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AFCRC TN-59-604

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^ KT\/^ NEW YORK UNIVERSITY

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

•- V-^ X Division of Electromagnetic Research

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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,



„ 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.



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- 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
h» An alternate method 13
B» The case of more than one bound state 17
6» 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 •

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, ■ 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 £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,» A >»J) < °-

This inequality may be written

(w,£w) < e 1 (u t ,u el ) 2 ^

where we have made use of the relation (w,n ..) ■ (u.,u ..). Thus, an upper
bound on A may be obtained from

A ± A t " / V u t dr + «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 i— 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 — 1 — 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 ±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 ° #
(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_, ■ -(2m/fo )E. V and u ... ■ v.. Since both sides of the inequality,

Eq. (2.8), are continuous at X ■ (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 ± A t " / \ l u t dr + I^ ( / U elt^ S*> 2 « ( 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 ,. » 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 ■ 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 ■ - - — J u ■, + * 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 ' •*« " 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 • 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

—2 —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

" ° 0" 1 J ( ^ u et )2 p " 1(ir -j V^V 1, ± 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.) ■ 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 ■ 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


1 3

Online LibraryTony RandallUpper bounds on scattering lengths when composite bound states exist → online text (page 1 of 3)