributional derivatives)

± ddx

m± = μ − z − m2±.

The claim now follows by considering (d/dx)(m+ + m−)(x, t).The other key notion is that of the ω limit set of a potential μ ∈ VC under

the shift map. This was already mentioned in the introduction, and we can nowgive the more precise definition

ω(μ) = {ν ∈ VC : There exist xn → ∞ so that d(Sxnμ, ν) → 0

}.

For the definition of the shifted measures Sxμ, see (1.1). Typically, μ will begiven as a half line potential V, but it is of course easy to interpret V asan element dμ = V dx of VC (such a μ automatically gives zero weight to(−∞, 0]).

The compactness of VC ensures that ω(μ) is non-empty, compact, andinvariant under {Sx : x ∈ R}. Moreover, and in contrast to the discrete case,ω(μ) is also connected because we now have a flow Sx.


4 Main Results and their Proofs

It will be useful to introduce VC+ as the set of all μ ∈ VC with |μ|((−∞, 0]) = 0.These measures will serve as the potentials of half line problems on (0, ∞). Wecan think of such a μ as a measure on (0, ∞) or on R.

Theorem 3 Let μ ∈ VC+ . Then ω(μ) ⊂ R(�ac).

Here, �ac denotes an essential support of the absolutely continuous part ofthe spectral measure of the half line problem on (0, ∞) (say).

Theorem 3 is proved in the same way as the analogous result (Theorem 1.4)from [19]. Therefore, we will only make a few quick remarks and then leavethe matter at that.

First of all, note that although the original result of Breimesser and Pearson[7, Theorem 1] is formulated for Schrödinger operators with locally integrablepotentials, the same proof also establishes the result for operators with mea-sures as potentials. Indeed, one never works with the potential itself but onlywith solutions to the Schrödinger equation (3.1) or with transfer matrices. Seealso [19, Appendix A].

As a second ingredient, we need continuous dependence of the (half line)m functions m± on the potential.

Lemma 1 Let μn, μ ∈ VC and suppose that d(μn, μ) → 0. Fix x ∈ R withμn({x}) = μ({x}) = 0. Then

m±(x, z; μn) → m±(x, z; μ),

uniformly on compact subsets of C+.

This follows because convergence in VC implies weak ∗ convergence of therestrictions of the measures to compact intervals, at least if the endpointsof these intervals are not atoms of μ. It then follows that the solutionsto the Schrödinger equation converge, locally uniformly in z. This is mostconveniently established by rewriting the Schrödinger equation as an integralequation. See, for example, [3, Lemma 6.3] for more details. One can now use(3.2) to obtain the Lemma. In fact, it is also helpful to approximate m± bym functions of problems on bounded intervals. This allows us to work withsolutions that satisfy a fixed initial condition.

To prove Theorem 3, fix ν ∈ ω(μ). By definition of the ω limit set, thereexists a sequence x j → ∞ so that Sx jμ → ν in (VC, d). Fix x ∈ R with μ({x +x j}) = ν({x}) = 0. By Lemma 1,

m±(x + x j, z; μ) → m±(x, z; ν) ( j → ∞),

locally uniformly in z. The Breimesser-Pearson Theorem [7, Theorem 1] (seealso [19, Theorem 3.1]) together with [19, Theorem 2.1] then yield a relationbetween m+(x, z; ν) and m−(x, z; ν) which turns out to be equivalent to the

366 C. Remling

condition from Definition 1, with A = �ac. This last part of the argument isidentical with the corresponding treatment of [19].

Honesty demands that I briefly comment on a technical (and relativelyinsignificant point) here: To run the argument in precisely this form, oneneeds a slight modification of either Lemma 1 or the original Breimesser-Pearson Theorem. The easiest solution would be to prove the Breimesser-Pearson Theorem for two half line m functions m± (in the original versionfrom [7, 8], m− refers to a bounded interval). Alternatively, one can use avariant of Lemma 1 where the approximating m functions may be associatedwith bounded (but growing) intervals.

Let us now show how Theorem 3 can be used to produce Denisov-Rakhmanov type theorems. We will automatically obtain the following slightlymore general version of Theorem 1.

Theorem 4 Let μ ∈ VC+ , and suppose that the half line operator generated by μ

satisfies

σess = �ac = [0, ∞).

Then d(Sxμ, 0) → 0 as x → ∞.

The proof will also depend on the following observation (whose discreteanalog was pointed out in [15]; see also [16]).

Proposition 1 Let μ ∈ VC+ and assume that ν ∈ ω(μ). Then

σ(ν) ⊂ σ+ess(μ).

Here, σ(ν) is the spectrum of −d2/dx2 + ν on L2(R), while σ+ess(μ) denotes

the essential spectrum of the half line operator −d2/dx2 + μ on L2(0, ∞) (say).

Proof In the discrete case, this followed from a quick argument using Weylsequences. In the continuous case, this device is not as easily implementedbecause of domain questions. The following alternative argument avoids theseissues and thus seems simpler: Suppose that d(Sx jμ, ν) → 0. Then the wholeline (!) operators associated with Sx jμ converge in strong resolvent sense tothe (whole line) operator generated by ν. To prove this fact, one can argue asin Lemma 1 above.

Since the operators with shifted potentials are unitarily equivalent to theoperator generated by μ itself, it follows from [18, Theorem VIII.24(a)] thatσ(ν) ⊂ σ(μ). The ω limit set does not change if μ is modified on a left halfline; any discrete eigenvalue, however, can be moved (or removed) by such amodification. Similarly, σess = σ+

ess ∪ σ−ess, and σ−

ess is completely at our disposal,so we actually obtain the stronger claim of the Proposition. ��

Proof of Theorem 4 We will show that ω(μ) consists of the zero potential only.This will imply the claim because the distance between Sxμ and ω(μ) must goto zero as x → ∞.


By Theorem 3 and Proposition 1, any ν ∈ ω(μ) must satisfy

σ(ν) = [0, ∞), ν ∈ R((0, ∞)). (4.1)

Here, we use the (well known) fact that Im m± > 0 almost everywhere on Aif the corresponding potential is reflectionless on this set. Indeed, (3.3) showsthat otherwise we would have m+ + m− = 0 on a set of positive measure, henceeverywhere, but this is clearly impossible.

What we will actually prove now is that only the zero potential, ν = 0,satisfies (4.1). This argument follows a familiar pattern; see, for example, [9]or [21] (especially Lemma 4.6 and the discussion that follows) for similararguments in somewhat different situations.

Suppose that ν ∈ VC obeys (4.1), and fix x ∈ R with ν({x}) = 0. Let m±be the m functions of Hν = −d2/dt2 + ν on L2(x, ∞) and L2(−∞, x),respectively, and consider the function

H(z) = m+(x, z) + m−(x, z) = − W( f+, f−)

f+(x, z) f−(x, z).

Here, W(u, v) = uv′ − u′v denotes the Wronskian. This last expression iden-tifies H as

H(z) = − 1

G(x, x; z),

the negative reciprocal of the (diagonal of the) Green function of Hν . Com-pare, for example, [10, Section 9.5]. (Of course, this reference does not discussSchrödinger operators with measures, but the rather elementary argumentbased on the variation of constants formula generalizes without any difficulty.)

The defining property of G is given by

((Hν − z)−1ϕ

)(x) =

∫ ∞

−∞G(x, y; z)ϕ(y) dy.

This holds for z /∈ σ(ν) = [0, ∞), ϕ ∈ L2(R). The spectral theorem shows thatif z = −t < 0, then

⟨ϕ, (Hν + t)−1ϕ

⟩ =∫

[0,∞)

d‖E(s)ϕ‖2

s + t> 0.

Since G is continuous in x, y, this implies that G(x, x, t) � 0, thus H(t) < 0 fort < 0. Furthermore, the fact that ν ∈ R((0, ∞)) implies that Re H(t) = 0 foralmost every t > 0.

So we know the phases of the boundary values of the Herglotz functionH almost everywhere. By the exponential Herglotz representation (or, syn-onymously, the Herglotz representation of ln H(z)), this determines H up toa (positive) multiplicative constant. Since H0(z) = (−z)1/2 has the propertiesdescribed above, this says that for suitable c > 0,

H(z) = c√−z = − c√

2+ c

π

∫ ∞

0

(1

t − z− t

t2 + 1

)

t1/2 dt. (4.2)

368 C. Remling

We will now need some information on the large z asymptotics of m functions.This subject has been analyzed in considerable depth; see, for example, [2, 13,14, 20]. Of course, the treatment of these references needs to be adjusted hereto cover the case of Schrödinger operators with measures, but this is easy todo, especially since we will only need the rather unsophisticated estimate

m±(x, −κ2

) = −κ + o(1) (κ → ∞).

Here, it is important that we assumed that ν({x}) = 0.Since H = m+ + m−, it now follows that c = 2 in (4.2). Furthermore, the

measures from the Herglotz representations of m± are absolutely continuouswith respect to the measure

dρ(t) = 2

πχ(0,∞)(t)t1/2 dt

from the Herglotz representation (4.2) of H. In fact, we can write

m±(x, z) = A± +∫ (

1

t − z− t

t2 + 1

)

g±(t) dρ(t),

with A± ∈ R, A+ + A− = −√2, and, more importantly, 0 � g± � 1 and

g++g− =1. More can be said here: Since ν is reflectionless on (0, ∞), wecan use (3.3) to deduce that Im m+(x, t) = Im m−(x, t) for almost everyt > 0. But for almost every t > 0, we have that Im m±(x, t) = g±(t)(2/π)t1/2,thus g+ = g− = 1/2 almost everywhere. It now follows that

m±(x, z) = √−z.

But m0 = √−z is the m function for zero potential, thus ν = 0, as desired. Thislast step is a basic result in inverse spectral theory for potentials (m determinesV); here, we of course need a version for measures, but this extension poses nodifficulties. See, for instance, [3, Theorem 6.2(b)] (this needs to be combinedwith the fact that m determines φ, but this is also discussed in [3]). ��

It remains to prove the Oracle Theorem. We prepare for this by making acouple of new definitions. First of all, put

RC(A) = R(A) ∩ VC.

Next, we consider again spaces of half line potentials, and we now think ofthese as restrictions of measures μ ∈ VC:

VC+ = {

χ(0,∞)μ : μ ∈ VC},

VC− = {

χ(−∞,0)μ : μ ∈ VC}

I emphasize that on VC± , we do not use the topology that is induced by VC ⊃ VC± .That would quite obviously be a bad idea because it would make the re-striction map μ �→ χ(0,∞)μ discontinuous; consider, for example, the sequenceμn = δ1/n. Instead, we just observe that in the notation from Section 2, wecan identify VC+ = VC

J , where J = (0, ∞), and we use the topology and metric


described in Section 2. In other words, if we denote this metric by d+, thend+(μn, μ) → 0 if and only if

∫f (x) dμn(x) →

∫f (x) dμ(x) (n → ∞)

for all continuous f whose support is a compact subset of (0, ∞). Similarremarks apply to VC− , of course.

Now the restriction maps VC → VC± are continuous, and the spaces (VC± , d±)

are compact.Finally, we introduce

RC+(A) = {

χ(0,∞)μ : μ ∈ RC(A)} ⊂ VC

+ ,

RC−(A) = {

χ(−∞,0)μ : μ ∈ RC(A)} ⊂ VC

− ,

and we use the same metrics d± on these spaces also. With this setup, we nowobtain statements that are analogs of [19, Proposition 4.1].

Proposition 2 Let A ⊂ R be a Borel set of positive measure, and fix C > 0.Then:

(a)(RC(A), d

)and

(RC±(A), d±

)are compact spaces;

(b) The restriction maps

RC(A) → RC+(A), μ �→ χ(0,∞)μ;

RC(A) → RC−(A), μ �→ χ(−∞,0)μ

are continuous and bijective (and thus homeomorphisms).(c) The inverse map

RC−(A) → RC(A), χ(−∞,0)μ �→ μ

is (well defined, by part (b), and ) uniformly continuous.

Proof

(a) Since RC(A) is a subspace of the compact space VC, it suffices to show thatRC(A) is closed. This can be done exactly as in [19, Proof of Proposition4.1(d)]; we make use of Lemma 1 of the present paper and Theorem 2.1,Lemma 3.2 of [19].The spaces RC±(A) are the images of the compact space RC(A) under thecontinuous restriction maps, so these spaces are compact, too.

(b) Continuity of the restriction maps is clear (and was already used inthe preceding paragraph). Moreover, these maps are surjective by thedefinition of the spaces RC±(A). Injectivity follows from equation (3.3): μ

on (0, ∞) determines m+(x, ·) for all x > 0. Fix an x > 0 with μ({x}) = 0.Since μ is reflectionless on A and |A| > 0, we have condition (3.3) on aset of positive measure, and this lets us find m−(x, ·). This m function, inturn, determines μ on (−∞, x).

370 C. Remling

Finally, recall that a continuous bijection between compact metric spacesautomatically has a continuous inverse.

(c) This is an immediate consequence of parts (a) and (b). ��

As in [19], the Oracle Theorem will follow by combining Proposition 2 withTheorem 3. In fact, in rough outline, things are rather obvious now: Propo-sition 2 says that a reflectionless potential can be approximately predicted ifit is known on a sufficiently large interval (recall how the topologies on thespaces RC(A), RC±(A) were defined), and Theorem 3 makes sure that Sxμ isapproximately reflectionless for sufficiently large x.

Some care must be exercised, however, if a continuous oracle � is desired.The following straightforward but technical considerations prepare for thispart of the proof. We again consider the spaces VC

J with metrics of the typedescribed in Section 2; in the applications below, the interval J will be boundedand open, but this is not essential here.

In a normed space, balls Br(x) = {y : ‖x − y‖ < r} are convex; Lemma 2below says that balls with respect to the metric (2.1) enjoy the followingweaker, but analogous property.

Lemma 2 If w j � 0,∑m

j=1 w j = 1 and μ, ν j ∈ VCJ satisfy d(μ, ν j) < ε with ε �

1/4 (say), then

d

⎛

⎝μ,

m∑

j=1

w jν j

⎞

⎠ < 6ε ln ε−1.

Proof Let N = max{n ∈ N : 2n+1ε � 1}, and abbreviate∑

w jν j = ν. Sinced(μ, ν j) < ε, it is clear from (2.1) that if n � N, then

ρn(μ, ν j) <2nε

1 − 2nε� 2n+1ε.

The definition of ρn shows that

ρn(μ, ν) �∑

w jρn(μ, ν j),

so we obtain that

ρn(μ, ν) < 2n+1ε (n � N).

This allows us to estimateN∑

n=1

2−n ρn(μ, ν)

1 + ρn(μ, ν)<

N∑

n=1

2−n · 2n+1ε = 2Nε <2ε ln ε−1

ln 2.

On the other hand, we of course have that

∑

n>N

2−n ρn(μ, ν)

1 + ρn(μ, ν)<

∑

n>N

2−n = 2−N < 4ε � 4ε ln ε−1

ln 4,

so we obtain the Lemma. ��


Proof of Theorem 2 We begin by introducing some notation that will proveuseful. Write

J− = (−L, 0), J+ = (a, b).

Our goal is to (approximately) predict the restriction of Sxμ to J+, and we aregiven the restriction of Sxμ to J−. We will use subscripts + and −, respectively,for such restrictions. So, for example, ν+ = χJ+ν, and this is now interpreted asan element of VC

J+ .Next, note that although a metric is explicitly mentioned in Theorem 2, by

compactness, it suffices to establish the assertion for some metric that generatesthe weak ∗ topology. We will of course want to work with the metric from (2.1)and Lemma 2. More specifically, denote this metric (on VC

J+) by d+. We use asimilar metric d− on VC

J− ; on VC, we also fix such a metric d, but, in addition,we demand, as we may, that d dominates d± in the following sense: If μ,ν ∈ VC, then

d−(μ−, ν−) � d(μ, ν), d+(μ+, ν+) � d(μ, ν). (4.3)

(The same notation, d±, was used for different purposes in Proposition 2; sincewe are not going to explicitly use those metrics here, that should not cause anyconfusion.)

With these preparations out of the way, the proof can now be accomplishedin four steps. Let A ⊂ R, |A| > 0, ε > 0, a, b ∈ R (a < b), and C > 0 be given.

Step 1: Use Proposition 2(c) and the definition of the topologies on RC(A),RC−(A) to find L > 0 and δ > 0 such that the following holds: For ν, ν ∈ RC(A),

d− (ν−, ν−) < 5δ =⇒ d+ (ν+, ν+) < ε2. (4.4)

We further assume that δ ≤ ε here. (The suspicious reader will have noticedthat it is at this point only that we can define d− and d.)

Step 2: The set

RCJ−(A) := {

μ− : μ ∈ RC(A)}

is compact by Proposition 2 (again, this is a continuous image of a compactspace). Since VC

J− is compact, it follows that the closed δ neighborhood

U δ = {μ− ∈ VC

J− : d−(μ−, ν−) ≤ δ for some ν ∈ RC(A)}

is also compact. Therefore, there exist ν1, . . . , νN ∈ RC(A) so that the 2δ ballsabout the ν j,− cover U δ . At these points, we can define a preliminary versionof the oracle in the obvious way as

�0(ν j,−

) = ν j,+.

372 C. Remling

However, this will be modified in the next step.

Step 3: We now define, for arbitrary σ ∈ U δ ,

�(σ) =∑

(3δ − d−(σ, ν j,−))�0(ν j,−)∑

(3δ − d−(σ, ν j,−)).

The sums are over those j for which d−(σ, ν j,−)< 3δ. It’s easy to seethat � : U δ →VC

J+ is continuous. Moreover, if j0 ∈{1,. . ., N} is such thatd−(σ, ν j0,−) < 2δ, then d−(ν j,−, ν j0,−) < 5δ for all j contributing to the sum.Therefore, (4.4) shows that d+(ν j,+, ν j0,+) < ε2 for these j. If ε > 0 wassufficiently small, then Lemma 2 now implies that

d+(�(σ), ν j0,+

)< 6ε2 ln ε−2 < ε, (4.5)

say. Recall that this holds for every j0 for which d−(σ, ν j0,−) < 2δ. Moreover,for every σ ∈ U δ , there is at least one such index j0.

The oracle � has now been defined on U δ , and this is all we need to dothe prediction. However, if a (somewhat specious) continuous extension toall of VC

J− is desired, one can proceed as above, by considering a suitablecovering and taking convex combinations. It is also possible, somewhat moreelegantly, to just refer to the extension theorem of Dugundji-Borsuk [4, Ch. II,Theorem 3.1].

Step 4: In this final step, we show that � indeed predicts μ. Given a potentialμ ∈ VC with �ac ⊃ A, first of all take x0 so large that

d (Sxμ, ω(μ)) < δ for all x � x0.

In other words, if we fix x � x0, we then have that

d(Sxμ, ν) < δ (4.6)

for some (in general: x dependent) ν ∈ ω(μ). By Theorem 3, ν ∈ RC(A).When we restrict to J±, then (4.3), (4.6) imply that

d− ([Sxμ]−, ν−) < δ, d+ ([Sxμ]+, ν+) < δ. (4.7)

In particular, this ensures that [Sxμ]− ∈ U δ , and thus there exists a j ∈{1, . . . , N} so that

d−([Sxμ]−, ν j,−

)< 2δ.

By (4.5),d+

(�([Sxμ]−) , ν j,+

)< ε. (4.8)

But by the triangle inequality, we also have that d−(ν−, ν j,−) < 3δ, so (4.4)shows that

d+(ν+, ν j,+

)< ε2.

If this is combined with (4.7), (4.8), we indeed obtain that

d+ (� ([Sxμ]−) , [Sxμ]+) < δ + ε + ε2 < 3ε

(say), as desired. ��


Acknowledgements I thank Sergey Denisov and Barry Simon for bringing [11] to my attentionand Lenny Rubin for useful information on extension theorems.

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