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AFCRC TN-59-604

*E ET PM. m***

?\ yt r"\ N

^ KT\/^ NEW YORK UNIVERSITY

"J IN " ' nst ' tuJe Â°f Mathematical Sciences

â€¢- V-^ X Division of Electromagnetic Research

^cccxx^

RESEARCH REPORT No. CX-44

Upper Bounds on Scattering Lengths

When Composite Bound States Exist

L. ROSENBERG, L. SPRUCH and T. F. O'MALLEY

Contract No. AF 19(604)4555

OCTOBER, 1959

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-hlt

UPPER BOUNDS ON SCATTERING IENGTHS

WHEN COMPOSITE BOUND STATES EXIST

L. Rosenberg, L. Spruch and T.F. O'Malley

'Qzay^A/uf

L. Rosenberg

v^xa^A jij,

â€ž SpruSh ^J

L. Sp

T.F. O'Malley *

The research reported in this document has been

sponsored by the Geophysics Research Directorate

of the Air .borce Cambridge Research Center, Air

Research and Development Command, under Contract

No. AF 19(60li)U555.

Requests for additional copies by Agencies of the Department of Defense,

their contractors, and other Government agencies should be directed to thei

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Department of Defense contractors must be established for ASTIA services

or have their 'need to know 1 certified by the cognizant military agency

of their project or contract.

All other persons and organizations should apply to the:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

- 1 -

Abstract

In the case of the zero energy scattering of one compound system by

another, where one real scattering length completely characterizes the problem,

(e.g., the reaction A+B -> C+D, in addition to A+B -> A+B, cannot take place)

it has previously been shown that the Kohn-Hulthen variational principle

provides an upper bound on the scattering length if no composite bound states

exist. The extension of this result to the case where one or more composite

bound states do exist is presented here. The inclusion of tensor forces,

exchange forces, and Coulomb forces is allowed. Several methods are given

for obtaining a rigorous upper bound on the scattering length, which involve

the addition of certain positive terms to the Kohn-Hulthen variational ex-

pression. The approximate information about the composite bound states which

is required to construct these additional terms can be found by standard methods.

As a consequence of one of the results obtained, it is shown that under certain

circumstances some ordinary variational calculations give a bound. Thus, an

analysis of a previous calculation in the light of the present results leads,

without further calculations, to a rigorous upper bound on the singlet electron-

hydrogen scattering length.

Table of Contents

Page

1. Introduction 2

2. The case of one bound state 3

3. Connection with forms of the variational principle 11

hÂ» An alternate method 13

BÂ» The case of more than one bound state 17

6Â» Discussion 21

References 2$

- 2 -

I. INTRODUCTION

The value of variational methods in scattering problems is greatly increased

when these methods are based on a minimum (or maximum) principle. This is especially

true for the difficult case of scattering by a compound system, where calculations

in which different trial functions are used can lead to quite different results.

Such a minimum principle, for zero energy scattering, has already been obtained for

the case in which the scattered particle cannot be bound to the scattering system .

The present paper is concerned with the extension of this result to the case in

which one or more composite bound states exist. For clarity of presentation the

detailed discussion is confined to the problem of the scattering of a spinless,

neutral particle, of zero orbital angular momentum, from a short range center of

force. The generalization to the case where the scattering is by a compound system,

and where long range Coulomb forces exist, is identical with the corresponding

2

generalization in the problem in which no composite bound states exist , and the

details will be omitted. Further, the inclusion of tensor forces is allowed.

The formulation, given in II, for the scattering of a particle by a compound system

can be extended, in a straightforward way, to treat the scattering of one compound

system by another. It will in fact be understood in the following that unless

otherwise stated (in particular, the result of Section IV is excepted) each of

the results obtained has, under the condition discussed in II, direct applicability

to this wider class of problems (the essential requirement is that the scattering

be completely characterized by one real scattering length). The problems which

may be treated include zero energy scattering of a neutron or a proton (or in

fact of nuclei) by nuclei, assuming realistic nuclear potentials. Similarly, the

scattering of electrons (or atoms^by atoms may be treated.

As discussed in II, the scattering cannot be characterized by a real

- 3 -

scattering length if the radiative capture process is possible. We shall, in

the following, make the approximation of ignoring the interaction of the

particle with the radiation field. The bound on the scattering length is then

rigorous only to the extent that this generally excellent approximation is

3

in fact valid.

In Section II a number of results are obtained which provide upper bounds

on the scattering length for the case where only one bound state exists. In

Section III it is shown that there are well defined circumstances for which some

ordinary variational calculations provide a bound. An alternate method for

obtaining bounds, based on the Kato formalism, is considered in Section IV. The

extension to the case of many bound states is treated in Section V. Since in

this case the exact scattering function has a number of nodes (even for the

scattering of a particle by a center of force), making the construction of an

accurate trial function more difficult, it would seem that a minimum principle

for the scattering length is even more valuable here.

II. THE CASE OF ONE BOUND STATE

We begin by considering the problem of the zero energy, zero orbital angular

momentum scattering of a spinless neutral particle of mass m by a short range

center of force whose strength is such that one, and only one, bound state exists.

As shown in I, the scattering length A may, in general, be written

A - A. - I u.Zu.dr + fw^wdr , (2.1)

where u. is a trial scattering function, w is the difference between u. and

the exact function, u, and A - r, for r -> oo â€¢

The trial function, u. , satisfies the same boundary conditions, with A replaced

W

by A. . It was shown in I that if no bound state exists, and if the normalization

given by Eqs. (2.2) (referred to in the following as the " appropriate"

normalization) is employed, then

f w*iat we shall refer

to as the " inappropriate" normalization, namely,

u -> 1 - r/A, for r -> oo. (2.U)

We assume in the following that the "appropriate" normalization is employed,,

(It is of course true that at non-aero energies one cannot generally say which

normalization is superior, and that even at zero energy the " inappropriate"

normalization provides a variational estimate for if not a bound on the

scattering length.)

- 5 -

In the case where one bound state exists, which is our present concern, the

inequality, Eq.(2.3), will not be valid unless the error function, w, is orthogonal

to the exact bound state function, denoted by u ., where (u n u ,) =1. [We have

oCu, â– e. u ., where e.. is related to the bound state energy E by e." - (2m/h )E^j

Since the exact solutions u and u . are orthogonal, the difference function

w will be orthogonal to u . provided u. and u - are orthogonal. We therefore

have the result that the Kohn-Hulthen variational principle gives an upper

bound on the scattering length provided a trial function is used which is

orthogonal to the exact bound state wave function*

It is easily shown that this statement is equivalent to the expression

for the bound given below in Eq. (2.5), where the only requirement on u. is that

it satisfy the correct boundary conditions. However, we will actually obtain

Eq. (2.5) in a slightly different way, which is the more natural generalization

of the method used for the no bound state case. Although w is in fact unknowi,

we may formally construct a function, namely,

[w - (w,u el )u Â£l ],

which is orthogonal ton .. It then follows, exactly as in the case for no

bound states, that

([w-(w,u 6l )u 6l ]/ fr-(w,Â» A >Â»J) < Â°-

This inequality may be written

(w,Â£w) < e 1 (u t ,u el ) 2 ^

where we have made use of the relation (w,n ..) â– (u.,u ..). Thus, an upper

bound on A may be obtained from

A Â± A t " / V u t dr + Â«1 ( J u t u el dr)2 ' &.*)

- 6 -

The difficulty with Eq. (2.5) is that u is not known in general .

Therefore unless the last term in Eq. (2.5) is small, for a given (presumably)

reasonably accurate trial function u, , and shows very small variation as the

accuracy of the trial bound state function, u , . , is increased, one cannot

claim to have a bound. In fact, preliminary calculations based on Eq. (2.5),

for the singlet e~H problem, gave negative results in the sense that the last

term in Eq. (2.5) was large and showed large variations for a series of functions

u ... ranging from a two parameter to an eleven parameter function with u. that

given by Seaton. The failure of this method in the particularly favorable

e~H case indicates that Eq. (2.5) is not a useful form. However, from

Eq. (2.5) one can obtain an expression which does not depend on the unknown

function u . by writing

htf Va dr ' 2 " \ E., if i < N. Here E. is the energy of the iâ€” bound state. For

i > N, we have e| K ' > 0.

Theorem 2: If the Hamiltonian matrix is constructed in the same way as

discussed in Theorem 1, but of rank K + 1 with the first K functions un-

altered, then the eigenvalues, E^ , satisfy the relations

- 8 -

E, < E< K+1 > < E f K) .

i â€” 1 â€” 1

We now apply these theorems to the case where only one bound state exists.

We introduce the orthonormal functions v. and v_ and form the 2x2 Hamiltonian

matrix

where

H Â±i b (v.,^), i,J -1,2.

It is assumed that v. is a sufficiently accurate ground state trial function

so that /_ -j

H ll = E l K Â° #

(If this last inequality is not satisfied, the method described below will

not be applicable. This, however, does not represent a serious limitation

since a trial function with the required accuracy can almost always be found.)

(2) (2)

It then follows that the eigenvalues of the above matrix, E^ and E^ ,

satisfy the relations

from Theorem 1, and

4 2) >o,

E< 2) < E* 1 ^ 0,

from Theorem 2, so that

e{ 2 >e 2 < 2) 0. We then have the inequality which is the new form of Eq. (2.7) >

Jw(X)/w(X)dr < JL (f u^ w(X)dr) 2 , (2.8)

2 (l)

with e_, â– -(2m/fo )E. V and u ... â– v.. Since both sides of the inequality,

Eq. (2.8), are continuous at X â– (see I) we have

/w^wdr < (I u -..oowdr) .

- e lt V elt

Therefore, since dfw ^aLu., the expression which provides an upper bound on the

scattering length becomes

A Â± A t " / \ l u t dr + I^ ( / U elt^ S*> 2 Â« ( 2 - 9 >

We have then succeeded in our purpose in that we have obtained a bound which

does not require a knowledge of the exact bound state function nor of its

energy, and which has at the same time avoided the use of the rigorous but

often very crude Schwarz inequality. (In the case where the scatterer is a

compound system, the ground state wave function of the scatterer must still

- 10 -

n

be known exactly in order to obtain a rigorous bound. ) Note that the

earlier results, Eqs. (2. 5) and (2.6), may be considered as special

cases of Eq. (2.9); the choice u ,. Â» u . in Eq. (2.9) yields Eq. (2.5), and

the use of the Schwarz inequality in the last integral of Eq. (2.9) leads

back to Eq. (2.6). It is of course clear that for a given form of the trial

functions u. and u ,. the optimum choice of the variational parameters is

such as to minimize the right hand side of Eq. (2.9), subject to the

requirement that j u -, + X u -. + dr > 0. .

For the case of arbitrary values of the orbital angular momentum, L, the

inequality which corresponds to that given in Eq. (2.9) has the same form as

the result for L â– 0. We merely replace A (with a similar replacement of A . )

by Aj., defined in terms of the phase shift 17 . as

A^ - - [lx3x...x(2L+l)] 2 (tan^ L /k 2L+1 ) kii0 .

Further, the trial function, u, . (r), satisfies the boundary conditions

u Lt (0) - ,

u u (r) -> A Lt r" L /(2L+l) - r L+1 , for r -> oo,

n

and^Cnow contains the additional term L(L+l)/r . Of course for a particular

value of L the bound state considered is one with orbital angular momentum L.

These results for arbitrary values of L can be extended in a straightforward

manner to the case of scattering of one compound system by another, where each

system may carry a net charge. The regular and irregular solutions of the

"free" wave equation in the limit of vanishing energy have previously been

given for this general case * The asymptotic form of the wave function is taken

as the "appropriate" linear combination of these solutions. The analysis

then proceeds in a manner similar to that already described in I and II and we

o

shall omit further details here.

- 11 -

III. CONNECTION WITH FORMS OF THE VARIATIONAL PRINCIPLE

The inequality, Eq. (2.9), has the particular consequence that there exist

well defined circumstances, which will now be described, under which an upper

bound on the scattering length may be obtained from an ordinary variational

calculation. (As in Section II the discussion in this section will be confined

to the case where only one bound state exists.) We consider first the Kohn-Hulthen

principle, which has the zero energy form

A~ A t - Ju t A t dr, (3.1)

where u. satisfies the boundary conditions given in Eqs. (2.2). Now suppose

u. may be written,

u t = u t +bu elt , . < 3 ' ?)

where u ... is a trial bound state function which is sufficiently accurate to

give binding, i.e. ,

e lt = / u elt^ u elt dr>0 -

i

Since u _. vanishes at the origin and at infinity u. satisfies the same boundary

condition as u, and can therefore itself serve as a proper trial scattering

function. Substituting this form for u. into the variational expression, Eq. (3.l),

i

and determining b variationally, with u. and u .. momentarily considered as

fixed, we find

b â– - - â€” J u â– , + * u. dr ,

e lt/ elt t

and

A * A t " / u t /u t dr + \rr ( / u ei/ V dr)2 - (3 - 3)

it

It follows from Eq. (2.9) that the variational estimate of A given bv Bq. (3.3)

is actually an upper bound. Conversely, the approximation to the zero energy

scattering function, obtained in the course of a calculation of an upper bound

- 12 -

on A based on Eq. (2.9), should be taken to be

\ ' ( Z7 / u elt^ u t dr)u <

"It ' â€¢*Â« " elt

rather than V, since it is the former function which can be interpreted as a

variationally determined trial function. We note that the prescription given

by Kohn to evaluate the variational parameters should be used rather than that

of Hulthen since, while both methods will provide an upper bound, for a

given form of the trial function the Kohn method will yield a lower (and

therefore better) estimate of the scattering length. ( See I.)

It is of interest to see if there are any variational calculations of

scattering lengths reported in the literature which were performed with trial

functions which may be written in the form given by Eq. (3.2-'. Such calculations

may be reinterpreted in the light of the present discussion. As an example

we note that Borowitz and Greenberg have recently performed a variational

calculation for electron-hydrogen scattering, using as a zero energy trial

function one which was explicitly constructed to be of the form of a trial

scattering function plus a multiple of a trial H~ bound state function. Since

the normalization used by these authors corresponds to what we have called

the inappropriate normalization [see Eq. (2.U)] neither the results for the

singlet case, where one bound state of H exists, nor for the triplet case,

where no bound states exist, is necessarily a bound. However, as described

in I, for those calculations which have been performed using the inappropriate

normalization the results may be converted, with a trivial amount of labor,

so that they c orrespond to the Kohn-Hulthen form and therefore do give a

bound under the circumstances considered. The method of conversion may be

briefly restated as follows. If A. and I = J u.cif u. dr have been evaluated,

- 13 -

with u. normalized as in Eq. (2.U), then the bound on A is given by

2

A < A. - A I,

If A. is positive and j A 1 1 is small compared to unity then this conversion may

be expected to give, in addition to a bound, an improved approximation to the

scattering length.

Returning to the work of Borowitz and Greenberg, we note that two separate

calculations of the singlet scattering length were performed, using two

12

trial H functions obtained by Chandrasekhar â€¢ One of these was a two parameter

function containing no dependence on the interelectronic distance, r ? , so that

no polarization was allowed for. The other function, which included a linear

r 12 dependence, contained three parameters. In each case the parameters used

where those given by Chandrasekhar. The two and three parameter functions

â€”2 â€”2 12

give values of e 1+ equal to 0.0266 a and 0.0518 a respectively

it o o

(a is the Bohr radius). The rigorous uppper bounds obtained by the conversion

process described above yielded, in both cases, slightly improved approximations

to the scattering length (see Table I).

IV. AN ALTERNATE METHOD

In this section we consider an alternate method for obtaining upper bounds

on the scattering length for the case of the scattering of a particle by a

center of force where one bound state exists. The method is based on the

formalism given by Kato for finding upper and lower bounds on the scattering

phase shift. (Since the Kato method, as opposed to the Kato identity, is

unrelated to the other methods discussed in this paper, this section may be

read independently of the others.)

The Kato identity for arbitrary scattering energy, (hkj/2m, of which

Eq. (2.1) is the zero energy form with the normalization < 9 < n ( i.e .,

9 / 0), is

- 1U -

k cot(^- 0) - k cot (^ t - 9) - Ju^c/u^dr + j w g ^w Q dr, (U.l)

where now

ancl

the boundary conditions

tt a (0) - ,

A - (U * 2)

u e (r) -> sin(kr+7 )/sin(7- 0) , f or r -> oo ,

where A is the exact phase shift and where 9 satisfies < 9 < n. The trial

function, u^. , satisfies boundary conditions of the form given in Eqs. (6.2),

but with A replaced by a trial phase shift, v 7 . , which is arbitrary; w_ is

defined as Mq. - u fl . The Kato inequalities are

" Â° 0" 1 J ( ^ u et )2 p " 1(ir -j V^V 1, Â± P9" 1 / ( ^ u 9t )2p " ldr# (U,3)

Here p(r) is a non-negative function which must fall off faster than l/r

but which is otherwise arbitrary and is chosen for convenience. The numbers

a Q and p fl are defined by the associated eigenvalue problem, in which the

equation

(/+Hp)0 Q (r) -

is considered. The asymptotic form of e (r) is characterized by a phase shift,

6(n.). The infinite, discreet set of eignevalues, j Ugl , is defined such that

for any integer, n,

6(0.^) - 9 + nn .

The smallest positive eigenvalue is a ft and the largest negative eigenvalue is -P Q .

For zero energy scattering the phase shift in the associated eignevalue

problem, 6(m.), is given, according to a theorem due to Levinson, by

- 15 -

6 ((J.) â– mn, if there is no state of exactly zero energy, where ra is the

number of bound states for the operator * + up. [if there are ra bound states

and there is one at zero energy in addition, the phase shift is (m+ xOn.J we

are considering the case for which there is one and only one bound state for

the actual physical system, i.e., for m- * 0. We choose 9 â– y> where

< y < ii. (The point is that we wish to exclude 9-0; note that the eigen-

values |i __. do not form a discreet set at zero energy.) It is then clear that

nu

-p is determined by the condition that the operator at -p o has associated with

AFCRC TN-59-604

*E ET PM. m***

?\ yt r"\ N

^ KT\/^ NEW YORK UNIVERSITY

"J IN " ' nst ' tuJe Â°f Mathematical Sciences

â€¢- V-^ X Division of Electromagnetic Research

^cccxx^

RESEARCH REPORT No. CX-44

Upper Bounds on Scattering Lengths

When Composite Bound States Exist

L. ROSENBERG, L. SPRUCH and T. F. O'MALLEY

Contract No. AF 19(604)4555

OCTOBER, 1959

NEW YORK UNIVERSITY

Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-hlt

UPPER BOUNDS ON SCATTERING IENGTHS

WHEN COMPOSITE BOUND STATES EXIST

L. Rosenberg, L. Spruch and T.F. O'Malley

'Qzay^A/uf

L. Rosenberg

v^xa^A jij,

â€ž SpruSh ^J

L. Sp

T.F. O'Malley *

The research reported in this document has been

sponsored by the Geophysics Research Directorate

of the Air .borce Cambridge Research Center, Air

Research and Development Command, under Contract

No. AF 19(60li)U555.

Requests for additional copies by Agencies of the Department of Defense,

their contractors, and other Government agencies should be directed to thei

ARMED SERVICES TECHNICAL INFORMATION AGENCY

DOCUMENTS SERVICE CENTER

ARLINGTON HALL STATION

ARLINGTON 12, VIRGINIA

Department of Defense contractors must be established for ASTIA services

or have their 'need to know 1 certified by the cognizant military agency

of their project or contract.

All other persons and organizations should apply to the:

U.S. DEPARTMENT OF COMMERCE

OFFICE OF TECHNICAL SERVICES

WASHINGTON 25, D.C.

- 1 -

Abstract

In the case of the zero energy scattering of one compound system by

another, where one real scattering length completely characterizes the problem,

(e.g., the reaction A+B -> C+D, in addition to A+B -> A+B, cannot take place)

it has previously been shown that the Kohn-Hulthen variational principle

provides an upper bound on the scattering length if no composite bound states

exist. The extension of this result to the case where one or more composite

bound states do exist is presented here. The inclusion of tensor forces,

exchange forces, and Coulomb forces is allowed. Several methods are given

for obtaining a rigorous upper bound on the scattering length, which involve

the addition of certain positive terms to the Kohn-Hulthen variational ex-

pression. The approximate information about the composite bound states which

is required to construct these additional terms can be found by standard methods.

As a consequence of one of the results obtained, it is shown that under certain

circumstances some ordinary variational calculations give a bound. Thus, an

analysis of a previous calculation in the light of the present results leads,

without further calculations, to a rigorous upper bound on the singlet electron-

hydrogen scattering length.

Table of Contents

Page

1. Introduction 2

2. The case of one bound state 3

3. Connection with forms of the variational principle 11

hÂ» An alternate method 13

BÂ» The case of more than one bound state 17

6Â» Discussion 21

References 2$

- 2 -

I. INTRODUCTION

The value of variational methods in scattering problems is greatly increased

when these methods are based on a minimum (or maximum) principle. This is especially

true for the difficult case of scattering by a compound system, where calculations

in which different trial functions are used can lead to quite different results.

Such a minimum principle, for zero energy scattering, has already been obtained for

the case in which the scattered particle cannot be bound to the scattering system .

The present paper is concerned with the extension of this result to the case in

which one or more composite bound states exist. For clarity of presentation the

detailed discussion is confined to the problem of the scattering of a spinless,

neutral particle, of zero orbital angular momentum, from a short range center of

force. The generalization to the case where the scattering is by a compound system,

and where long range Coulomb forces exist, is identical with the corresponding

2

generalization in the problem in which no composite bound states exist , and the

details will be omitted. Further, the inclusion of tensor forces is allowed.

The formulation, given in II, for the scattering of a particle by a compound system

can be extended, in a straightforward way, to treat the scattering of one compound

system by another. It will in fact be understood in the following that unless

otherwise stated (in particular, the result of Section IV is excepted) each of

the results obtained has, under the condition discussed in II, direct applicability

to this wider class of problems (the essential requirement is that the scattering

be completely characterized by one real scattering length). The problems which

may be treated include zero energy scattering of a neutron or a proton (or in

fact of nuclei) by nuclei, assuming realistic nuclear potentials. Similarly, the

scattering of electrons (or atoms^by atoms may be treated.

As discussed in II, the scattering cannot be characterized by a real

- 3 -

scattering length if the radiative capture process is possible. We shall, in

the following, make the approximation of ignoring the interaction of the

particle with the radiation field. The bound on the scattering length is then

rigorous only to the extent that this generally excellent approximation is

3

in fact valid.

In Section II a number of results are obtained which provide upper bounds

on the scattering length for the case where only one bound state exists. In

Section III it is shown that there are well defined circumstances for which some

ordinary variational calculations provide a bound. An alternate method for

obtaining bounds, based on the Kato formalism, is considered in Section IV. The

extension to the case of many bound states is treated in Section V. Since in

this case the exact scattering function has a number of nodes (even for the

scattering of a particle by a center of force), making the construction of an

accurate trial function more difficult, it would seem that a minimum principle

for the scattering length is even more valuable here.

II. THE CASE OF ONE BOUND STATE

We begin by considering the problem of the zero energy, zero orbital angular

momentum scattering of a spinless neutral particle of mass m by a short range

center of force whose strength is such that one, and only one, bound state exists.

As shown in I, the scattering length A may, in general, be written

A - A. - I u.Zu.dr + fw^wdr , (2.1)

where u. is a trial scattering function, w is the difference between u. and

the exact function, u, and A - r, for r -> oo â€¢

The trial function, u. , satisfies the same boundary conditions, with A replaced

W

by A. . It was shown in I that if no bound state exists, and if the normalization

given by Eqs. (2.2) (referred to in the following as the " appropriate"

normalization) is employed, then

f w*iat we shall refer

to as the " inappropriate" normalization, namely,

u -> 1 - r/A, for r -> oo. (2.U)

We assume in the following that the "appropriate" normalization is employed,,

(It is of course true that at non-aero energies one cannot generally say which

normalization is superior, and that even at zero energy the " inappropriate"

normalization provides a variational estimate for if not a bound on the

scattering length.)

- 5 -

In the case where one bound state exists, which is our present concern, the

inequality, Eq.(2.3), will not be valid unless the error function, w, is orthogonal

to the exact bound state function, denoted by u ., where (u n u ,) =1. [We have

oCu, â– e. u ., where e.. is related to the bound state energy E by e." - (2m/h )E^j

Since the exact solutions u and u . are orthogonal, the difference function

w will be orthogonal to u . provided u. and u - are orthogonal. We therefore

have the result that the Kohn-Hulthen variational principle gives an upper

bound on the scattering length provided a trial function is used which is

orthogonal to the exact bound state wave function*

It is easily shown that this statement is equivalent to the expression

for the bound given below in Eq. (2.5), where the only requirement on u. is that

it satisfy the correct boundary conditions. However, we will actually obtain

Eq. (2.5) in a slightly different way, which is the more natural generalization

of the method used for the no bound state case. Although w is in fact unknowi,

we may formally construct a function, namely,

[w - (w,u el )u Â£l ],

which is orthogonal ton .. It then follows, exactly as in the case for no

bound states, that

([w-(w,u 6l )u 6l ]/ fr-(w,Â» A >Â»J) < Â°-

This inequality may be written

(w,Â£w) < e 1 (u t ,u el ) 2 ^

where we have made use of the relation (w,n ..) â– (u.,u ..). Thus, an upper

bound on A may be obtained from

A Â± A t " / V u t dr + Â«1 ( J u t u el dr)2 ' &.*)

- 6 -

The difficulty with Eq. (2.5) is that u is not known in general .

Therefore unless the last term in Eq. (2.5) is small, for a given (presumably)

reasonably accurate trial function u, , and shows very small variation as the

accuracy of the trial bound state function, u , . , is increased, one cannot

claim to have a bound. In fact, preliminary calculations based on Eq. (2.5),

for the singlet e~H problem, gave negative results in the sense that the last

term in Eq. (2.5) was large and showed large variations for a series of functions

u ... ranging from a two parameter to an eleven parameter function with u. that

given by Seaton. The failure of this method in the particularly favorable

e~H case indicates that Eq. (2.5) is not a useful form. However, from

Eq. (2.5) one can obtain an expression which does not depend on the unknown

function u . by writing

htf Va dr ' 2 " \ E., if i < N. Here E. is the energy of the iâ€” bound state. For

i > N, we have e| K ' > 0.

Theorem 2: If the Hamiltonian matrix is constructed in the same way as

discussed in Theorem 1, but of rank K + 1 with the first K functions un-

altered, then the eigenvalues, E^ , satisfy the relations

- 8 -

E, < E< K+1 > < E f K) .

i â€” 1 â€” 1

We now apply these theorems to the case where only one bound state exists.

We introduce the orthonormal functions v. and v_ and form the 2x2 Hamiltonian

matrix

where

H Â±i b (v.,^), i,J -1,2.

It is assumed that v. is a sufficiently accurate ground state trial function

so that /_ -j

H ll = E l K Â° #

(If this last inequality is not satisfied, the method described below will

not be applicable. This, however, does not represent a serious limitation

since a trial function with the required accuracy can almost always be found.)

(2) (2)

It then follows that the eigenvalues of the above matrix, E^ and E^ ,

satisfy the relations

from Theorem 1, and

4 2) >o,

E< 2) < E* 1 ^ 0,

from Theorem 2, so that

e{ 2 >e 2 < 2) 0. We then have the inequality which is the new form of Eq. (2.7) >

Jw(X)/w(X)dr < JL (f u^ w(X)dr) 2 , (2.8)

2 (l)

with e_, â– -(2m/fo )E. V and u ... â– v.. Since both sides of the inequality,

Eq. (2.8), are continuous at X â– (see I) we have

/w^wdr < (I u -..oowdr) .

- e lt V elt

Therefore, since dfw ^aLu., the expression which provides an upper bound on the

scattering length becomes

A Â± A t " / \ l u t dr + I^ ( / U elt^ S*> 2 Â« ( 2 - 9 >

We have then succeeded in our purpose in that we have obtained a bound which

does not require a knowledge of the exact bound state function nor of its

energy, and which has at the same time avoided the use of the rigorous but

often very crude Schwarz inequality. (In the case where the scatterer is a

compound system, the ground state wave function of the scatterer must still

- 10 -

n

be known exactly in order to obtain a rigorous bound. ) Note that the

earlier results, Eqs. (2. 5) and (2.6), may be considered as special

cases of Eq. (2.9); the choice u ,. Â» u . in Eq. (2.9) yields Eq. (2.5), and

the use of the Schwarz inequality in the last integral of Eq. (2.9) leads

back to Eq. (2.6). It is of course clear that for a given form of the trial

functions u. and u ,. the optimum choice of the variational parameters is

such as to minimize the right hand side of Eq. (2.9), subject to the

requirement that j u -, + X u -. + dr > 0. .

For the case of arbitrary values of the orbital angular momentum, L, the

inequality which corresponds to that given in Eq. (2.9) has the same form as

the result for L â– 0. We merely replace A (with a similar replacement of A . )

by Aj., defined in terms of the phase shift 17 . as

A^ - - [lx3x...x(2L+l)] 2 (tan^ L /k 2L+1 ) kii0 .

Further, the trial function, u, . (r), satisfies the boundary conditions

u Lt (0) - ,

u u (r) -> A Lt r" L /(2L+l) - r L+1 , for r -> oo,

n

and^Cnow contains the additional term L(L+l)/r . Of course for a particular

value of L the bound state considered is one with orbital angular momentum L.

These results for arbitrary values of L can be extended in a straightforward

manner to the case of scattering of one compound system by another, where each

system may carry a net charge. The regular and irregular solutions of the

"free" wave equation in the limit of vanishing energy have previously been

given for this general case * The asymptotic form of the wave function is taken

as the "appropriate" linear combination of these solutions. The analysis

then proceeds in a manner similar to that already described in I and II and we

o

shall omit further details here.

- 11 -

III. CONNECTION WITH FORMS OF THE VARIATIONAL PRINCIPLE

The inequality, Eq. (2.9), has the particular consequence that there exist

well defined circumstances, which will now be described, under which an upper

bound on the scattering length may be obtained from an ordinary variational

calculation. (As in Section II the discussion in this section will be confined

to the case where only one bound state exists.) We consider first the Kohn-Hulthen

principle, which has the zero energy form

A~ A t - Ju t A t dr, (3.1)

where u. satisfies the boundary conditions given in Eqs. (2.2). Now suppose

u. may be written,

u t = u t +bu elt , . < 3 ' ?)

where u ... is a trial bound state function which is sufficiently accurate to

give binding, i.e. ,

e lt = / u elt^ u elt dr>0 -

i

Since u _. vanishes at the origin and at infinity u. satisfies the same boundary

condition as u, and can therefore itself serve as a proper trial scattering

function. Substituting this form for u. into the variational expression, Eq. (3.l),

i

and determining b variationally, with u. and u .. momentarily considered as

fixed, we find

b â– - - â€” J u â– , + * u. dr ,

e lt/ elt t

and

A * A t " / u t /u t dr + \rr ( / u ei/ V dr)2 - (3 - 3)

it

It follows from Eq. (2.9) that the variational estimate of A given bv Bq. (3.3)

is actually an upper bound. Conversely, the approximation to the zero energy

scattering function, obtained in the course of a calculation of an upper bound

- 12 -

on A based on Eq. (2.9), should be taken to be

\ ' ( Z7 / u elt^ u t dr)u <

"It ' â€¢*Â« " elt

rather than V, since it is the former function which can be interpreted as a

variationally determined trial function. We note that the prescription given

by Kohn to evaluate the variational parameters should be used rather than that

of Hulthen since, while both methods will provide an upper bound, for a

given form of the trial function the Kohn method will yield a lower (and

therefore better) estimate of the scattering length. ( See I.)

It is of interest to see if there are any variational calculations of

scattering lengths reported in the literature which were performed with trial

functions which may be written in the form given by Eq. (3.2-'. Such calculations

may be reinterpreted in the light of the present discussion. As an example

we note that Borowitz and Greenberg have recently performed a variational

calculation for electron-hydrogen scattering, using as a zero energy trial

function one which was explicitly constructed to be of the form of a trial

scattering function plus a multiple of a trial H~ bound state function. Since

the normalization used by these authors corresponds to what we have called

the inappropriate normalization [see Eq. (2.U)] neither the results for the

singlet case, where one bound state of H exists, nor for the triplet case,

where no bound states exist, is necessarily a bound. However, as described

in I, for those calculations which have been performed using the inappropriate

normalization the results may be converted, with a trivial amount of labor,

so that they c orrespond to the Kohn-Hulthen form and therefore do give a

bound under the circumstances considered. The method of conversion may be

briefly restated as follows. If A. and I = J u.cif u. dr have been evaluated,

- 13 -

with u. normalized as in Eq. (2.U), then the bound on A is given by

2

A < A. - A I,

If A. is positive and j A 1 1 is small compared to unity then this conversion may

be expected to give, in addition to a bound, an improved approximation to the

scattering length.

Returning to the work of Borowitz and Greenberg, we note that two separate

calculations of the singlet scattering length were performed, using two

12

trial H functions obtained by Chandrasekhar â€¢ One of these was a two parameter

function containing no dependence on the interelectronic distance, r ? , so that

no polarization was allowed for. The other function, which included a linear

r 12 dependence, contained three parameters. In each case the parameters used

where those given by Chandrasekhar. The two and three parameter functions

â€”2 â€”2 12

give values of e 1+ equal to 0.0266 a and 0.0518 a respectively

it o o

(a is the Bohr radius). The rigorous uppper bounds obtained by the conversion

process described above yielded, in both cases, slightly improved approximations

to the scattering length (see Table I).

IV. AN ALTERNATE METHOD

In this section we consider an alternate method for obtaining upper bounds

on the scattering length for the case of the scattering of a particle by a

center of force where one bound state exists. The method is based on the

formalism given by Kato for finding upper and lower bounds on the scattering

phase shift. (Since the Kato method, as opposed to the Kato identity, is

unrelated to the other methods discussed in this paper, this section may be

read independently of the others.)

The Kato identity for arbitrary scattering energy, (hkj/2m, of which

Eq. (2.1) is the zero energy form with the normalization < 9 < n ( i.e .,

9 / 0), is

- 1U -

k cot(^- 0) - k cot (^ t - 9) - Ju^c/u^dr + j w g ^w Q dr, (U.l)

where now

ancl

the boundary conditions

tt a (0) - ,

A - (U * 2)

u e (r) -> sin(kr+7 )/sin(7- 0) , f or r -> oo ,

where A is the exact phase shift and where 9 satisfies < 9 < n. The trial

function, u^. , satisfies boundary conditions of the form given in Eqs. (6.2),

but with A replaced by a trial phase shift, v 7 . , which is arbitrary; w_ is

defined as Mq. - u fl . The Kato inequalities are

" Â° 0" 1 J ( ^ u et )2 p " 1(ir -j V^V 1, Â± P9" 1 / ( ^ u 9t )2p " ldr# (U,3)

Here p(r) is a non-negative function which must fall off faster than l/r

but which is otherwise arbitrary and is chosen for convenience. The numbers

a Q and p fl are defined by the associated eigenvalue problem, in which the

equation

(/+Hp)0 Q (r) -

is considered. The asymptotic form of e (r) is characterized by a phase shift,

6(n.). The infinite, discreet set of eignevalues, j Ugl , is defined such that

for any integer, n,

6(0.^) - 9 + nn .

The smallest positive eigenvalue is a ft and the largest negative eigenvalue is -P Q .

For zero energy scattering the phase shift in the associated eignevalue

problem, 6(m.), is given, according to a theorem due to Levinson, by

- 15 -

6 ((J.) â– mn, if there is no state of exactly zero energy, where ra is the

number of bound states for the operator * + up. [if there are ra bound states

and there is one at zero energy in addition, the phase shift is (m+ xOn.J we

are considering the case for which there is one and only one bound state for

the actual physical system, i.e., for m- * 0. We choose 9 â– y> where

< y < ii. (The point is that we wish to exclude 9-0; note that the eigen-

values |i __. do not form a discreet set at zero energy.) It is then clear that

nu

-p is determined by the condition that the operator at -p o has associated with

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