annphys97

File: DISTIL 573701 . By:DS . Date:19:11:97 . Time:10:52 LOP8M. V8.0. Page 01:01Codes: 4113 Signs: 2181 . Length: 51 pic 3 pts, 216 mm

Annals of Physics � PH5737

Annals of Physics 261, 180�218 (1997)

Unfolding the Quartic Oscillator

Eric Delabaere and Fre� de� ric Pham

Laboratoire de Mathe� matiques, UMR CNRS 6621, Universite� de Nice-Sophia Antipolis,Parc Valrose, 06108 Nice Cedex 2 France

Received December 27, 1996; revised July 30, 1997

The `èxact WKB method'' is applied to the general quartic oscillator, yielding rigorousresults on the ramification properties of the energy levels when the coefficients of the fourthdegree polynomial are varied in the complex domain. Simple though exact ``model forms'' aregiven for the avoided crossing phenomenon, easily interpreted in terms of complex branchpoints in the `àsymmetry parameter.'' In the almost symmetrical situation, this gives ageneralization of the Zinn�Justin quantization condition. The analogous ``model quantizationcondition'' near unstable equilibrium is thoroughly analysed in the symmetrical case, yieldingcomplete confirmation of the branch point structure discovered by Bender and Wu. Thenumerical results of this analysis are in excellent agreement with those computed by Shanley,overtaking the most optimistic expectations of the realm of validity of semiclassical models.� 1997 Academic Press

The aim of this article is to present exact results on how the energy levels of theone-dimensional quartic oscillator,

_&�2 d 2

dq2+V(q)& 9=E9, (0)

where V is the general fourth degree polynomial1

V(q)=q4+:q2+;q, (V)

depend on the coefficients of the polynomial V.By the ``quasihomogeneity'' property of Eq. (0), which is invariant under the

substitution

q [ *q, � [ *3�, : [ *2:, ; [ *3;, E [ *4E, (V)

no generality would be lost by setting �=1. But � will play a prominent role in ourtechniques of proof, which will be based on the complex WKB method, or moreprecisely its `èxact asymptotic'' version initiated by Voros [Vo1] and further

Article No. PH975737

1800003-4916�97 �25.00Copyright � 1997 by Academic PressAll rights of reproduction in any form reserved.

1 One can get rid of the constant term, and let the coefficient of q4 be 1 (assuming it was positive),by an affine rescaling of the energy. One gets rid of the q3 term by a suitable change of origin in q.


worked out by the powerful methods of Ecalle's resurgence theory [E1, E2, E3,DDP1].

First focusing on what happens for real values of the coefficients, we shall thenallow for complex deformations of V, which will shed light on the ramificationproperties of the energy levels. For instance the energy levels of the doubleoscillator have square root branch points near the real axis of the complexified``asymmetry parameter'' ;, which can be considered as the source of the ``avoidedcrossing'' phenomenon. Most interesting are the situations close to equilibrium(where double turning points ave involved in the WKB analysis). For instance inthe case of the symmetric double well, analyzing the quantization condition nearthe bottom of the well (stable equilibrium) leads to a rigorous derivation of theZinn�Justin quantization condition [ZJ1, ZJ2, ZJ3, ZJ4, ZJ5], the solutions ofwhich can be expressed as series of multi-instanton expansions (cf. [DDP2]). Weshall give here a more general version of this condition, allowing the asymmetryparameter ; to be of order O(�) and not just 0. Near unstable equilibrium (the topof the hump in the double well), our analysis will yield a more subtle branch pointstructure in the complex plane of the variable |crit��, where |crit is the actionintegral along the critical (lemniscate shaped) trajectory. In the symmetrical casethis will give a rigorous justification of the complex branch point structure dis-covered a long time ago by Bender and Wu [BW1] and much commented uponsince (cf. [Si1, Si2, Sh1, Sh2, Wi, Al.Si, HS, BW2, BHS]).

The main idea of our approach can be summarized as follows: the (E, :, ;) spacecan be covered by various analytic charts, in each of which the quantization condi-tion is given exactly by an explicit analytic relation between the coordinates of thechart. Such an analytic relation, which we call a model quantization condition, looksvery similar to what one would get by a lowest order semi-classical approximation(for instance our ``model equation'' in Section 3.1 has exactly the same form asBender and Wu's ``secular equation'' in [BW1, Section 4]). But in contradistinctionto semiclassical approximations our quantization conditions are exact. The pricewe have to pay is that instead of the (E, :, ;) variables we must work with newvariables (the coordinates of the chart), deduced from the former by resummingWKB-type expansions in powers of � (the resummability being ensured byresurgence theory, as we explain elsewhere; cf. bibliographical comments hereafter).

All the necessary material for deriving our model quantization conditions isextensively described in our common work [DDP2] with Herve� Dillinger.2 In thepresent paper the reader is required to take these conditions for granted3 and con-centrate on their consequences:

1. Theoretical consequences, e.g., rigorous statements on the Riemann sheetstructure of the energy eigenvalues as multivalued analytic functions of the complexvariables (:, ;).

181UNFOLDING THE QUARTIC OSCILLATOR

2 Two preliminary versions of [DDP2] have appeared in preprint form in April 1991 and Jan. 1996.The final version will appear in J. Math. Phys.

3 Perhaps their similarity with usual WKB approximations will make them not too hard to accept.


2. Numerical consequences, e.g., computing numerical values of the branchpoints (of E as a function of (:, ;)). The numerical scheme is divided in two steps.

(a) Solve the ``model'' (transcendental) relations between the ``modelvariables''; since the relations are explicit this is just classical analysis, and thenumerical scheme (Newton's method) converges rapidly.

(b) Go from the ``model'' variables to the ``physical'' ones. This step involvesresummation of divergent series. In the present paper we made the very crudeapproximation of replacing the series by their lowest order terms. Surprisingly, theresults thus obtained are already in very good agreement with those obtained byother methods (cf. [Sh1]). At this point one should mention that the ``resurgencerelations'' (cf. [DDP2]) which led to our model quantization conditions giveprecisely the information required for resumming numerically the series by the ``hyper-asymptotic methods'' of Berry and Howls [BeH]. One should thus be able to getextremely accurate numerical values.

Genesis of this Paper and Acknowledgments

The proof of the Zinn�Justin conjecture was announced in our 1990 note withHerve� Dillinger [DDP0]4 and was detailed in Delabaere and Dillinger's thesis(Nice, 1991). But only recently did we realize how these techniques could be usedin a geometrical spirit; the idea is to use the implicit resurgent function theorem toreduce the exact quantization condition to a simple ``model form'' before tryingto solve it. In the simple oscillator case this was suggested to us by Y. Colin deVerdie� re. Further ``unfolding'' of this idea was greatly stimulated by our partici-pation to the ``exponential asymptotics'' semester at the Newton Institute inCambridge (Spring 1995).5 We thank Carl Bender for most helpful discussions, andJ. Lascoux who first introduced us to the Bender and Wu problem.

Bibliographical Comments

For the more mathematically minded reader, detailed justifications of the techni-ques presented in [DDP2] are given in [DDP1] (an exposition of the generic-energy case, in a geometrical spirit), and in [DP]. The first part of [DP] startswith a self-contained digest of Ecalle's resurgence theory and shows how it appliesto one-dimensional WKB expansions; the second part of [DP] is a detailed deri-vation of how to cope with double turning points, with a thorough study of the``confluence'' phenomenon.

182 DELABAERE AND PHAM

4 In Section 1 of that note the sign \ in the monodromy factor has been forgotten.5 In particular our treatment of avoided crossings was inspired by a talk of S. Slavyanov.

File: 595J 573704 . By:XX . Date:14:11:97 . Time:15:45 LOP8M. V8.0. Page 01:01Codes: 2019 Signs: 1341 . Length: 46 pic 0 pts, 194 mm

0. CLASSICAL APPROXIMATION AND SEMI-CLASSICAL INVARIANTS

0.1. The Bohr�Sommerfeld sketch

For a given value of (:, ;, E ) we shall denote by L:, ;, E the classical ``energyshell'' in phase space, i.e., the curve in the ( p, q)-plane defined by the equationp2+V(q)=E.

Answering the following questions is a well-known exercise for beginners inThom's ``catastrophy theory'' (cf., e.g., [PS]):

1. How does the shape of V depend on (:, ;)?

2. How does the shape of the energy shell L:, ;, E depend on (:, ;, E)?

The answers are the following:

1. The ``bifurcation curve'' 8:3+27;2=0 splits the (:, ;)-plane into tworegions (cf. Fig. 1), the simple well region and the double well region.

2. In the three-dimensional (:, ;, E )-space (with vertical E-axis, orientedupwards), the set of points for which the energy shell L:, ;, E is singular is the well-known ``swallowtail surface'' (V(q)&E=(dV(q)�dq)=0); cf. Fig. 2a). Below thatsurface lies the classically forbidden region. Above it lies the simple oscillator region,where the energy shell L:, ;, E is a connected oval. The spearhead shaped region inthe middle is the double oscillator region, where the energy shell L:, ;, E splits intotwo disconnected ovals.

Fig. 1. The bifurcation curve, separating the ``simple well region'' (on the right) from the ``doublewell region'' (on the left).



Fig. 2. (a) The swallowtail. (b) Plane sections of the swallowtail for :<0. The stable layer (1), theunstable layer (2), the metastable layers (3), the Maxwell edge (4), the cuspidal edge (5).

The forbidden and simple oscillator regions are separated by the stable layer(that part of the swallowtail where E is an absolute minimum of the potential),Fig. 2b); the boundary of the double oscillator region has three two-dimensionalstrata: the upper stratum is the unstable layer (where E is the local maximum valueof the double well potential); the two lower ones are the metastable layers (whereE is only a local minimum of the double well).

The stable and metastable layers meet along the Maxwell edge (where E is theminimum of a symmetrical double well), which is a line of transverse self-inter-section of the swallowtail. The unstable layer meets the metastable layers along thecuspidal edge (where V has a horizontal stationary tangent at a triple turning point).



v In the simple oscillator region we get approximate energy levels En as func-tions of (:, ;) by writing the Bohr�Sommerfeld quantization condition

|#=(n+ 12) h, n=0, 1, 2, ..., (0.1)

where h=2?� and |# :=�# p dq is the action integral over the oscillator cycle #,defined as being the oval L:, ;, E , so oriented that |#>0.

v In the double oscillator region two analogous Bohr�Sommerfeld quantizationconditions can be written

|#&=(n&+ 1

2) h (resp. |#+=(n++ 1

2) h), (0.2)

corresponding respectively to the ``left oscillator cycle'' #& and ``right oscillatorcycle'' #+ . We shall denote by E &

n&and E +

n+the resulting approximate energy levels.

Altogether these quantization conditions give a pattern of surfaces in (:, ;, E)-space, which we shall call the Bohr�Sommerfeld sketch. Typical plane sections ofthis sketch are represented on Figs. 3a, b, and c).

The Relation between the double oscillator actions. Using the fact that the func-tion p(q) has a pole at infinity, with residue \(i;�2) one easily shows (deformingcontours on the Riemann sphere of the complex q-variable) that

|#&&|#+

=?; (0.3)

(cf. [KL], where physical implications of this remark are discussed). As a conse-quence, the value of (:, ;) at which two energy levels E &

n&and E +

n+cross each other

depends only on the difference n&&n+ of their quantum numbers.

0.2. WKB Invariants on the Complex Energy Shell

For a given value of (:, ;, E ) let LC:, ;, E be the complexified energy shell. It can

be considered as a twofold covering of the complex q-plane, ramified at the turningpoints where p=0. Outside the turning points this covering carries formal solutionsof Eq. (0), the so-called WKB expansions: defined locally on the covering, up to anarbitrary normalization factor, they read

P(q, �2)&1�2 e(i��) �qq0

P(q$, � 2) dq$,

where

P(q, �2)= p(q)+ p1(q) �2+ p2(q) �4+ } } }

is a formal powerseries in �2. Analytic continuation along any closed path #~ of thecovering therefore multiplies them by some factor which does not depend on theinitial normalization point q0: we call it the monodromy factor of #~ .



As an important example, let #~ be deduced from a real ``oscillator cycle'' #(cf. Section 0.1) by distorting it away from its turning points as shown on Fig. 4.

Noticing that analytic continuation along #~ multiplies P(q, �2)&1�2 by &1, onesees that the monodromy factor of #~ reads &a#, where

a#=e(i��) 0#

(0.4)0#=|

#P(q, �2) dq=|#+0(�2).

The formal expansion a# is the Voros multiplier of the cycle # (cf. [Vo1, Vo2]).Of course the same kind of construction applies to any cycle # tying two turning

points in LC:, ;, E , provided these turning points are simple (i.e., simple zeros of

Fig. 3. Plane sections of the Bohr�Sommerfeld sketch: (a) :<0; (b) :>0; (c) ;=0.



Fig. 3��Continued

E&V(q)). Among such cycles # (which geometers call vanishing cycles, because theyvanish when one makes the pair of simple turning points coalesce), an importantrole will be played in the sequel by what we call fading cycles, i.e., those vanishingcycles along which p dq is pure imaginary (positive), so that the Voros multiplier isexponentially small. The simplest example of these is the tunnel cycle of the doubleoscillator, which travels both ways along the classically forbidden segment of thereal q-axis, so that dq is real and p is pure imaginary.

1. EXACT QUANTIZATION OF SIMPLE OSCILLATORS

1.1. Quantization in the Simple Oscillator Region

In the simple oscillator region we just have one oscillator cycle #. Let us defineits monodromy exponent s by

s=0#

2?�&

12

(1.0)

Fig. 4. The distorted vanishing cycle.



so that the monodromy factor reads

&a#=e2i?s.

A careful study shows that one must distinguish two cases, splitting the simpleoscillator region into two subregions, the lower region and the higher region,separated by the broken lines of Figs. 5a and 5b (``Stokes separatrix''; cf. alsoFig. 5c).

v The lower region is that region where the complexified energy shell has nofading cycle. The Stokes pattern in the q-plane then looks as shown on Fig. 5c.I.From the general results in [DDP1] it therefore follows that the Voros multipliera# (and its logarithm 0#) is Borel resummable. Understanding now a#, 0# , s as thetrue functions of (:, ;, E) deduced from the corresponding formal symbols by Borelresummation, the exact quantization condition demands that the monodromy factorshould be equal to 1:

s(:, ;, E )=n # N (the set of natural integers). (1.1)

Reexpressing s in terms of 0# , one recognizes the Bohr�Sommerfeld quantizationcondition with |# replaced by its (Borel resummed ) semiclassical incarnation 0# .

Remembering that (�0#��E)|�=0=(�|#��E )=(T#�2) (half the time-period of theclassical particle along #), the implicit function theorem allows us to consider(:, ;, E) [ (:, ;, s) as an analytic change of variables (analytic diffeomorphism); it

Fig. 5. Stokes splitting of the simple oscillator region: (a) :<0; (b) :>0. (c) Stokes pattern in theq-plane, for (:, ;, E ) in the lower region (I), upper region (II), or ``separatrix'' (III).



Fig. 5��Continued

transforms the collection of energy levels into the lattice of horizontal planess=n # N.

v The higher region is that part where the complexified energy shell has onecomplex fading cycle #0 , which can be seen as the continuation of the tunnel cycleacross the unstable layer (crossing the unstable layer makes the middle pair of realturning points become complex conjugate). The Stokes pattern in the q-plane thenlooks as shown on Fig. 5c.II. Due to the fact that the oscillator cycle # has a non-zero intersection index with this fading cycle #0 (cf. [DDP1]), the corresponding



Voros multiplier a# (and therefore s) is not Borel-resummable any more; its Boreltransform has singularities on the real axis at integral multiples of u, where wedenote by iu the (pure imaginary) action integral along #0 . But like any resurgentseries it can be resummed by ``right'' or ``left'' resummation (analogous to Borelresummation but for the fact that the integration axis runs slightly above (resp.below) the real axis), or more conveniently here by Ecalle's median resummation(kind of a ``geometrical mean'' between the two), which has the advantage ofpreserving reality properties.6

Understanding s now as the median resummation of the corresponding symbol,one can prove (using arguments analogous to those developed in [DDP2]) thatthe exact quantization condition reads

sin ?s+cos(?;�2�)

- 2 cosh(U�2�)=0, (1.1)$

where U=u+O(�2) is the (Borel resummed) real valued resurgent function definedby

0#0=iU (1.2)

(that the corresponding series is Borel-resummable follows from Theorem 2.5.1 of[DP], because #0 has a zero intersection index with itself !).

1.2. Regularity near the Stable Layer

The action integral |# is holomorphic near the stable layer, where it vanisheswith a simple zero (the time-period T#=2(�|#��E) has a strictly positive limitT(:, ;) when # vanishes). The same is of course true for 0# . In addition, thedependence of 0# on the parameters (:, ;, E ) is regular7 near the stable layer. Thisimplies that such operations as

v rescaling the energy near the (minimal ) critical value Vcrit(:, ;) of the potentialfunction, i.e., making the substitution

E=Vcrit(:, ;)+Eresc�, (1.3)

v replacing the rescaled energy by a resurgent expansion

Eresc=E0+E1�+E2 �2+ } } } , (1.4)

when performed at the formal level, can be directly interpreted as operations onfunctions through the resummation operations.


6 For the definition of median resummation see [DP, Section 0.5], for instance. Some examples ofcomputation of median symbols are given in [DDP2, Section IV].

7 In the sense of [DP] (regular dependence of a resurgent function on parameters). Here regularitystems from the fact that the cycle #~ is not pinched by the pair of coalescing turning points.


Noticing that (�s��Eresc)|�=0=(1�4?) T(:, ;) differs from zero (and is independenton Eresc), it is easy to see that for every natural integer n Eq. (1.1) has a uniqueformal solution of the form (1.4) with E0=(4?�T(:, ;))(n+ 1

2) (the Rayleigh�Schro� dinger series). By the ``implicit resurgent function theorem'' the aboveregularity property allows us to conclude that the Rayleigh�Schro� dinger series isresurgent (in our case, Borel resummable) and that its Borel sum is an exact solutionof the quantization condition; cf. [DDP2].

2. EXACT QUANTIZATION OF DOUBLE OSCILLATORS

The exact quantization condition for the double oscillator can be written in thegeneral form

(sin ?s+)(sin ?s&)=cos ?(s++s&)(1+a#0)1�2&1

2, (2.0)

where s& and s+ are the monodromy exponents of the left and right oscillatorcycles #& , #+ , whereas a#0 is the Voros multiplier of the tunnel cycle #0 . Here agains& and s+ are not Borel resummable, their Borel transforms having singularities onthe real axis at integral multiples of u (the action integral along #0 , divided by i).It must be understood that the symbols in this formula should be interpretedthrough median resummation (which, in the case of a#0, is just Borel resummation).

Such a relation is useful only inasmuch as all its ingredients depend regularly on(:, ;, E). This is always the case for s+ and s&. This is also the case for a#0 in thedouble oscillator region (Section 2.1 hereafter), but not near the metastable layeror the Maxwell edge, where our demand for regular dependence will force us toreexpress a#0 in terms of other variables, not so simply related to it as the variableU� =0#0

�i�=&ln(a#0) which we use in Section 2.1.

The Relation between left and right exponents. The same ``Cauchy integral''reasoning as in Subsection 0.1 holds with |#\

replaced by 0#\, so that

0#&&0#+

=?;,

or equivalently,

s&&s+=;�2�. (2.1)

2.1. Quantization in the Double Oscillator Region

In the double oscillator region none of the cycles #+ , #& , #0 vanishes. Definingthe (Borel-resummed) real valued power expansion U as previously by 0#0

=iU,and setting U� =U��, the quantization condition can be written as

(sin ?s+)(sin ?s&)=cos ?(s++s&) =(U� ), (2.2)



Fig. 6. The avoided crossing phenomenon in the double oscillator region.

where =(U� ) is the exponentially small function defined by

=(U� )=(1+e&U� )1�2&1

2 \t14

e&U� + .

Considering s+ , s& , U� as free variables, we call (2.2) the model form of the quantiza-tion condition. For fixed U� it defines a doubly periodic lattice of curves exhibitingthe avoided crossing phenomenon shown on Fig. 6, where ;� =;��=2(s&&s+), ands=(s&+s+)�2.

Notice that for large U� this lattice of curves depends very slowly on U� (whendifferentiating Eq. (2.2), the differential of U� is multiplied by the small expo-nential e&U� ).

To translate this picture in terms of the (:, ;, E ) variables one just has to under-stand it as the image of the ``true'' picture through the following changes ofvariables (analytic diffeomorphisms):

1. replace (:, ;, E) by (|#+, |#&

, u=|#0�i) (the Jacobian of this change of

variables is a nonzero constant; cf. [DDP1, Section 2.1]);

2. replace (|#+, |#&

, u) by their semiclassical incarnations (0#+, 0#&

, U),interpreted as true functions of (:, ;, E ) by Borel resummation; by the implicitresurgent function theorem this is a resurgent change of variables, tangent to iden-tity as � � 0;

3. get (U� , ;� , s) by obvious linear rescaling.

Complex branch points of the energy levels. By straightforward trigonometry wecan rewrite Eq. (2.2) as

(1+=(U� )) sin2(?s)=sin2 ?;�2

+=(U� ). (2.2)$



It is clear from this equation that s can be seen as a multivalued analytic functionof (U� , ;� ) (periodic in ;� of period 2), with square root branch points at

;� =2&\2i?

sinh&1- =(U� ) (2.3)

(& # Z, the set of integers). We can interpret these branch points by saying that

the ``avoided crossing'' phenomenon is but the real trace of ``truecrossings'' at complex values of ;� , exponentially close to the real axis.

Getting back to the (:, ;, E ) picture, a similar statement holds for E as a multi-valued function of (:, ;) with the following caution.

Caution ! In comparing the branch points in the (U� , ;� , s) and (:, ;, E ) picturesone must not forget that projecting on (U� , ;� ) parallel to the s-axis and projectingon (:, ;) parallel to the E-axis are not exactly equivalent operations, because U�depends on E and not only on (:, ;). But as already noticed the quantization con-dition (2.2)$ depends on U� only through the exponentially small function =(U� )=t

14 e&U� , whose derivative with respect to E is also exponentially small:

\ ddE

=(U� )=dU�dE

=$(U� )t&14

dU�dE

e&U� + .

Taking this fact into account one checks that the branch points of E are given byan expression which almost coincides with (2.3), but for the fact that to the small=(U� ) term in it must be added a still smaller correction, of the order of magnitude&(1�16?2)(dU� �dE)2 =$(U� )2.

2.2. Quantization Near the Metastable Layer

What we call the metastable layer corresponds to the vanishing of one of the twooscillator cycles in the double oscillator. Since the tunnel cycle is pinched by thisvanishing cycle, a#0 no longer depends regularly on (:, ;, E), so that the result of anenergy rescaling

E=Vcrit(:, ;)+Eresc�

is not so directly computable from the formal expression of a#0. As shown in[DDP2] such a rescaling results in a formal expression for a#0 which is no longergiven by an integral power series of � but by an expansion of the form (2.4) whichwe call the critical Voros multiplier (we assume here that the vanishing cycle is theone on the right, so that s+ is no longer large),

a#0=&- 2?

1(1+s+) \c+

� +s++1�2

e&ucrit ��(1+O(�)), (2.4)



where ucrit=ucrit(:, ;) is the value of u=|#0�i for E=Vcrit(:, ;) and c+=c+(:, ;)

is the ``critical action multiplier,'' exactly definable in terms of classical mechanics(cf. [DDP2, Section IV]). The (1+O(�)) factor is a resurgent integral power seriesexpansion in �, depending regularly on (:, ;, Eresc).

Consequently, defining the (large) parameter

V� +=ucrit

�+\s++

12+ ln

ucrit

c+

+0(�)

by the implicit equation

(1+a#0)1�2&1=&- ?�2

1(1+s+)V� s++1�2

+ e&V� + (2.5)

and using Euler's complement formula we can rewrite the quantization condition(2.0) in the form

- 2?1(&s+)

sin ?s&=12

cos ?(s++s&) V� s++1�2+ e&V� +, (2.6)

where one should remember that s+ and s& are linked by the relation s&&s+=;�2�.Here again one gets an ``avoided crossing'' phenomenon (cf. Fig. 7), which can beanalysed in complete analogy with Subsection 2.0 (the details are left to the reader).

2.3. Quantization Near the Maxwell EdgeThe Maxwell edge corresponds to the vanishing of both oscillator cycles. This

means that ; is assumed to be of order O(�), and it is therefore convenient to work

Fig. 7. The avoided crossing phenomenon near the metastable layer. The oscillations around thestraight lines have been exagerated, just for the fun of making them visible.



with the rescaled asymmetry parameter ;� =;�� (throughout this paper putting ahat on a letter will mean making it a dimensionless quantity by dividing it by thesuitable power of ��e.g., by � for an `àction-like'' quantity).

Since the tunnel cycle is pinched by both vanishing cycles, the critical Vorosmultiplier a#0 is now given by an expansion of the form (cf. [DDP2])

a#0=2?

1(1+s+) 1(1+s&) \c�+

s++s&+1

e&ucrit ��(1+O(�)). (2.7)

Consequently, defining the (large) parameter

W� =ucrit

�+(s++s&+1) ln

ucrit

c+O(�) (2.8)

by the implicit equation

(1+a#0)1�2&1=?

1(1+s+) 1(1+s&)W� s++s&+1e&W� (2.9)

and using Euler's complement formula we can rewrite the quantization condition(2.0) under the form

2?1(&s+) 1(&s&)

=cos ?(s++s&) W� s++s&+1e&W� (2.10)

which again exhibits the `àvoided crossing'' phenomenon (cf. Fig. 8). Since theright-hand side is exponentially small (for large W� ), the solutions of that equationare close to the zeros of the left-hand side, which occur for all natural integralvalues of s+ or s&; near a crossing point where both s+ and s& are natural integers(s+=n+ , s&=n&), a local model for the `àvoided crossing'' can be obtained byexpanding both sides of (2.10) in Taylor series of the variables

x=;�4

+n+&n&

2

y=s&n++n&

2.

After factoring out the numerical coefficients, one gets

y2&x2=W� n++n&+1e2y ln(W� )

2?n+! n&!e&W� (1+O(x, y)) (2.11)

(where the O(x, y) term does not depend on W� ).



Fig. 8. The avoided crossing phenomenon near the Maxwell edge. The oscillations around thestraight lines have been exagerated, just for the fun of making them visible.

The Zinn�Justin quantization condition. In the special case of the symmetricaldouble oscillator (;� =0), Eq. (2.10) becomes the Zinn�Justin quantization condition(cf. [ZJ5])

- 2?1(&s)

=\- cos 2?s W� s+1�2e&W� �2. (2.12)

Again expanding both sides in Taylor series of y=s&n (n a natural integer) theresulting equation can be solved by iteration, yielding s as a convergent expansion

s=n\=̂n+ :�

k=2

rn, k(\=̂n)k, (2.13)

where \ must be understood as & for even eigenstates and + for odd eigenstates,and

=̂n=W� n+1�2e&W� �2

- 2? n !,

whereas rn, k=rn, k(ln W� ) is a polynomial in ln(W� ) of degree <k.Convergent expansion (2.13) is our ``model form'' for the Zinn�Justin ``multi-

instanton expansion.'' To obtain the latter from our ``model form'' all we have todo is replace W� by its (resurgent) expansion (2.8) and express E as a (resurgent)power expansion in s by inverting the relation s=s(E, �) (here again we use the``implicit resurgent function theorem.'' For the details, cf. [DDP2, Section V]).



Fig. 9. Complex branch points of the energy levels (bold lines are cuts).

Complex branch points of the energy levels. Fixing W� at some large value, thelocal branch point structure of s as a function of ;� (near the crossing point of then+ and n& levels) is easily read on Eq. (2.11). One finds that s has a pair of square-root branch points at complex conjugate positions ;� =bn&, n+

or b� n&, n+, where

Re(bn&, n+)&2(n&&n+) and Im(bn&, n+

)&4W� (n++n&+1)�2e&W� �2�- 2?n+! n& !.Looking at Fig. 8 it is easy to see how these local data patch together in a

neighbourhood of the real ;� -axis. Start with the ground state for which st&|;� |�4for real ;� ; this (real analytic) function prolongs in the complex ;� cut plane, witha pair of complex conjugate cuts starting from b0, 0 and b� 0, 0 (cf. Fig. 9 0)). Crossingany of these cuts leads us to a new determination of s, corresponding for real ;� tothe first excited state, holomorphic in the cut plane shown on Fig. 9i with two newpairs of cuts starting from b0, 1 , b� 0, 1 and b1, 0 , b� 1, 0 . Crossing again any of these newcuts leads to the second excited state (cf. Fig. 9ii), etc. Getting back to the (:, ;, E)picture, similar statements hold for E as a multivalued function of ; (for fixed :).Of course, proving them requires the same precautions as explained at the endof 2.0.

3. EXACT QUANTIZATION NEAR THE UNSTABLE LAYER

The unstable layer is characterized by the vanishing of the fading cycle #0 . Equiv-alently, it is the surface where the real analytic function U=0#0

�i vanishes (with asimple zero). For similar reasons as those expounded in 1.2 this (resurgent) func-tion is the Borel sum of the corresponding formal symbol, and depends regularly on(:, ;, E).

In a small neighbourhood of the unstable layer, of width O(�), a convenientsystem of coordinates is (|&

crit , |+crit , U� =U��), where |&

crit (resp. |+crit) is the action



integral along the left part #&crit (resp. right part #+

crit) of the critical trajectory (a lem-niscate, with an ordinary double point at the origin). The first two coordinates(|&

crit , |+crit) deserve being called shape parameters, because they only depend on the

shape of the potential (i.e., on (:, ;)) and not on E. Of course they can be replacedby (;, |crit=|&

crit+|+crit) (recall that ;=(|&

crit&|+crit)�?). It will turn out more

convenient to replace them by (;� =;��, X ), where X� =(|crit ��)+ } } } is a (Borelresummed) resurgent function to be defined below.

3.1. The Model Equation

As shown in [DDP2], the critical trajectory #+crit (resp. #&

crit) does not carry onecanonically defined ``critical Voros multiplier'' but two such, complex conjugates toeach other, which we shall denote by a+, a� + (resp. a&, a� &). They are related to oneanother by the relations

a+a� +=a&a� &=1+eU�

(3.0)a&=ei?;� a+,

so that knowing a+ is knowing all of them. One has

a+=- 2? eU� �4

1(1�2+i(U� �2?)) \c�+

i(U� �2?)

ei|+crit ��(1+O(�)), (3.1)

where c=c(:, ;) is the ``critical action multiplier,'' and the (1+O(�)) factor is aresurgent (Borel resummable) integral power series of �, depending regularly on(:, ;, Eresc).

The exact quantization condition reads

1+e&i?;� +a++a� &=0. (3.2)

To rewrite it in more workable form, let us introduce the parameter

X� =|crit

�&

U�?

ln|crit

c+O(�)

defined implicitly by the relation:

a+=- 2? eU� �4

1(1�2+i(U� �2?))X� i(U� �2?)ei((X� &?;� )�2). (3.3)

Setting

.(U� )=arg 1 \12

+iU�2?+ (3.4)



(the determination of the argument being so chosen that .(0)=0) and noticingthat

- 2? eU� �4

1(1�2+i(U� �2?))=- 1+eU� ei.(U� ),

we thus get the ``model form'' for our quantization condition,

cos?2

;� +- 1+eU� cos(%&.(U� ))=0, (3.5)

where % stands for the function of (X� , U� ):

%=12

X� +U�2?

ln X� . (3.6)

This is an analytic relation between three dimensionless parameters X� , ;� , U� , the firsttwo of which we shall keep on calling shape parameters, although X� depends on E(but infinitely slowly, through O(�) terms).

Illustrations in the symmetrical case (;=0). Figure 10 illustrates the ``quantiza-tion of U� ,'' for fixed X� >>0 and ;� =0: the ``quantized values'' can be read as theabscissae of the intersection points of the two curves y=&1�- 1+eU� and y=cos(X� +(U� �2?) ln X� &.(U� )). Notice that for U� >>0 the values are evenly spaced

Fig. 10. Quantization near unstability, for the symmetrical oscillator. For large X� the frequency ofthe oscillations is of the order ln(X� )�2? for limited values of U� ; it tends to a constant as |U� | tends toinfinity (but this is not visible at the scale of the picture).



Fig. 11. Quantization in the (X� , U� )-plane.

(as could be expected for a simple oscillator), whereas for U� <<0 they tend tocluster in pairs (as expected for a symmetrical double oscillator).

Figure 11 shows how these quantized values depend on X� ; the correspondinggraph can be deduced from the periodic lattice of curves cos(%&.(U� ))=&1�- 1+eU� (cf. Fig. 12) by the transformation

(%, U� ) W (X� , U� )

(an analytic diffeomorphism for large enough X� ). Therefore it is invariant under thediscrete group of transformations generated by (X� , U� ) [ (gU� (X� ), U� ), where gU� isthe X� variable interpretation of the translation % [ %+2? (for fixed U� ).

Fig. 12. Quantization in the (%, U� )-plane.



Another interesting fact which can be read from Eq. (3.5) is the density of levelsfor small �. The angle inside the cos reads 3��, where

3=X2

+U2?

ln \X� +&�. \U

� + .

For small U (of the order O(�)) one has

�3�U

& &1

2?ln \X

� +so that the density diverges logarithmically (in conformity with Weyl's rule), asemphasized by Colin de Verdie� re and Parisse [CdVP]. When |U | is not smallWeyl's rule predicts that the density of level should have a finite limit, and it canbe checked indeed that the above logarithmic divergence is then cancelled by thederivative of the &�.(U��) term (use the fact that .$(U� )tln( |U� | )�2? when|U� | � �, by Stirling's formula).

3.2. Complex Branch Points of U� as a Function of X� ( for ;=0)

In order to study the complex analytic continuation of our ``model'' quantizationcondition (3.5), it is convenient to rewrite it under the following equivalent form8

ei(%&?�(?�4))=G�(U� )1�2, (3.7)�

where

G�(U� )=1(1�2+i(U� �2?))

1(1�2&i(U� �2?))tan \?

4\i

U�4 + . (3.8)�

Here and in the sequel the symbol � (resp. \) must be understood as + for evenstates, & for odd states (resp. the other way round: mnemotechnically, just rememberthat even states stand below odd states).

G� is a meromorphic function, with double poles only, distributed along thepositive imaginary axis at the following positions u�

l (indexed by l=1, 2, 3, ...):

u+l =4i?(l& 3

4) (=i?, 5i?, 9i?, ...)(3.9)

u&l =4i?(l& 1

4) (=3i?, 7i?, 11i?, ...).

Since complex conjugation changes G� into the complex conjugate of its inverse,its zeros are double zeros, at the complex conjugate positions u� �

l .Notice that G� has modulus 1 along the real axis, with G�(0)=1. If the

determination of the square root in (3.7)� is so chosen that G�(0)1�2=1, the nth


8 Remember that .(U� )=(1�2i) ln[1(1�2+i(U� �2?))�1(1�2&i(U� �2?))] and that % stands for the right-hand side of (3.6).


bound state is characterized by the following analytic relations between the realquantities X� , U� :

%={\n+

34+ ?+

12i

ln G+(U� )

\n+14+ ?+

12i

ln G&(U� )

(even n)

(odd n).(3.10)n

Notice that these relations for different even (resp. odd) values of n are connectedto one another by analytic continuation in the complex U� -plane; more precisely,turning clockwise around a pole of G� (resp. anticlockwise around a zero of G�)results in adding 2 to the quantum number n. In other words, all the even nbranches of the curve in Fig. 11 are the real traces of one connected complexanalytic curve in C2, and the same statement holds for odd n branches.

Problem. Describe this complex curve, as a Riemann surface over the X� -complexplane, for ``large enough'' X� .

By ``large enough'' X� we mean the following: for |IU� |<4?L, with L any giveninteger, we demand that |X� |>CL (some constant depending on L), with |arg X� |<(?�2).

Answer. The following statements give an almost complete answer (since ourRiemann surface is invariant by complex conjugation, we only describe it in thelower half plane of X� , which corresponds to the upper half plane of U� ).

1. Position of the singularities.

Even case. The constant CL can be so chosen (as a function of L) that forevery L the above Riemann surface is a connected branched covering of theX� -plane, with only square root branch points (x (m)

l , u (m)l ), doubly indexed by natural

integers l, m subject to the only following restrictions:l=1, 2, ..., L;m is larger than some integer depending on l; and

m has the same parity as l. (+)

For every l as above, let m go to infinity; then the sequence (u (m)l ) tends to u+

l , theargument of u (m)

l &u+l tending to ?�2 from the >?�2 side; the sequence (x (m)

l ) tendsto infinity in the positive real direction, with

Rx (m)l =2? \m+

34++O \ln m

m +Ix (m)

l =&4 \l&34+ ln _2? \m+

34+&+ } } }

(the beginning of a convergent asymptotic expansion for large m).



Fig. 13. The n th inner sheet with n=8 (lines are cuts).

Odd case. The statements are analogous, with the only following differences:

v condition (+) should be replaced by

m and l have opposite parity ; (&)

v u+l should be replaced by u&

l ;

v and

Rx (m)l =2? \m+

14++O \ln m

m +Ix (m)

l =&4 \l&14+ ln _2? \m+

14+&+ } } } .

2. Sheet structure. Starting from the n th quantum level on the real X� axis (asgiven by Eq. (3.10)n), analytic continuation in the lower half X� -plane is possible inthe cut plane represented on Fig. 13. Let that cut plane be called the nth inner sheet.Then for every (l, m) the vertical cut starting from x (m)

l downwards is a borderline9

between the (m&l)th inner sheet and the (m+l)th inner sheet.


9 Free from singularities.


Proof of Statement 1. (Say, in the even case; G is short for G+). Logarithmicdifferentiation of (3.7) yields

\12

+U�

2?X� + dX� +ln X�2?

dU� =12i

d ln(U� ). (3.11)

For large |X� | this equation implies that dU� �dX� is everywhere small, except perhapsin a neighbourhood of the singular points of the right-hand side, namely u+

l

(hereafter we only consider the upper half U� -plane). Near every such singular pointthe ramification condition is obtained by equating the coefficients of dU� in (3.11),i.e.,

ln X�2?

=12i

G$(U� )

G(U� ). (3.12)

Since the right-hand side has a simple pole at u+l (with residue i), relation (3.12)

for U� close enough to u+l and R ln X� large enough defines U� as a holomorphic

function of ln X� , which we shall denote by U� l(ln X� ):

U� l(ln X� )=u+l +

2i?ln X�

+ } } } (3.13)

(the sum of a convergent series of inverse powers of ln X� ). For large enough Crelation (3.12) thus defines a conformal transformation between the domain[ |X� |>C, &(?�2)<arg X� �0], i.e., [R ln X� >ln C, &(?�2)<I ln X� �0], andsome tapered neighbourhood of u+

l , contained in a small disc [ |U� &u+l |<\l] and

satisfying ?�2�arg(U� &u+l )<(?�2)+o( |U� &u+

l | ).Substituting this function U� l in (3.7)+ (and taking the logarithms of both sides)

yields an implicit equation for X� , which can be written as

12

X� +U� l(ln X� )

2?ln X� =\m+

34+ ?+

12i

ln G(U� l(ln , X� )), (3.14)

where m is an even integer and the determination of the logarithm has been chosento be real along the real X� -axis.10

Key Lemma. For any given l and large enough C, relation (3.14) has a uniquesolution X� =x (m)

l in the domain ( |X� |�C, &(?�2)�arg X� �0).

Proof. Solving Eq. (3.14) is solving a fixed point problem X� =F(X� ), where

F(X� )=2? \m+34+&

U� l(ln X� )?

ln X� +1i

ln G(U� l(ln , X� )) (3.15)


10 G(U� ) is positive real all along the positive imaginary U� -axis.


and we shall try to chose our domain in such a way that |F $(X� )|<1. Remembering(3.12), we check that

F $(X� )=&U� l(ln X� )�?X� .

For large enough C we can assume that the image by U� l of our domain satisfies|U� l&u+

l |�\l<3? (say), so that |U� l |<4?l, and consequently,

|F $(X� )|<4l�C.

For C>4L(L�l) this will be smaller than 1, allowing us to solve our fixed pointproblem by successive iteration provided our domain is stable by F. To check thatthis is the case, it is more convenient to work with the variable Y=ln X� . Ourdomain becomes the horizontal band [RY�ln C, &(?�2)�IY�0], and the fixedpoint problem now reads Y=ln 8(Y ), with

8(Y )=2? \m+34+&

U� l(Y )?

Y+1i

ln G(U� l(Y ))

=2? \m+34+&

u+l

?Y+

2i

ln Y+bl(Y ), (3.16)

where bl(Y ) is a bounded function of Y in the band (remember that |U� l&u+l |�

some small constant \l and notice that this implies |ln G(U� l(Y ))&2 ln Y |<someconstant cl).

Denoting by bl some bound of the function bl(Y ), one thus easily checks thata sufficient condition for our band to be stable is

2? \m+34+�ln C+

?ln C

+(4l+1)+bl . K (3.17)

Having thus found x (m)l , the corresponding value u (m)

l of U� is given by u (m)l =

U� l(ln X� ). The square root nature of the branch point follows from the fact thatderiving (3.12) with respect to U� gives an equation 0=(1�i(U� &u+

l )2)+ } } } whichis not satisfied for U� =u (m)

l .Upper bounds on |(dU� �dX� )| warrant that U� cannot go to infinity for finite values

of X� .

Proof of Statement 2. Saying that x (m)l is a square root branch point of U� (X� ),

with value u(m)l , is equivalent to saying that u (m)

l is a quadratic critical point of thereciprocal function X� (U� ), with critical value x (m)

l . As we just saw, this critical pointis close to u+

l (for m large), and the direction in which it is seen from u+l is close

to the pure imaginary direction. Consequently, since dX� �dU� is real along the pureimaginary U� -axis, the steepest descent lines of IX� going through the saddlepointu(m)

l will be close to the pure imaginary axis and will tend to the logarithmicsingularity u+

l or u+l+1 when IX� goes to &�; in other words, they make up a



smooth arc from u+l to u+

l+1, close to the pure imaginary axis, the image of whichin the X� -plane is the vertical half-line hanging from x (m)

l downwards (countedtwice).

We now claim that the cut half-plane of Fig. 13 is nothing but the conformalimage of the U� -upper half plane cut along the union (for l=1, 2, ...) of all such

Fig. 14. The right (resp. left) path in the U� -plane (top) is conformally mapped onto the left (resp.right) path in the X� -plane (bottom).



``steepest descent arcs.'' To understand why precisely those x (m)l shown on Fig. 13

mark the boundary of this conformal image, let U� climb up the imaginary axis,avoiding all logarithmic singularities by small half-circular detours (always on thesame side, never crossing the cuts); the corresponding paths in the X� -plane areshown on Fig. 14.

To understand how ``crossing to another sheet'' justs amounts to changing thevalue of n, notice that crossing ]ul , ul+1[ from left to right in the U� -plane increasesn by 2l.

Outer sheets. Completing the cut half-plane of Fig. 13 by symmetry, we get acut plane containing the real axis (which more fully deserves being called ``the nthinner sheet,'' rather than just its lower half part). Consider in this cut plane thevertical band Rx (n&1)

1 <X� <Rx (n+1)1 , which we shall call the nth band.

Along the right border of this band, consider the vertical cut hanging down fromx(n+1)

1 , or the complex conjugate cut soaring above x� (n+1)1 . Crossing either cut

leads us into the (n+2)th band, from which we can reiterate the procedure. Wethus see that the union of all the n th bands, when n runs over the set of integersof the given parity, is the cut plane represented on Fig. 15 (please ignore the brokenlines). We shall call it the main (or zeroth) outer sheet.

A similar construction with the vertical band between x (n+1)1 and x (n+2)

2 yieldsthe cut plane also shown on Fig. 15 (full and broken lines), which we shall call thefirst outer sheet. More generally, the lth outer sheet will have two series of cutsindexed by an even (resp. odd) integer n ; the smaller cuts [x (n+l)

l , x� (n+l)l ] separate

it from the (l&1)th outer sheet; the larger ones [x (n+l+1)l+1 , x� (n+l+1)

l+1 ] separate itfrom the (l+1)th outer sheet.

Fig. 15. The main and first outer sheets.



3.3. Back to the Energy and ``Shape'' Variables

Still assuming that ;=0, let us examine which information the above study givesus about quantization in the (:, E, �) variables. By the quasihomogeneity propertyrecalled in the Introduction, the quantization condition depends only on the coupleof variables

(:̂=:��2�3, E� =E��4�3). (3.18)

The relation between these variables and our ``model'' variables (X� , U� ) can be writ-ten in the form

U� =?E� (&:̂)&1�2+\&3?8

E� 2(&:̂)&5�2+3?8

(&:̂)&3�2++\35?

64E� 3(&:̂)&9�2&

85?64

E� (&:̂)&7�2++ } } } = :p�0

U� p(E� , :̂), (3.19)

where U� p is quasihomogeneous of weight p (:̂ being given weight 23 and E� weight 4

3),

X� =43

(&:̂)3�2+U�?

ln 6+ :p�1

X� p(U� )(&:̂)&3p�2. (3.20)

The series (3.19) and (3.20)11 are ``resurgent'' in the following sense: for (3.20) wemean ``resurgence'' in the usual (Ecalle) sense, the resurgence variable being(&:̂)3�2=(&:)3�2��, and the dependence on U� being regular; for (3.19) what wemean is that after the substitution (3.18) one gets a resurgent object in 1��, dependingregularly on (:, E).12

One should notice that X� in formula (3.20) does not just depend on the ``shape''parameter :̂ but also on U� , so that the ramification condition (3.12), which givesthe branch points of the quantized levels when projected on the complex X� -plane,does not exactly give the branch points of the projection on the complex :̂-plane;rewriting dX� in Eq. (3.11) as a combination of d:̂ and dU� (obtained by differentiating(3.20)), one sees that the condition for the dU� terms to cancel out now reads underthe modified form

ln X�2?

+ln 6

? \12

+U�

2?X� + (1+ } } } )=12i

G$(U� )

G(U� ). (3.21)

For large X� this modification of the ramification condition (3.12) is numericallysmall and does not affect the conclusions of Section 3.2.


11 To understand the first two terms in formula (3.20), notice that for ;=0 we have |crit=43 (&:)3�2

and c=6|crit , where c is the ``critical action multiplier'' in Eq. (3.1).12 The domain in which this regularity property holds will be made precise in the next section. Apart

from this regularity property, which they do not mention, this is what Sternin and Shatalov call a``resurgent function of two variables'' [StSh].


All our numerical computations (leading to Figs. 13, 14, and 15) have been madewith this modified equation (modifying Eq. (3.14) accordingly). Numerical evidenceshows that our iteration scheme for solving Eq. (3.14) converges not only for largeenough m (as proved in the ``key-lemma'' of Section 3.2) but for every l, m withm�l.

Translating these numerical results back to the (:̂, E� )-variables raises the difficulttask of computing numerically the Borel sums of the resurgent series (3.19) and(3.20). For large enough |:̂|, the crude approximation consisting of just keeping thefirst few terms in Eqs. (3.19), (3.20) should already give good results. As we shallsee, comparison with data from other sources shows that the agreement is goodindeed, not only qualitatively but to a large extent also quantitatively, even for smallvalues of |:̂|.

A good understanding of the multivaluedness of the energy levels should not belimited to the domain in (:̂, E� )-space, where our ``unstable layer model'' is valid. Ithas to be set globally in C2, using the various charts at our disposal. This will bethe aim of the next (and last) section.

4. GLOBAL STUDY OF THE SYMMETRICAL CASE

Up to this point our study was a purely local one and not applicable at placesinvolving turning points of multiplicity higher than 2, namely near the cuspidaledge (Fig. 2) and, of course, near the origin of the swallowtail. In order to illustratehow the results of this local study can be patched together, we shall restrict ourattention to the symmetrical case ;=0. Partly because of its relative simplicity andpartly for physical reasons, this case has received much attention from physicists,so that one may find special interest in understanding how our methods give apractically complete description of it.

By the quasihomogeneity property recalled in the Introduction, the quantizationcondition can be considered in this case as a relation between two variables only,namely

:̂=:��2�3, E� =E��4�3.

Using the methods of [DDP2] we can write this relation under various modelforms, valid in different domains of C2 (the complex (:̂, E� )-space). As exemplifiedin the previous sections, we can build two kinds of models:

v the generic models, which use WKB analysis with only simple turningpoints; typical examples are Eqs. (1.1) and (1.1)$ of Section 1.1, and Eq. (2.2) ofSection 2.0 (specialized here by setting ;� =0, since we deal with the symmetricalcase);



v the critical models, which focus near critical values Ecrit of the energy suchthat the WKB analysis involves double turning points: Ecrit=&:2�4 as in the``Zinn�Justin'' model (Eq. (2.12) of Section 2.2); Ecrit=0 as in the `ùnstable layer''model of Section 3.

At this stage it is natural to ask what are the respective domains of these models,and which part of C2 they cover altogether. One may also wonder whether ourinitial emphasis on real values of the parameters did not lead us to forget someimportant pieces in our collection of models. To answer the latter question, let usrecall that the collection of all models provided by [DDP2] is labelled by allpossible topological types of ``Stokes patterns''13 in q-space, having only simple ordouble turning points. Since the critical Stokes patterns are the most easily drawn(because of their symmetries), we shall focus our attention on the critical models,which will turn out carrying enough information to give us a clear global pictureof the ramification properties of the energy levels.

4.1. A Complete List of the Critical Models

The critical models are of two kinds, depending on which component of the criti-cal curve E(E+:2�4)=0 one focuses.

1. For Ecrit=0, :{0 one has one double turning point q=0 (and two simpleones, symmetrical with respect to the origin). It is easily checked that the topologi-cal type of the Stokes pattern depends only on how : is situated with respect to thethree half-lines arg(:)=0 mod(2?�3): the ``left sector,'' which contains the negativereal axis, bears the `ùnstable layer model'' of Section 3; the two ``right sectors,''separated by the positive real axis, bear two quantization conditions which areformally similar to Eq. (1.1),

s(:, Eresc�)=n # N, (4.1)

but which differ analytically in a subtle way. The Borel sum of the symbol s forarg(:)>0 is not the analytic continuation of its Borel sum for arg(:)<0, becauseof a Stokes phenomenon which occurs at arg(:)=0 (Borel resummability of sbreaks down there); still, no Stokes phenomena occur at arg(:)=0 for the solutionsof (4.1), which are Borel-resummable for &2?�3<arg(:)<2?�3 (cf. [DDP2,Section V.A.2, Remark]).

2. For Ecrit=&:2�4, :{0 one has two double turning points (symmetricalwith respect to the origin). The topological type of the Stokes pattern depends onlyon how : is situated with respect to the three half-lines arg(:)=? mod(2?�3): the``right sector'' &?�3<arg(:)<?�3 bears an `èmpty'' model, where the quantization


13 We shall say that two Stokes patterns have the same topological type if one can transform one intothe other by a homeomorphism of the complex q-plane preserving the asymptotic directions of Stokeslines at infinity (e.g., the positive real direction is transformed into itself ).


condition is impossible; the two ``left sectors,'' separated by the negative real axis,bear two quantization conditions which read

- 2?1(&s)

1ei?s=\\c

�+s+1�2

e&ucrit �2�(1+O(�)) (4.2)

for the `ùpper left'' sector, and the complex conjugate condition for the ``lower left''sector. The \ sign on the left-hand side selects the parity of quantum levels (cf.[DDP2, Section V.B]). For real negative : a Stokes phenomenon occurs, and thelack of Borel resummability suggests using median symbols, leading to Eq. (2.12).

4.2. Domains of the Critical Models

For every critical model the ``model variables'' are resurgent functions of 1��,depending regularly on the shape parameter : and the ``rescaled'' energy Eresc=(E&Ecrit)��, for every Eresc # C and every : in the sector ``bearing'' the model(cf. Section 4.1). Our problem here is to translate this information in terms of thevariables :̂=:��2�3 and E� =E��4�3. Resurgence with respect to 1�� implies that Borelresummation yields holomorphic functions of � for &?�2<arg(�)<+?�2, |1��|>{(arg �), where the `ìndicatrix'' function { is the function of %=arg(�) giving theexponential type of the Borel transform in the direction arg(!)=&% (we denoteby ! the dual variable of 1��). In the case of WKB expansions there are goodreasons to believe that the Borel transforms are bounded at infinity,14 so that { isidentically 0. This implies that our ``model variables,'' inasmuch as they are givenas entire functions of WKB expansions, are holomorphic functions of (:̂, E� ) for allE� # C and all :̂ in a sector twice as big as the :-sector. For instance the expansions on the left-hand side of Eq. (4.1), Borel-resummed for 0<arg(:)<2?�3, yields aholomorphic function, defined for &?<arg(:̂)<?�3; the expansion U� of Section 3,Borel-resummed for |arg(:)&?|<?�3, yields a holomorphic function of (:̂, E� ) for|arg(:̂)&?|<2?�3. Notice that the domain of holomorphy may be smaller in thecase of a ``model variable'' whose definition is implicit, such as the X� variable ofSection 3.1. Although the quantization condition (3.2) is well defined in the fullsector |arg(:̂)&?|<2?�3, its ``model form'' (3.5), (3.6) is only valid in that sectorfor large enough |:̂|, because the implicit definition (3.3) of X� only makes sense forsmall enough |�|.

4.3. Matching the Models

It follows from the above discussion that apart from some boundedneighbourhood of :̂=0 the whole complex (:̂, E� )-space is covered by the domainsof the three critical models focused around Ecrit=0, namely:

v the `ùnstable layer'' model ( |arg(:̂)&?|<2?�3);

v the two ``simple well'' models (&?<arg(:̂)<?�3; &?�3<arg(:̂)<?).


14 This was conjecture by Voros [Vo1] on the basis of numerical evidence. Then J. Ecalle outlined ageneral proof, unfortunately unpublished yet.


The domains of the latter two models contain the positive real :̂ region, where thepotential function is indeed a ``simple well.'' Their union covers the complex :̂ planeminus the negative real-axis.

Looking at Eq. (4.1) it is not difficult to check that no ramification of the energylevels can occur outside some neighbourhood of the negative real axis (whose sizemay depend on E� ). Just write

�s�Eresc

=�|#

�E+O(�)=T#+O(�)

and notice that T# (the time period of the oscillator cycle for E=0, i.e., at thebottom of the well) is a nonvanishing function of :. Of course the rate of decreaseof the O(�) correction when � � 0 depends on : and Eresc and can be controlleduniformly only when : is kept inside a compact (closed bounded) subset of itssector (and |Eresc | is bounded).

On the other hand, our analysis in Section 3 precisely describes the ramificationof the energy levels in a neighbourhood of the negative real axis, at least for largeenough |:̂| (depending on |E� | ).

Of course precise bounds should be computed in order to evaluate the sizes ofthe above-mentioned neighbourhoods and to check that the neighbourhood, where�s��Eresc is not known to differ from 0, is contained in the neighbourhood whereour analysis of Section 3 applies. We have not done it, but as we shall now show,taking this for granted leads to conclusions which are in complete agreement withwhat is known from other sources, thus giving a posteriori support to ourhypothesis.

To get a complete portrait of the global ramification, all we have to do now is``match'' the solutions of Eq. (4.1) with those of Section 3. This ``matching problem''is easily solved by the following complexified version of the Bohr�Sommerfeldquantization rule. Near infinity in the :̂ plane, not too close to the singularities, theexact quantization condition in any chart can be rewritten under the form

|̂#=n+ 12+O(�) (n=0, 1, 2, ...), (4.4)

where |̂#=|#�� is the (rescaled) classical action integral along a suitable complexcycle # (depending continuously on (:̂, E� ) in the given chart).

Comparing leading terms, it is thus easy to see that the solution of Eq. (3.5)corresponding to what we called the ``n th outer sheet'' (Fig. 15) is the analytic con-tinuation of the solution of Eq. (4.1) with the same quantum number n. The resultsof Section 3 thus lead us to the following portrait of the ramification.

4.4. Global Ramification of the Energy Levels

v Outer continuation. Starting along the positive real :̂ axis (the simple well),consider the ground state energy (n=0). It prolongs as a holomorphic function inthe cut domain shown on Fig. 16, the left part of which (in the ( |arg(:̂)|>(2?�3))



Fig. 16. The Bender and Wu main sheet.

sector) is just the conformal image of the ``main outer sheet'' by the change ofvariable X� [ :̂. Forgetting about the existence of an ``unknown zone'' around :̂=0,where our proofs do not apply, we see that our cut domain looks exactly like thecut plane predicted by Bender and Wu. We shall call it the Bender and Wu mainsheet.

The cuts bordering the Bender and Wu main sheet form an infinite sequence(indexed by the integer m of Subsection 3.2); the distance of the m th cut to theorigin grows like m2�3, whereas its length tends slowly to zero like ln(m)�m1�3. Lookingat the picture near infinity by the change of variable :̂ [ &1�:̂, what we see is the``horn of singularities'' predicted by Bender and Wu.

Leaving the Bender and Wu main sheet through any of the cuts will take us intoa smaller domain, with an additional series of cuts intertwinned between the pre-vious ones as shown on Fig. 17, the ``Bender and Wu first offspring,'' as we call it.The left part of this domain is just the conformal image of what in Section 3 wecalled the first outer sheet, whereas its right part (for positive real :) correspondsto the first even excited state (quantum number n=2).

Fig. 17. The Bender and Wu first offspring.


File: 595J 573735 . By:SD . Date:17:11:97 . Time:09:30 LOP8M. V8.0. Page 01:01Codes: 2168 Signs: 1428 . Length: 46 pic 0 pts, 194 mm

Crossing any of the new cuts will lead us into the next even state n=4, etc.,and similar statements hold for odd states if we start with the n=1 state. Thecorrespondence between the outer sheets of Subsection 3.3 (( |arg(:̂)|>(2?�3) sector)and the quantum number n of the simple well (positive real :̂ axis) may besummarized by the following table:

Even states Odd states

Main outer sheet n=0 n=1First outer sheet n=2 n=3

Second outer sheet n=4 n=5} } } etc.

v Inner continuation. Suppose now that, instead of bypassing the ``unknownzone,'' we cross straight through it along the real axis (all energy levels are, ofcourse, known to be analytic functions of :̂ along the real axis). Then our discus-sion in 3.2 leads to the following conclusion.

For any given quantum number n (starting on the positive real :̂ side) the nthenergy level is holomorphic in a cut domain looking as shown on Fig. 18, the leftpart of which is just the conformal image of the ``n th inner sheet'' of Section 3.

Figure 20 shows the union for all even n of the corresponding sets of branchpoints; it looks like a regular lattice filling the ( |arg(:̂)|>(2?�3)) sector, as pre-dicted by Shanley (who determined it very precisely for |:̂|<20).

The fact that our picture for fixed n (Fig. 18) does not look like Shanley's [Sh1]should not be a surprise, because our conventions for choosing the cuts are different

Fig. 18. The domain of inner continuation.



Fig. 19. An avatar of Fig. 13.

Fig. 20. The lattice of all branch points.


. .


Table I. Comparison of P. Shanley's Results [Sh1, Section 6, Table VII] andWKB Estimates for the Singularities with the Smallest Imaginary Part

Eigenvalues labels Shanley's results WKB estimates

n=0, 2 &4.1937+2.1697i &4.1177+2.2275in=2, 4 &6.8432+1.8945i &6.8101+1.9072in=10, 2 &14.5470+1.4974i &14.5395+1.4985in=50, 52 &39.0374+1.0652i &39.0363+1.0652in=100, 102 &61.2629+0.9037i &61.2625+0.9037in=140, 142 &76.4159+0.8324i &76.4156+0.8324i

from his (his cuts were chosen in the radial direction from the origin). The simplestway to see that our result agrees with Shanley's is to get back to Fig. 13 and tryto figure out what happens if one changes the slopes of its cuts to &ln n (say)instead of &�. A little thought shows that the only singularities to be seen arexn&1

1 , xn&22 , xn&3

3 , ... and xn+11 , xn

2 , xn&13 , ...; the resulting conformal image in the

:̂-plane looks as shown on Fig. 19, and this is just Shanley's picture.

4.5. Conclusion

Our results are therefore in complete qualitative agreement with Shanley's (aswell as with Bender and Wu's), and it would be interesting to check to which extentthis agreement is also quantitative. Of course Shanley's results [Sh1] are limited toa finite range of values of n, whereas ours are asymptotic, and we have presentlyno theoretical proof that their domains should overlap.

Nevertheless Table I shows surprisingly good agreement even for small valuesof n. And yet, in translating our numerical results from the X� -plane to the :̂-plane,we made the very crude approximation of forgetting the O(�) corrections inEq. (3.20) (Section 3.3).

One should get still better agreement by replacing this lowest order WKBapproximation by the exact WKB series (which can be computed by the `èxactmatching method'' of [DDP2, Section IV]), and applying to this formal series suchnumerical resummation procedures as ``resummation to the least term,'' or its``hyperasymptotic'' refinements `à� la Berry and Howls'' [BeH].

Crude as they may be, our computations already suggest that the `ùnknownzone'' where our methods do not apply consists only of an `ùninteresting'' neigh-bourhood of zero, containing no branch points. The subsistence of such an`ùnknown zone'' is due to our technical inaptitude for handling turning points ofmultiplicity higher than 2.15 On the other hand, it is shown in [Ph] that thisinaptitude is not an inherent shortcoming of exact WKB analysis, but is due to our


15 Notice that the results of Voros [Vo1] on the potential q4 and the completely new `èxact quanti-zation'' scheme which he proposed more recently in [Vo3] for the potential q2M stand, so to speak, onthe opposite side to ours, in the sense that they tell us what happens in the very center of what we calledthe `ùnknown zone.''


poor knowledge of the special functions attached to turning points of higher multi-plicity. If one believes (as conjectured in [Ph]) that these special functions are newtranscendentals, not expressible in terms of known special functions, it is notastonishing that investigating the vicinity of :̂=0 would need to resort to com-pletely different methods (theoretical ones like Simon's [Si1], L#ffel and Martin's[LM], or numerical ones like Shanley's). Conversely the main theorem in [Ph]implies that such investigations, pushed to a complete understanding of the generalquartic oscillator (not just the symmetrical one), would provide us with thecapacity of exactly handling turning points of multiplicity 4 in any asymptoticproblem. In other words, working on the general quartic oscillator is not justinvestigating an interesting model in quantum physics, it is perfecting a tool for thewhole community of asymptoticians (in the same way as Stokes' works on the Airyfunction went far beyond just improving our understanding of the rainbow!). But,of course, special functions of three complex variables (:̂, ;� , E� ) are not easy tostudy, and the first task beforehand would be to study turning points of order 3,i.e., the general cubic oscillator (where only two complex variables are involved).This is also an interesting physical model, the simplest one exhibiting resonances;some of the illustrative examples in [DDP2] deal with it.

REFERENCES

[AlSi] G. Alvarez and H. J. Silverstone, Large-field of the LoSurdo�Stark resonances in atomichydrogen, Phys. Rev. A 50, No. 6 (1994), 4679�4699.

[BW1] C. M. Bender and T. T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969), 1231�1260.[BW2] C. M. Bender and T. T. Wu, Anharmonic oscillator. II. A study of perturbation theory in

large order, Phys. Rev. D 7 (1973), 1620�1636.[BHS] C. M. Bender, H. J. Happ, and B. Svetitsky, Phys. Rev. D 9 (1974), 2324.[BeH] M. V. Berry and C. J. Howls, Hyperasymptotics for integral with saddles, Proc. Roy. Soc.

Lond. A 434 (1991), 657�675.[CNP0] B. Candelpergher, C. Nosmas, and F. Pham, Premiers pas en calcul e� tranger, Ann. Inst.

Fourier 43, No. 1 (1993), 201�224.[CdVP] Y. Colin de Verdie� re and B. Parisse, Equilibre instable en re� gime semi-classique. II. Condi-

tions de Bohr�Sommerfeld, Ann. Inst. H. Poincare� Phys. The� or. 61, No. 3 (1994), 347�367.[DDP0] E. Delabaere, H. Dillinger, and F. Pham, De� veloppements semi-classiques exacts des niveaux

d'e� nergie d'un oscillateur a� une dimension, C. R. Acad. Sci. Paris Se� r. I 310 (1990), 141�146.[DDP1] E. Delabaere, H. Dillinger, and F. Pham, Re� surgence de Voros et pe� riodes des courbes

hyperelliptiques, Ann. Inst. Fourier 43, No. 1 (1993), 163�199.[DDP2] E. Delabaere, H. Dillinger, and F. Pham, Exact semi-classical expansions for one dimen-

sional quantum oscillators, J. Math. Phys. (1997), to appear.[DP] E. Delabaere and F. Pham, Resurgent methods in semi-classical asymptotics, Ann. Inst.

H. Poincare� Phys. The� or., to appear.[E1] J. Ecalle, ``Les fonctions re� surgentes,'' Publ. Math. D'Orsay, Universite� Paris-Sud 1981.05,

1981.06, and 1985.05.[E2] J. Ecalle, ``Cinq applications des fonctions re� surgentes,'' Publ. Math. D'Orsay, Universite�

Paris-Sud 84T 62, Orsay.[E3] J. Ecalle, Weighted products and parametric resurgence, in ``Analyse alge� brique des pertur-

bations singulie� res I: Me� thodes re� surgentes,'' Travaux en cours, Hermann, Paris, 1994,pp. 7�49.



[Fr] N. Fro� man, The energy levels of double-well potentials, Ark. Fys. 32 (1966), 79�97.[FHWW] K. W. Ford, D. L. Hill, M. Wakano, and J. A. Wheeler, Quantum effects near a barrier

maximum, Ann. Phys. 7 (1959), 239�258.[HS] E. Harrel and B. Simon, The mathematical theory of resonances whose widths are exponen-

tially small, Duke Math. J. 47 (1980), 845.[KL] D. Khuat-duy and P. Leboeuf, Multiresonance tunneling effect in double-well potentials,

Appl. Phys. Lett. 63, No. 14 (1993), 1903�1905.[LM] J. J. Loeffel and A. Martin, ``Proceedings of the R.C.P.,'' No. 25, Vol. II, Math. Department,

Strasbourg, 1970.[Ph] F. Pham, Confluence of turning points in exact WKB analysis, in ``The Stokes Phenomenon

and Hilbert's 16th Problem'' (B. L. J. Braaksma, G. K. Immink, and M. van der Put, Eds.),pp. 215�235, World Scientific, Singapore, 1996.

[PS] J. Poston and I. Stewart, ``Catastrophe Theory and Its Applications,'' Pitnam, London, 1978.[Si1] B. Simon, Coupling constant analyticity for the anharmonic oscillator, Ann. of Phys. 58

(1970), 76�136.[Si2] B. Simon, Large orders and summability of eigenvalue perturbation theory: A mathematical

overview, Int. J. Quant. Chem. 21 (1982), 3�25.[Sh1] P. E. Shanley, Spectral properties of the scaled quartic anharmonic oscillator, Ann. of Phys.

186 (1988), 292�324.[Sh2] P. E. Shanley, Nodal properties of the scaled quartic anharmonic oscillator, Ann. of Phys.

186 (1988), 325�354.[StSh] B. Sternin and V. Shatalov, ``Borel�Laplace Transform and Asymptotic Theory (Intro-

duction to Resurgent Analysis),'' CRC Press, Boca Raton, FL, 1995.[Vo1] A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst.

H. Poincare� Phys. The� or. 39 (1983), 211�338.[Vo2] A. Voros, Re� surgence quantique, Ann. Inst. Fourier 43, No. 5 (1993), 1509�1543.[Vo3] A. Voros, Exact quantization condition for anharmonic oscillator (in one dimension),

J. Phys. A: Math. Gen. 27 (1994), 4653�4661.[Wi] A. S. Wightman, ,4

& and generalized Borel summability, in ``Mathematical Quantum FieldTheory and Related Topics, Canadian Math. Soc. Conference Proceedings, 1988,'' Vol. 9,pp. 1�13.

[ZJ1] J. Zinn-Justin, The principles of instanton calculus: A few applications, in ``Recent Advancesin Field Theory and Statistical Mechanics, Les Houches, Session 39, 1982,'' North-Holland,Amsterdam, 1984.

[ZJ2] J. Zinn-Justin, Multi-instanton contributions in Quantum Mechanics (II), Nucl. Phys. B 218(1983), 333�348.

[ZJ3] J. Zinn-Justin, Instantons in Quantum Mechanics: Numerical evidence for a conjecture,J. Math. Phys. 25, No. 3 (1984), 549�555.

[ZJ4] J. Zinn-Justin, ``Quantum Field Theory and Critical Phenomena,'' Oxford Sci., Oxford, 1989.[ZJ5] J. Zinn-Justin, From instantons to exact results, in ``Analyse alge� brique des perturbations

singulie� res I: Me� thodes re� surgentes,'' Travaux en cours, pp. 51�68, Hermann, Paris, 1994.


annphys97

Documents

energy levels of theone

exact results

exact wkb method

complex wkb method

complex deformations

polynomial v

wkb analysis

complex domain