Isochronous oscillatory motions and the quantum

spectrum

ABD RAOUF CHOUIKHA

University Paris-Sorbonne, Paris-Nord

Institut Galilee, LAGA

4 Cour des Quesblais, 35430 SAINT-PERE

FRANCE

Abstract: Necessary and suﬃcient conditions for isochrony of oscillatory motions introduced in the paper

”Physica Scripta vol 94, N 12” are discussed. Thanks to the WKB perturbation method expressions are

derived for the corrections to the equally spaced valid for analytic isochronous potentials.

In this paper, we bring some improvements and we suggest another quantization of the quantum spectrum.

These results will be illustrated by several examples

Key–Words: oscillatory motions, isochronicity, WKB method, quantum spectrum

1 Introduction

Consider the Schrodinger equation

HΨ = EΨ

Ψ being the wave function associated with the

state of the particle and

H=p2

m+G

the Hamiltonian operator who describes this evo-

lution.

In the sequel we are interested in the case of

isochronous potentials. This means the frequency

of the classical motion in such potentials is energy-

independent, it is natural to expect their quan-

tum spectra to be equally spaced. However, as it

has already been shown in some speciﬁc examples,

this property is not always true.

This second order partial diﬀerential equation

is linear and homogeneous. This is not trivial

to solve in the case of complex potentials, apart

from numerical resolution. There is however an

approximate method of resolution, the WKB ap-

proximation, named after the physicists Wentzel,

Kramers and Brillouin. This approximation

is based on the fact that the solutions of the

Schrodinger equation can be approximated

by a function comprising usually conventional

quantities, provided that the potential does not

vary strongly over distances of the order of the

length of wave.

These is a connection between classical and

quantum transformations. This fact has been

established by Eleonskii and al. [2]. They

show that the classical limit of the isospectral

transformation for the Schrodinger equation

is precisely the isochronicity preserving the

energy dependence of the oscillation frequency.

In quantum mechanics, the energy levels of a

parabolic well are regularly spaced by a certain

quantity. Moreover, it is possible to construct

potentials, essentially diﬀerent from the parabolic

well, whose spectrum is exactly harmonic.

The semiclassical WKB method is one of

powerful approximations for computing the

energy eigenvalues of the Schrodinger equation.

The ﬁeld of its applicability is larger than

standard perturbation theory which is restricted

to perturbing potentials with small coupling

constants. In particular, it permits to write the

quantization condition as a power series in ¯h

(such series are generally non convergent). The

solvable potentials are those whose series can be

explicitly summed. This problem has motivated

a lot of authors who highlight some exactly

solvable, in a sense that the exact eigenenergies

and eigenfunctions can be obtained explicitly, see

[3] for example. Our method described below

permits also another approach of two-dimensional

superintegrability, see [4].

Received: April 17, 2022. Revised: November 11, 2022. Accepted: December 12, 2022. Published: December 31, 2022.

International Journal on Applied Physics and Engineering

DOI: 10.37394/232030.2022.1.10

Abd Raouf Chouikha

E-ISSN: 2945-0489

Volume 1, 2022

Consider the scalar equation with a center at

the origin 0

¨x+g(x) = 0 (1)

or its planar equivalent system

˙x=y, ˙y=−g(x) (2)

where ˙x=dx

dt ,¨x=d2x

dt2and g(x) = dG(x)

is analytic on Rwhere G(x) is the potential of (1).

Suppose system (2) admits a periodic orbit

in the phase plane with energy Eand g(x) has

bounded period for real energies E. Given G(x),

Let T(E) denotes the minimal period of this pe-

riodic orbit. Its expression is

T(E) = 2 Zb

p2E−2G(x).(3)

T(E) is well deﬁned and there is a neighborhood

of the real axis for which T(E) is analytic.

We suppose that the potential G(x) has one

minimum value which, for convenience locate

at the origin 0 and d2G(x)

dx2(0) = 1. The turning

points a, b of this orbit are solutions of G(x) = E.

Then the origin 0 is a center of (2). This center

is isochronous when the period of all orbits

near 0 ∈R2are constant (T=2π

√g0(0) = 2π).

The corresponding potential G(x) is also called

isochronous.

Since the potential G(x) has a local minimum

at 0, then we may consider an involution Aby

G(A(x)) = G(x)and A(x)x < 0

for all x∈[a, b]. So, any closed orbit is A-

invariant and Aexchanges the turning points:

b=A(a).

We proved the following results in [1]

Theorem 1-1 Let g(x)be an analytic func-

tion and G(x) = Rx

0g(s)ds and Abe the analytic

involution deﬁned by G(A(x)) = G(x). Suppose

that for x6= 0, xg(x)>0. Then the equation

¨x+g(x) = 0 (1)

has an isochronous center at 0if and only if the

function d

dx[G(x)/g2(x)]

is A-invariant i.e. d

dx [G/g2](x) = d

dx [G/g2](A(x))

in some neighborhood of 0..

Theorem 1-2 Let G(x) = Rx

0g(s)ds be

an analytic potential. Suppose that for x6=

0, xg(x)>0. Then the equation

¨x+g(x) = 0 (1)

has an isochronous center at 0if and only if

x−2G

g=F(G) (4)

where Fis an analytic function deﬁned in some

neighborhood of 0.

2 An alternative result

As consequences we prove the following

Theorem 2-1 Let G(x) = Rx

0g(s)ds be an

analytic potential deﬁned in a neighborhood of 0.

Suppose equation

¨x+g(x) = 0 (1)

has an isochronous center at 0. Let g(n)(x)be the

n-th derivative of the potential (with respect to x):

g(n)(x) = dn

dxnG(x), n ≥1then these derivatives

may be expressed under the form

g(n)(x) = an(G)x+bn(G), n ≥0 (5)

where anand bnare analytic functions with re-

spect to G.

In fact, as we had see in [1], the functions an

and bnare only dependent on G1the odd part of

G=G(x).

Proof By Proposition 3-4 of [1], condition

x(G) = √2G+P(G) with P=P(G) is a non-zero

analytic function implies that equation (1) has an

isochronous center at 0. Deriving with respect to

Gone obtains

dG =1

x+P0(G) = 1

or equivalently

x=1

1 + xP 0(G)=a1(G) + b1(G)

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Abd Raouf Chouikha

E-ISSN: 2945-0489

Volume 1, 2022

with

a1(G) = −1

2GP 02−1and b1(G) = 2GP 0

2GP 02−1.

Notice that by hypothesis Gis deﬁned deﬁned in

a neighborhood of 0 then 2GP 02−1 is necessary

non zero.

The functions a1(G) and b1(G) are analytic since

Pand P0they are too.

Derive now g0(x) it yields

g0(x) = dg

dx =d

dxa1(G)x+a1(G) + d

dxb1(G) =

dGa1(G)gx +d

dGb1(G)g+a1(G).

g0(x) =  d

dGa1x+d

dGb1(a1x+b1) + a1(G)

where the symbol prime 0means d

dG and a1or b1

stands for a1(G) or b1(G).

After replacing g(x) = a1(G)x+b1(G) one

obtains

2Ga0

1a1+a0

1xb1+a12+1/2a1√2b1

√G+b0

1a1x+b0

1b1

By simplifying one ﬁnd the expression of

g0(x) = a2(G)x+b2(G) with

a2(G) = a0

1b1+a1b1

2G+b0

1a1

b2(G) = 2Ga1a0

1+a2

1+b1b0

Here a1b1

2G=P0

(2GP 02−1)2

which is analytic. Then the functions a2(G) and

b2(G) are analytically dependent on the functions

a1(G), b1(G) and their derivatives.

By recurrence we easily prove that

g(p)(x) = ap(G)x+bp(G)

where the function ap(G) and bp(G) are analytic

with respect to G. Thank to Maple we are able

to carry out the calculations.

Theorem 2-2 Let G(x) = Rx

0g(s)ds be an

analytic potential and φ(x)a function deﬁned in

a neighborhood of 0.Abe the analytic involution

deﬁned by G(A(x)) = G(x). Then for a < 0<

b=A(a)and G(a) = G(b) = Ethe following

integrals equality holds

φ(x)

pE−G(x)g(x)dx =Zb

φ(x)−φ(A(x))

pE−G(x)g(x)dx

In particular, if we may expressed φ(x) = u(G)x+

v(G)then

φ(x)

pE−G(x)g(x)dx =Zb

2u(G)x

pE−G(x)g(x)dx

Proof It suﬃces to split the integral

φ(x)

pE−G(x)g(x)dx =Z0

φ(x)

pE−G(x)g(x)dx+

φ(x)

pE−G(x)g(x)dx

Recall that a < 0< b. By deﬁnition when

x∈[a, 0] then A(x)∈[0, b] and conversely. By a

change of variable x=A(y) the integral becomes

φ(x)

pE−G(x)g(x)dx =−Zb

φ(A(y))

pE−G(y)g((A(y))A0(y)dy =

−Zb

φ(A(y))

pE−G(y)g(y)dy

since g((A(y))A0(y) = g(y). Therefore

φ(x)

pE−G(x)g(x)dx =Zb

φ(x)

pE−G(x)g(x)dx−

φ(A(y))

pE−G(y)g(y)dy =.

On the other hand, suppose φ(x) = u(G)x+v(G).

Then the following integral may be written

φ(x)

pE−G(x)g(x)dx =Zb

u(G)x+v(G)

pE−G(x)g(x)dx =

u(G)x

pE−G(x)g(x)dx +Zb

v(G)

pE−G(x)g(x)dx

The last integral can be written

v(G)

pE−G(x)g(x)dx =ZE

v(G)

√E−GdG = 0

since v(G) is analytic. The other integral can be

written

u(G)x

pE−G(x)g(x)dx =Z0

u(G)x

pE−G(x)g(x)dx+

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DOI: 10.37394/232030.2022.1.10

Abd Raouf Chouikha

E-ISSN: 2945-0489

Volume 1, 2022

2u(G)x

pE−G(x)g(x)dx

=Zb

u(G)y

pE−G(y)g(A(y))A0(y)dy =

−u(G)y

pE−G(y)g(y)dy =Zb

u(G)y

pE−G(y)g(y)dy

since y=A(x).Finally,

φ(x)

pE−G(x)g(x)dx =Zb

u(G)x

pE−G(x)g(x)dx+

u(G)x

pE−G(x)g(x)dx

We may also derive

Corollary 2-3 Under hypotheses of Theo-

rem 2-1, consider the derivatives of g:g(j)(x) =

djg

dxjThen the analytic function

Vm,ν (x) =

j=1 djg

dxj!ν

may be expressed under the form :

Vm,ν (x) = um,ν (G)x+vm,ν (G) (6)

where ν= (ν1, ν2, ...., νm)and um,ν and vm,ν

are analytic functions with respect to G.

Proof By Theorem 2-1 any derivative of g

may be written when Gis isochronous

g(n)(x) = an(G)x+bn(G), n ≥0

where anand bnbeing analytic functions. It is

easy to realize that it is the same for any power

of any derivative (g(n)(x))ν

n. We may prove that

by recurrence

(g(n)(x))ν

n=an,ν (G)x+bn,ν (G).

More generally, we may also prove by recurrence

that a product of power of derivatives have the

similar expression

(g(n)(x))ν

1(g(p)(x))ν

2=an,p,ν (G)x+bn,p,ν (G).

Thus we may write for any product

Vm,ν (x) =

j=1 djg

dxj!ν

=um,ν (G)x+vm,ν (G)

3 Applications to the WKB

quantization

3.1 The quantum spectrum

Consider the Schrodinger equation

"−¯h2

dx2+G(x)#ψ(x) = Eψ(x).(7)

The Hamiltonian of the system is given by

H=p2

m+G(x)

where the mass m= 1.

This Hamiltonian is a constant of motion, whose

value is equal to the total energy E.

The wave function can always be written as

ψ(x) = exp(i

hσ(x))

The WKB expansion for the phase is a power se-

ries in ¯h:

σ(x) =

∞

(¯h

i)kσk(x).

Following [1], [5] rewrite the quantisation condi-

tion as

∞

I2k(E)=(n+1

2)¯h, n ∈N

1where

I2k(E) = 1

2π(¯h

i)2kZγ

dσ2k, k ∈N. (8)

When G(x) is analytic and xg(x)>0, it has

been proved that the contour integrals can be re-

placed by equivalent Rieman integrals between

the two turning points. More precisely,

I2(E) = −¯h2

24√2π

∂2

∂E2Zb

g2(x)

pE−G(x)dx

and

I4(E) = −¯h4

4√2π[1

120

∂3

∂E3Zb

g02(x)

pE−G(x)dx

−1

288

∂4

∂E4Zb

g2(x)g0(x)

pE−G(x)dx]

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DOI: 10.37394/232030.2022.1.10

Abd Raouf Chouikha

E-ISSN: 2945-0489

Volume 1, 2022

One establishes the following

Iγ

dσm= 2 X

L(ν)=m

2−1+|ν|i

(m−3+2|ν|)!!

∂m

2−1+|ν|

∂E m

2−1+|ν|×

UνG(ν)

√E−Gdx

where

G(ν)(x) =

j=1 djG

dxj!ν

Coeﬃcients Uνare deﬁned by a recurrence

equation.

Notice that higher order corrections quickly

increase in complexity and we know only few

cases where a WKB expansion can be worked to

all orders. Which resulting in a convergent series

whose sum is identical to the exact spectrum.

3.2 Applications

Here are some easy examples in order to verify

Theorem 2-1.

3.2.1 The isotonic potential

It is known, the spectrum of a potential is gener-

ally not strictly regularly spaced, except for the

harmonic G(x) = 1

2x2and the isotonic ones,

which are very particular cases :

G(x) = 1

8α2[αx + 1 −1

αx + 1]2

There are isochronous potentials with a strictly

equally spaced (harmonic) spectrum. As we have

seen in [1] its inverse is

x=√2G+P(G) = √2√Gα2−1−√2Gα2+ 1

α=

√2G−1 + √2Gα2+ 1

α.

3.2.2 A generalization

Let us consider now a three-parameters family of

potentials more general than the isotonic case

x=2G

g−2α−1 + √1 + βG

b√1 + βG ,

where αand βare real parameters such that

2α2≤β.

Here

x=√2G+P(G) = −2α+√2Gβ +p4α2+ 4 Gβα2

A resolution of these equations yields

G(x) = 8α2+ (β+ 2α2)(4αx +βx2)

2(β−2α2)2−

(4α2+ 2αβx)p2(2 + βx2+ 4αx)

2(β−2α2)2

Then, the above potential is isochronous accord-

ing to Theorem 2-1. Applying scaling property

of isochronous potentials. The potentials G(x)

and 1

γ2G(γx) have the same period. That means

the following three-parameters potentials family

is isochronous

G(x) = 1

2γ2X2(γx) =

[2α+βγx −αp2(2 + βγ2x2+ 4αγx)]2

2γ2(β−2α2)2

So the case 2α2=βand γ= 1 yields the isotonic

potential.

3.2.3 WKB corrections

By Theorem 2-1, we may write g=dG

dx =a(G)x+

b(G) . Writing

I2(E) = −¯h2

24√2π

∂2

∂E2Zb

g2(x)

pE−G(x)dx =

−¯h2

24√2π

∂2

∂E2Zb

g(x)

pE−G(x)g(x)dx

and by Theorem 1-2 we may express

−¯h2

24√2π

∂2

∂E2Zb

2a(G)x

pE−G(x)g(x)dx =

−¯h2

24√2π

∂2

∂E2ZE

2a(v)px(v)

√E−vdv.

Then making the change of variables u=v

E(we

suppose here ω= 1)

I2(E) = −¯h2

24π

∂2

∂E2[EZ1

2a(uE)x(u)

√1−udu].

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E-ISSN: 2945-0489

Volume 1, 2022

A similar calculation gives the fourth order

correction

I4(E) = −¯h4

4√2π[1

120

∂3

∂E3Zb

g02(x)

g(x)pE−G(x)g(x)dx

−1

288

∂4

∂E4Zb

g(x)g0(x)

pE−G(x)g(x)dx].

By Theorem 1-1 one gets g=dG

dx =a(G)x+b(G)

and g0=dg

dx =a1(G)√2G+b1(G). However, we

may easily prove

g02

g=a1,2(G)x+b1,2(G)

where a1,2(G) and b1,2(G) are analytic functions.

As well as

g(x)g0(x) = c1,2(G)x+d1,2(G)

where c1,2(G) and c1,2(G) are analytic functions.

Similar to I2, I4may be expressed through Abel

integrals :

I4(E) = −¯h4

4π[E−3

120 ZE

(x(v))5

√E−v

∂3

∂v3a1,2(v)dv

−E−4

288 ZE

(x(v))7

√E−v

∂4

∂v4c1,2(v)dv].

Thus one can choose I2(E) and deduce the

corresponding analytic isochronous potential

such that, its asymptotic decay is faster than

the asymptotic decay of I4(E). Therefore, I2(E)

and I4(E) grow exponentially fast as Egrows

to ∞. We will get the similar for higher order

corrections.

Turn now to upper order WKB correction.

Following [1], [3] the explicit expression for I2n(E)

is given by

I2n(E) = −√2

π¯h2nX

L(ν)=2n

2|ν|

(2n−3+2|ν|)!!Jν(E)

(9)

where

Jν(E) = ∂n−1+|ν|

∂En−1+|ν|Zb

UνG(ν)

√E−Gdx

where

G(ν)(x) =

j=1

(djG

dxj)ν

and where ν= (ν1, ν2, ..., ν2n), νj∈N, L(ν) =

P2n

j=1 jνjand |ν|=P2n

j=1 νj. The coeﬃcients

Uνsatisfy a certain recurrence relation.

By Corollary 2-3 G(ν)may be expressed un-

der the form

G(ν)(x) =

j=1 djG

dxj!ν

=un,ν (G)x+vn,ν (G)

where un,ν and vn,ν are analytic functions with

respect to G. Therefore,

Jν(E) = ∂n−1+|ν|

∂En−1+|ν|Zb

Uνun,ν (G)x

√E−Gg(x)dx

By Corollary 1-1, we can write

Jν(E) = ∂n−1+|ν|

∂En−1+|ν|Zb

2Uνun,ν (G)x

√E−Gg(x)dx

Jν(E) = ∂n−1+|ν|

∂En−1+|ν|ZE

2Uνun,ν (v)x(v)

√E−vdv

Another equivalent formulation via Abel integrals

Jν(E) = A(n,ν)ZE

(x(v))n−2+|ν|

√E−v

∂n−1+|ν|

∂En−1+|ν|un,ν (v),

where A(n,ν)= 2UνE−n+1−|ν|.

Similar to I2and I4the nth correction I2nwill

be expressed through Abel integrals :

I2n(E) = −¯h2n

πX

L(ν)=2n

2|ν|+1UνE−n+1−|ν|

(2n−3+2|ν|)!! ×

(x(v))n−2+|ν|

√E−v

∂n−1+|ν|

∂En−1+|ν|un,ν (v)dv

where un,ν (G) is such that

G(ν)(x) =

j=1 djG

dxj!ν

=un,ν (G)x+vn,ν (G).

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Abd Raouf Chouikha

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100

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4 Conclusion

It is well known that the quantum Schrodinger

solvable potentials is rather small because quan-

tum exactly solvability is a very strong condition.

We have therefore highlighted another no less

complicated but interesting expression of I2n(E)

in the general case. The natural question is which

of the two is more appropriate to use. In fact,

it will depend on the type of isochronous po-

tentials that we consider. In some situations, it

may be easier to use either of these formulas. As

we remarked higher order corrections quickly in-

crease in complexity. Here too the WKB correc-

tions I2n(E) grow exponentially fast as Egrows

to ∞. The WKB series should be summed for

any isochronous potential and would be ﬁnite as

Egrows to ∞.

References:

[1] Chouikha A.R. 2019, On isochronous an-

alytic motions and the quantum spectrum

Physica Scripta, Volume 94, Number 12.

[2] Eleonskii VM, Korolev VG and Kulagin NE

1997, On a classical analog of the isospec-

tral Schrodinger problem, JETP Lett., 65

(11), 889.

[3] J. F. Carinena, A. M. Perelomov, M. F.

Ranada, and M. Santander, A quantum ex-

actly solvable nonlinear oscillator related to

the isotonic oscillator, Journal of Physics A,

vol. 41, no. 8, Article ID 085301, 2008.

[4] C. Gonera, J. Gonera, New superintegrable

models on spaces of constant curvature, An-

nals of Physics Volume 413, February 2020,

168052.

[5] Robnik M and Salasnich L 1997, WKB to

all orders and the accuracy of the semiclassi-

cal quantization, J. Phys. A: Math. Gen., 30,

1711.

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