Semi-classical correspondence and exact results : the case of the spectra of homogeneous Schrödinger operators [erratum: J. Physique Lett. 43 (1982) p.159]

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Semi-classical correspondence and exact results : the

case of the spectra of homogeneous Schrödinger

operators [erratum: J. Physique Lett. 43 (1982) p.159]

André Voros

To cite this version:

André Voros. Semi-classical correspondence and exact results : the case of the spectra of homogeneous

Schrödinger operators [erratum: J. Physique Lett. 43 (1982) p.159]. Journal de Physique Lettres,

Edp sciences, 1982, 43 (1), pp.1-4. �10.1051/jphyslet:019820043010100�. �jpa-00232000�

L-1

Semi-classical

_{correspondence}

and

exact

results :

the

case

of the

_spectra

of

_homogeneous

Schrödinger

_{operators (*)}

A. Voros

_(**)

Service de

_{Physique Théorique,}

CEN

_Saclay,

91191 Gif sur Yvette Cedex, France

(Reçu

le

13 octobre

_{1981, accepte le 5 novembre 1981 )}

Résumé. 2014 La méthode

semi-classique

BKW est

_susceptible

en toute _{généralité} de

_produire

des

résul-tats exacts. Ceux-ci, dans le cas de

_{l’opérateur}

de

_Schrödinger

avec

_{potentiel homogène}

q2M,

_prennent

la forme d’une

_équation

fonctionnelle _{pour le} déterminant de Fredholm, ou bien d’identités

arithmé-tiques

satisfaites par la

fonction

Zeta du _spectre.

Abstract. 2014 The semi-classical WKB method is liable to

yield

exact results in full

_generality.

Such

results, in the case of the

_Schrödinger

_operator with a

_{homogeneous potential}

q2M,

_appear as a

func-tional

_{equation obeyed by}

the Fredholm determinant, or as arithmetical identities satisfied

_by

the Zeta function of the _spectrum.

LE

JOURNAL

DE

_{PHYSIQUE-LETTRES}

J.

_Physique

- LETTRES 43

(1982)

L-.1 -

L-4 Classification

Physics

Abstracts 03.65 - 02.90 ler JANVIER 1982,

We consider the

_following

_{Schrodinger equation}

on the real axis :

and we start more

specifically

with the case M = 2

(homogeneous quartic oscillator).

From the

_eigenvalue

_spectrum

_{{ Àn }neN}

of

_equation

_(1),

we build an entire

_function,

the Fred-holm determinant

_J(~),

and a

meromorphic

function,

the Zeta function

~(s) :

L-2 JOURNAL DE _{PHYSIQUE -} LETTRES

Analytic properties

and some exact results

_concerning

those functions

_already

appear in

[1-3]

(all

results

_quoted

as known are drawn from

[2]

unless stated

otherwise,

but the Zeta function

defined in

_{[2] equals}

C -4s/3

_’(s)

where C =

r(l/4)~(2/7c)~/3,

due _{to a} different normalization of

the

_spectrum).

One interest of such results is to extend classical formulas of

_{analysis :}

in the _case M = 1

(harmonic oscillator)

the

spectrum

consists of the

positive

odd

integers

and the Zeta function becomes

_{(1 -}

2-S)

times the Riemann Zeta function

_[4].

-This

note

_presents

a new

class

of exact results that we have drawn from the semi-classical

corres-pondence

between the

_{quantal equation (1)}

and the classical motion

_{governed by}

the

Hamilto-nian

_{( p2}

+

q2M).

This

_{correspondence}

is

usually

exploited

to build

_{asymptotic expansions}

of the

spectrum

for A -+ oo

_through

the WKB method. The

divergence

of the

_resulting

series

_[5]

streng-thens the belief that the semi-classical

_approach

is doomed to be

_approximate.

But this is not the

case,

as the mere extension of the WKB method to

_complex

variables

_{[6] actually gives}

control over

the

_summability

of semi-classical

_{expansions by}

the Borel

_{procedure [7],}

thus

_giving

rise to certain exact

_analytical

results. For

instance,

we have found that the determinant L1

_(À)

for M = 2 satisfies

the functional

_{equation (where j}

=

e2i 1t/3) :

of which we see no other direct

_{proof (the}

same calculation for M = 1 would _restore the reflection formula for the Euler Gamma

_function).

Since our derivation of

₍₃₎

is

_{lengthy [7]}

and

relies at a

point

on a

_yet

_{unproved regularity}

pro-perty,

a direct check of

(3)

is desirable to confirm the

_validity

of our

_approach,

which could then

find numerous

_applications

in

_physics.

Since

equation (3)

arises

by

the Borel resummation of a

divergent

series around A = _{oo, a} reasonable test is to

reexpand

it around A = 0

using

the

,

following

obvious

_Taylor

_series,

_convergent

for 0

I À

I

~o ~

It then follows that

_{equation (3)}

identifies each

_{~(3 n)}

to a

polynomial

in the

’(k)

(1 k

3

n) ;

this

_polynomial

is

_homogeneous

of

_degree

3 n if

_((k)

is

_assigned

the

_degree

k. For instance :

The annexed

table,

computed

by

the

numerical

method of reference

_[2],

_agrees

_perfectly

with the latter formulas for M = 2. Remark : _as

and as

_~(2)

is the sum of a

_{generalized hypergeometric sF 4}

series

(a

result

found jointly

with D. and

G.

_Chudnovsky

_[3]),

_{equation (5)}

constitutes a closed

analytical

expression

for

~(3)

in the case

M = 2.

Table I

and to consider

_jointly

the

_alternating

determinant and Zeta

functions,

which arise

_naturally

through

the reflection

_symmetry

of the even

_potential

q2M

_{[2] :}

The WKB

_analysis

of a Borel transform

_ofJ~(A)

around À = _oo likewise leads _to the functional relations :

...

The functional

_equation

for

_D(À)

itself is now

_just

a

_consistency

condition for the

_system

₍₁₁₎

written

_form

_{e41till À,}

etc... :

(this

could also be rewritten as a

_polynomial

in the

_D’s).

Now the

_expansion

of

₍₁₁₎

around A = 0

yields :

,

- at zeroth order : the relation

~(0)

=

7c~, which restores a known result :

-

at _any

_higher

order n,

an

expression

for

as a

_polynomial

either in the

_’(k)

alone or

_(at

one’s

_choice)

in the

_(P(k)

_alone,

for k n

_(those

L-4 JOURNAL DE _{PHYSIQUE -} LETTRES

of (M

+

_1),

(P(n)

_disappears

from the left hand side which then

_only

contains

_((n),

and vice versa

when M and 2

_n/(M

_{+ 1)}

are odd

_integers.

The first relations read :

We have checked

_numerically

several of the relations

(16-17)

and

_following,

_up to M = 4. The

_explicit

formulas known for

_{~(2 n)}

and

_{(P(2 n}

+

₁₎

in the harmonic case

_{[4] (M}

=

1),

_as well as

_{equations (5-6)}

for M =

2,

are just

special

instances of those

(the

existence of such relations also

suggests

that the

_spectrum

of

_{equation (1)}

has rich arithmetical

_properties).

Our results confirm that the

_complex

WKB method can be

_implemented

in an exact form for

analytic potentials

in one

_degree

of freedom.

Applied

to the

_{inhomogeneous}

anharmonic oscillator

( -

d2/dq2

+

Kq 2

+

q4)

as our last

example,

it leads to a

generalization

of

(3) :

with definition

₍₈₎

for D. It seems however more difficult to extract from

₍₁₈₎

than from

₍₃₎

nume-rical formulas that would be at the same time exact and

_interesting.

Another still harder

_problem

would be to see how much of this exact

_{approach persists}

in the

case of several

degrees

of freedom

(possibly nothing

for a

_{non-integrable system).}

Acknowledgments.

- We thank R.

Balian,

M. Gaudin and J. M. Maillard for fruitful discus-sions.

References

[1] PARISI, G., Trace Identities

_for

the _{Schrödinger Operator} and the WKB _{Method, Preprint} LPTENS _78/9

(Ecole Normale Supérieure, 1979).

[2] VOROS, A., Nucl.

_Phys.

B 165 ₍₁₉₈₀₎ 209.

[3]

VOROS, A., Oscillateur _Quartique et Méthodes

_{Semi-Classiques,}

Séminaire Goulaouic-Schwartz 1979-1980,

exposé

n° VI _(Ecole

_{Polytechnique).}

[4] ABRAMOWITZ, M., STEGUN, I. A.,

(Eds.),

Handbook _of Mathematical Functions,

Chap.

23

_(NBS

_1964, Dover _1965).

[5]

DINGLE, R. _{B., Asymptotic Expansions.} Their Derivation and _{Interpretation (Acad. Press.) 1973 ;} BALIAN, R., PARISI, G., VOROS, A., Quartic Oscillator, in _Feynman Path

_Integrals

_{(Proceedings,} Marseille

1978), Springer Lecture Notes in

_Physics,

Vol. 106 _(1979). [6] BALIAN, R., BLOCH, C., Ann.

_Phys.

85

₍₁₉₇₄₎

514 ;

KNOLL, J., SCHAEFFER, R., Ann.

_Phys.

97 ₍₁₉₇₆₎ 307.