A sufficient condition for the existence of bound states in a potential without spherical symmetry

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A sufficient condition for the existence of bound states

in a potential without spherical symmetry

K. Chadan, A. Martin

To cite this version:

K. Chadan, A. Martin.

A sufficient condition for the existence of bound states in a potential

without spherical symmetry. Journal de Physique Lettres, Edp sciences, 1980, 41 (9), pp.205-206.

�10.1051/jphyslet:01980004109020500�. �jpa-00231760�

L-205

A

sufficient

condition for the

existence

of bound

states

in

a

potential

without

_spherical

_symmetry

(*)

K. Chadan and A. Martin

_(**)

Laboratoire de Physique Théorique et Hautes _{Energies (***),}

Université de Paris Sud, 91405 Orsay, France

(Reçu le _{23 janvier} 1980, accepte le 6 mars ₁₉₈₀₎

Résumé. 2014 Dans cette

note, nous donnons une condition suffisante pour qu’un

potentiel

purement attractif

sans

_{symétrie sphérique}

ait au moins un état lié.

Abstract. 2014 In this note _we

give a sufficient condition for a

_purely

attractive

_potential

without

_spherical

_symmetry

to have at least one bound state.

Tome 41 N^ 9 1 er MAI 1980

LE JOURNAL

DE

_{PHYSIQUE -}

LETTRES

J. _Physique - LETTRES 41 (1980) L-205 - L-206 ler _MAl 1980,

Classification

Physics Abstracts 03.65 - 03.80

In

_principle,

it is not difficult to find sufficient conditions for the existence of bound states of a

particle

in a

_potential.

If the wave function satisfies the tridimensional

_{Schrodinger equation}

with

_{~(X) -~}

0

_as

_{I X I ~}

_oo, it would be sufficient

to find a trial wave function

_Ø(X)

such that

_(Ø,

_HØ)

is

_negative.

_However,

it would be

_preferable

to obtain conditions in terms of

_simple

_integrals

of the

_potential.

For a

spherically symmetric

and

_purely

attractive

potential, Calogero

[1]

obtained,

some time _ago, the sufficient condition

in which the choice of R is

_arbitrary.

In this formula the constants are

_optimal

as one can see on the

example - Vo

8(r - ro)

by

choosing R

close to _ro.

We

_give

in this letter the

_{generalization}

of

₍₂₎

to

the non

spherically symmetric

case, the first

step

toward the

_{generalization}

to the

_N-body

case,

assum-ing

that V is

everywhere

attractive. One of us

(K.C.

[2])

has

_already

obtained sufficient conditions for the

non

_spherical

case, for

example

where

which,

unfortunately,

necessitates the calculation of

an

_{9-uple integral,}

and is also not

_optimal.

We start here from the remark

_that,

if there are no

bound states in a

_potential

which decreases fast

enough

at

_infinity

and if this

_potential

is

_attractive,

the solution of the

_{Schrodinger equation}

at zero

energy

is

_necessarily

_positive;

_indeed,

the Bom series

_gives

the solution of

_(5),

and all its terms are

_positive

(see

also

_[3]).

_Therefore,

if we can show

_that,

under certain conditions on the

_potential,

the

_equation

₍₅₎

does not admit a

_{strictly positive}

_solution,

we have

proven the existence of bound states. For this

pur-pose, we consider the

_sphere

_{I X}

_I

=

R,

and divide the

space in two

_regions.

Let

(*) La version _francaise de cet article a été acceptée aux _Comptes

Rendus de 1’Academie des Sciences et est inseree dans le n° du 18 février 1980.

(**) Permanent address : _Theory Division, CERN, 1211 Genevc 23, Suisse.

(***) Laboratoire Associe au C.N.R.S.

L- 206 JOURNAL DE _{PHYSIQUE -} LETTRES

It is now _easy to obtain from

_{(5), by remembering}

that the

_potential

is

_everywhere

_attractive,

and

To go

further,

we must use the two

_following

lemmas :

Lemma 1. - If

W(Y) >

0,

and

f (r)

is a

_decreasing

function of r.

Lemma 2. -

f (r)

defined

by (9)

is such that

r f (r)

is an

_increasing

function of r.

To _prove lemma

1,

it suffices to notice that

_F(X)

defined

_{by (9)}

is a

_{superharmonic}

function

_[4]

_simply

because AF = - W 0.

Therefore,

the same is

true

_{for f (r)}

defined as the infimum of such functions. This _proves

_{that f (r)}

is a

decreasing

function.

For the

_proof

of lemma

2,

we first assume that W is

sufficiently

regular

(double

differentiability).

It follows then that

_F(X)

and

_{f (r)}

are also

regular,

and the

superharmoniticity of fer) implies

Integrating

this

_equation

from r to

_infinity,

we obtain

Now,

it is _easy to see that if W is

sufficiently

_decreasing

at

_infinity,

the r.h.s. of

₍₁₁₎

vanishes. This _proves the lemma 2. In the

_general

case where W is

_only

an

L 2

function,

one reduces the same conclusion

_by

approximating

it

_by

C2

functions.

Starting

from the above

lemmas,

we deduce that the

infimums of the

_integrals

in

₍₇₎

and

₍₈₎

are reached for

I X I = R. If we

denote these infimums

_{by Ji}

and

_J2,

(7)

and

₍₈₎

are written

_simply

and

Therefore,

a _necessary condition for the absence of bound states would

_{be J,}

+

_RJ2

1. It follows that a sufficient condition for

having

a bound state

will be

In the case of a

_spherically

_symmetric

potential,

this

gives

back the condition

(2)

of

_Calogero.

We see

therefore that

₍₁₂₎

is also

_optimal

_by

_considering

again

the Dirac ~-function

_potential.

Also,

one can

simplify

somewhat the calculations and

_replace

₍₁₂₎

by

a _stronger condition

_by

noticing

_that,

for

_{1 X}

₌

_R, we

_have

_{I X - Y 1 R + I Y I.}

One obtains then the sufficient condition

The above conditions can

_easily

be

_generalized

for

the case of m > 3 dimensions

_by

_using

the

_appropriate

Green’s function

_[4].

One obtains then the sufficient condition

where

_em

=

(4

7~)/r(~/2 -

1).

Acknowledgments.

- We would like to thank

P. C. Sabatier for a useful remark and discussions.

References

[1] CALOGERO, F., J. Math. _Phys. 6 (1965) 161.

[2] CHADAN, K., Colloque Interactions Fondamentales, Marseille

(1979). To appear (H. Bacry, Editor).

[3] MARTIN, A. and _SABATIER, P. _C., J. Math. Phys. 18 (1977) 1623.

[4] HAYMAN, W. K. and KENNEDY, P. B., Subharmonic Functions,

(Academic Press, London) 1976. A _{superharmonic}