PHYSICAL REVIEW C VOLUME 36, NUMBER 4 OCTOBER 1987

Unified formulation of variational approaches and separable expansions
for the solution of scattering equations

Sadhan K. Adhikari
Departamento de Fisica, Universidade Federal de Pernarnbuco, 50.000 Recife, Pernambuco, Brazil

Lauro Tomio
Instituto de Ft'sica Teorica, Universidade Estadual Paulista, 01.405 Sicko Paulo, Sd'o Paulo, Braz&l

(Received 9 March 1987)

Both the variational approach and the method of separable expansion have proved to be very
useful for the solution of scattering equations. Connections are established between these two ap-
proaches. It is shown that an underlying variational principle is responsible for the good conver-
gence obtained with the method of separable expansions. It is pointed out that a proper choice of
expansion functions is essential for the rapid convergence of the method of separable expansion.
Several types of separable expansions are tested numerically for the Reid 'So soft-core potential.
It is shown that with proper choices of expansion functions these separable expansions yield more
rapid convergence than those obtained previously in other studies of separable expansions. In the
case of the three-body scattering equations the usual separable expansions are shown to follow
from the general expansion schemes studied here with special choices of expansion functions.

I. INTRODUCTION

Variational approaches' have proved to be very use-
ful for solving scattering equations which usually have
the Lippmann-Schwinger (LS) form:

T= V+ VGOT,

where T is the transition (t) matrix, V the potential, and
Go the free resolvent operator. There are different forms
of variational principles which have been used in the
solution of the LS equation. Similar variational princi-
ples have also been used in the solution of few-body
equations of the Faddeev type. Here we classify the
different variational principles in certain classes and also
establish simple connections among some of them.

We classify the existing variational principles in essen-
tially two classes: the Hulthen-Kohn —type and the
Schwinger-type variational principles. A Hulthen-
Kohn —type variational principle is essentially a varia-
tional principle for the operator T —V. The simplest
variational principle of this class is the usual Kohn '

variational principle. A Schwinger-type' variational
principle is, on the other hand, a variational principle
for the transition matrix T. The simplest variational
principle of this class is the usual Schwinger variational
principle. Starting from the simplest variational princi-
ple in each class, one can generate complicated varia-
tional principles within each class simply by considering
a more complicated variational function. This estab-
lishes a simple connection between variational principles
within the same class. No such connection exists be-
tween a variational principle of the Hulthen-Kohn type
and another of the Schwinger type.

The usual Kohn variational principle does not in-
volve the resolvent operator and is the simplest of all the

available variational principles. This variational princi-
ple, however, has the unpleasant feature of requiring tri-
al functions having the same asymptotic behavior as the
scattering wave functions. All variational principles of
the Schwinger type and the more complicated variation-
al principles of the Hulthen-Kohn type explicitly use the
resolvent operator and can employ normalizable L trial
functions.

More complex classes of variational principles can be
constructed for the operator (T —V —VGOV) using the
idea of Schwinger and of Hulthen and Kohn. These
will, in general, involve explicitly the resolvent operator
Go and one can use normalizable trial functions.

Degenerate kernel schemes and methods of separable
expansions have also been successfully used for the solu-
tion of scattering equations for both two- and three-
particle systems. These methods essentially rely upon
approximating the kernel by a sum of separable terms.
In different ways of the application of the method one
generates a separable expansion for the operators T,
T —V, or T —V —VGOV, etc. Many such separable ex-
pansions have been suggested ' in the literature. Se-
parable expansions for the two-body t matrix T are par-
ticularly very useful because of the simplicity they bring
when used in the kernel of three body equations. ' The
same is true for separable expansion of three body t ma-
trix in the context of the kernel of four body scattering
equations. '

In an early study of the Schwinger and the Hulthen-
Kohn variational principles by Gerjuoy, Moe, and Sax-
on, " the variational t matrix was written essentially in a
separable form. Later, for an appropriate choice of vari-
ational functions, equivalence was established between
the Schwinger variational t matrix and the usual separ-
able expansion for the t matrix. Similarly, we show that
a Hulthen-Kohn —type variational principle leads to a se-

1275 Qc 1987 The American Physical Society



1276 SADHAN K. ADHIKARI AND LAURO TOMIO 36

parable expansion for the operator T —V.
Starting from a variational principle for T —V, Sloan

and Brady derived a separable expansion for T —V. We
shall see that, though they called this variational princi-
ple one of the Schwinger type, the variational principle
studied by them is intimately related to the usual Kohn
variational principle and hence should be termed one of
the Hulthen-Kohn type. They also deduced two separ-
able expansions for the t matrix which they termed non-
variational. We show that these two results essentially
follow from the Schwinger-type variational principle.

In this paper we present a unified approach to the
problem of separable expansions and show their relation
to the variational principles. We emphasize that a
correct choice of expansion functions is essential for a
rapid convergence. We study in detail several types of
separable expansions for the t matrix using two types of
form factors. The first type of form factor depends ex-
plicitly on the potential and the second type does not de-
pend on the potential. It is pointed out that in the case
of the three body t matrix all the separable expansions
proposed so far should be considered as special cases of
separable expansions discussed in this paper.

In order to test how these expansions converge in
practice, we performed numerical calculations in the
case of four expansion schemes for the Reid Sp poten-
tial. ' We found that with appropriate choice of expan-
sion functions each of these expansions converges faster
than expansions studied before.

The plan of the paper is as follows. In Sec. II we
present a new unified approach to the degenerate kernel
method for LS equations and present various approxima-
tion schemes for the t matrix. In Sec. III we establish
relations between the degenerate kernel method of Sec.
II and the variational principles of the Schwinger and
the Hulthen-Kohn-type for operators T and T —V, re-
spectively. In Sec. IV a brief discussion of our approach
is presented. In Sec. V we present numerical results for
four of the approximation schemes of Sec. II for the
Reid 'Sp potential. We end the paper with a brief sum-
mary in Sec. VI.

II. DEGENERATE KERNEL APPROACHES

In this paper we shall be limited to a discussion of the
present approach for solving a LS-type equation written
in the form

T= V+ VGpT, (2.1)

though many of the ideas presented in this paper are
applicable in the case of a general Fredholm integral
equation. In Eq. (2.1) T is the t matrix, V the potential,
and Gp the resolvent operator defined by

Go (E Ho+i@)—— —

6= AB 'C,
by the following rank N approximation

(2.2)

6~= g A ~g;)DJ(fl ~C, (2.3)

with

(D ')J; ——(f, ~B ~g;) . (2 4)

between A and B ' and

~ fJ)(fJ i~ 00

between B ' and C. If the functions g; and f~ are as-
sumed to form a complete set of states, then in the limit
N ~ oo in the space spanned by fj and g;,
i,j =1,2, . . . , N, 6 of Eq. (2.2) can be taken to be iden-
tical to the approximation 6~ of Eq. (2.3):

(f, ~6~g, )=(f, ~6~~g, ), i j=1,2, . . . , N (2.5)

because we have the following assumed completeness re-
lations:

N
1= lim g ~g;)(g;

~X~ oo

N

lim g ( fJ )(f)
~N~ oo J=

which have been used to arrive at Eq. (2.3) from Eq.
(2.2). When N is large it is expected that g; and f~ will
span the relevant parts of the Hilbert space and 6& will
be a good approximation to 6. In practice, however,
one works with a large but finite N.

The above idea leads to various approximations for
the t matrix T. One can easily write T of Eq. (2.1) in the
forms

T= V(V —VGDV) 'V

=1(V ' —G ) '1

= V(1 —GO V) '1

=1(1—VG0) 'V,

(2.6)

(2.7)

(2.8a)

(2.8b)

etc. Using the above idea in Eq. (2.6), one has the ex-
pansion (with f =g)

Here,
~ g; ) and (fJ ~

are two sets of arbitrary chosen
functions. The approximation (2.3) is independent of
normalization of functions

~ g; ) and ( f~ ~

. The approx-
imation (2.3) is achieved from (2.2) by inserting a set of
complete states

N
lim y ~g )(g

~N~ oo

where Hp is the relative kinetic energy operator and
E+ie is the complex energy parameter with a small
positive imaginary part e~O.

The present approach to the separable expansion is
based on approximating the operator

T = g Vif)D (f iV,

with

(D )p ——(f, i
V —VGOVi f;),

(2.9a)

(2.9b)



36 UNIFIED FORMULATION OF VARIATIONAL APPROACHES. . . 1277

which is the approximation scheme of Ref. 7, where it
has been used with good success. It has also been
shown that expansion (2.9) follows from the Schwinger
variational principle. Similarly, Eq. (2.7) leads to

sion which results from

T= VGo(Go —GoVGo)

Using the method elaborated above we have

Ig;&D;J&g, I, (2.10a)
N

TN= y «0
I g;»j&gj I, (2.15a)

')j =B,; —&gj I
Go

I g &

B;=(g I1V '1lg;&

(g, lf &c &f lg;&,
l, m =1

where

« '} I = &f I
V

I fi & .

(2.10b)

(2.10c)

(2.10d)

(2.10e)

Here the technique of Eqs. (2.2) —(2.4) has been applied
twice; first, in relation to the operator (V ' —Go) ' and
then in relation to V ' explicitly, and two types of ex-
pansion functions f and g are used in the two expan-
sions. It is easy to verify that Eqs. (2.9) and (2.10) are
solutions of the LS equation (2.1) with the following ap-
proximate potentials:

but now with

')j, = &g, I «o —Go VGo)
I a & . (2.15b}

T = V+ V(E H+i e) —' V, (2.16)

It is easy to realize that Eq. (2.15) follows from Eq.
(2.13) by changing

I f; & to Go Ig; &. Expansion (2.15)
was studied before by Sloan and Brady. Thus, by writ-
ing complicated expressions for T or by replacing the ex-
pansion functions by more complex ones, one can derive
many expansions for T. We shall see that all of them
are intrinsically related to the Schwinger variational
principle.

In a similar manner it is possible to make separable
expansion for T —V which is equivalent to writing T as
the sum of V and a separable expansion. For example,
applying the technique of (2.2) —(2.4) to the last term of

VX= y V lf; &~,J&f, I
V (2.11) one gets the following approximate t matrix,

Tx = V+ g V
I f; &D;, &fj I

V (2.17a)
N

vx= & Ig;&&ij&g, I, (2.12) where

with B given by (2.10c). The approximation schemes so
defined are independent of the normalization of the arbi-
trary functions f and g.

Finally, Eq. (2.8a) leads to the expansion

Tx = & V
I f &D, & gj I

(2.13a)

with

(D '),; =(g,
I
(1 —GoV)

I f; &, (2.13b)

which is the simplest separable expansion that does not
involve V ' or Go '. Expansion (2.13}results if we solve
the LS equation (2.1) with the approximate potential

v~= g vlf;», &gj I
(2.14a)

where

(J '),;=(g, I f; & (2.14b)

In this way it is possible to generate many approxima-
tion schemes. We shall mention finally one more expan-

respectively. Now J and K of Eqs. (2.11) and (2.12) are
defined by

(J ')j, =(fj I vl f;&

and

(D ')j, ——(f l(E —H)lf &. (2.17b)

N

T~=V+ y «. Ig, »,, (g, IG.V, (2.19a)

Expansion (2.17) was obvious from the work of Takatsu-
ka and McKoy. Equation (2.17) does not involve a
resolvent operator and has the problems associated with
the vanishing of the determinant of the matrix D ' in
(spurious) positions other than bound states and reso-
nances. Though this expansion is the simplest, it may
not be very useful from a practical point of view. It has
later been pointed out that very special care is needed
to choose the expansion functions f and f. We shall
show that expansion (2.17) is a consequence of the usual
form of Kohn variational principle.

In the simplest choice, of course, one uses f =f in Eq.
(2.17), though other choices are possible in which one
can avoid the problem of spurious vanishing of the
determinant of the matrix D '. One interesting choice
is (fj I

= (gj I
Go, when Eq. (2.17) yields

T~ ——V+ g V
I f; &Dj(gj I

Go V, (2.18a)
ij = I

where

(D-'), , =&g, I(1—G, V) If, & . (2.18b)

Another possibility is to take f =f in Eq. (2.17) and re-
place

I f; & by Go Ig;& and (f, I
by (gj I Go, while Eq.

(2.17) yields



1278 SADHAN K. ADHIKARI AND LAURO TOMIO 36

T= V+ VGoV+ VGo(V ' —Go) 'Go V, (2.20)

one gets the following approximate t matrix,

N

TN V+ VGo V+ g VGo
I f &D J &gj IGo V . (2.21a)

with

(D ')p ——(g~ I(V ' —Go)If &. (2.21b)

Equation (2.21) is the simplest expansion of this type.
the next section we present a relation of the above ex-
pansions with variational principles.

III. VARIATIONAL PRINCIPLES

It has been shown in cases of some of the degenerate
kernel approaches presented in the preceding section
that they can be derived from some variational princi-
ples which are responsible for good convergences of
these approaches. ' Some of the degenerate kernel
schemes of the preceding section have been previously
termed nonvariational and this has been made responsi-
ble for the poor convergence obtained with these
schemes without making an adequate search for an ap-
propriate expansion function. We shall show that all the
degenerate kernel schemes presented in the preceding
section can be derived from some variational principle,
and the poor convergence rate obtained in previous nu-
merical calculations using some of these schemes result-
ed because of the difficulty of finding an appropriate ex-
pansion function.

It has been shown in Ref. 7 that expansion (2.9) can be
obtained from the usual Schwinger variational form for
the t matrix

&p I
T

I
p'&=&I

I

V
I 0,'+'&+&0,' 'I v

I

p'&

—&q,'-'I (v —VG, v)
I q,'+'&, (3.1)

which is stationary with respect to small variations
of

I

P'+ '
& and ( g' '

I

around their correct values.
P P

The stationary property of this expression follows easily
from the LS equations for tP'+, ' and g'

P P

where

(D '),;=(g, I
(Go —GoVGo) Ig; & . (2.19b)

Expansion (2.19) was studied by Sloan and Brady, ' who
derived Eq. (2.19) from a variational principle which
they termed as one of the Schwinger type. We do not
agree with this conclusion of theirs for reasons we dis-
cuss in the next section. Now it is obvious that, by mak-
ing special choices of

I f; & and (f, I

in Eq. (2.17), one
gets a different expansion, which will be related to (2.17).

Finally, it is possible to make separable expansions for
more complicated operators such as T —V —VGQ V. For
example, applying the technique of (2.2) —(2.4) to the last
term of

Using the following variational expressions for the wave
functions in (3.1),

N

I q', +, '& = & a;(p')
I f; &,

i =1
(3.3a)

(3.3b)

—(X' '
I
(V ' —G )

I

X'+, '&, (3.4)

which is stationary with respect to small variations of
I

X'+, '
& and (X~ '

I
around their correct values, and

where

Ix',+'&=vIq'+'& and &x',-'I =(q,'- I
v.

The stationary property of (3.4) follows from the follow-
ing equations satisfied by the 7's:

Ix',+'&=vIp'&+vG, Ix',+'&,

(X' 'I =(p
I

V+(X' 'I G V .

(3.5a)

(3.5b)

Next, making the following variational expansions for
the functions,

I

x',+'
& = g ~;(p')

I g; &, (3.6a)

&x,'-'I = y &g, Ib, (p),
j=1

(3.6b)

and demanding that expression (3.4) be stationary with
respect to variations of coefficients a;(p') and bj(p), one
can easily solve for a; and b~ and, using Eqs.. (3.4) —(3.6),
one recovers expansions (2.10a)—(2.10c). Then technique
of Eqs. (2.2) —(2.4) has again been applied to evaluate
(2.10c). This has the advantage of making the result
(2.10d) —(2.10e) independent of the normalization of the
expansion functions. Expansion (2.10) has the advantage
of having energy independent analytic simple form fac-
tors if the f 's have that property, even if V is energy
dependent.

Next, we consider the separable expansion (2.13). It
can easily be related to expansion (2.9) if one takes
(g~ I

= (f, I
V, when the two expansions become identi-

cal. Now it is easy to realize that expansion (2.13)
should follow from the following stationary form for the
t matrix,

and demanding that expression (3.1) be stationary with
respect to variations of a;(p') and b, (p), one can solve
for a;(p') and b~(p) and, using Eqs. (3.1)—(3.3), one easi-
ly recovers expansion (2.9).

Expansion (2.10) is easily related to the Schwinger
variational expansion (2.9). This can be seen by using

I g; &= V
I f; & and (fj I

V=(g,
I

in Eqs. (2.10a)
—(2.10c). In other words, Eq. (2.10) can be derived from
the following stationary form for the t matrix:

& p I
T

I

p' & =
& p I x,'+, ' &+ & xp'

'
I
p' &

I
op+'& —

I
p'&+G V

I

0'+'&

&q,'-'I =&pI+&q', -'I vG. .

(3.2a)

(3.2b)

&p I
T

I
p'&=&pl V

I 0,'+ &+&xp 'I p &

—(X(p '
I
(1—Go V)

I

@'+. ' &, (3.7)



36 UNIFIED FORMULATION OF VARIATIONAL APPROACHES. . . 1279

which is stationary with respect to small variations of
f&+' and Xz

' and where X and g are defined by (3.5b)
and (3.2a), respectively. The stationary property of (3.7)
follows from Eqs. (3.2) and (3.5).

Though expansions (2.9), (2.10), and (2.13) all follow
essentially from the Schwinger variational forms (3.1),
(3.4), and (3.7) for the t matrix, respectively, in practical
numerical application the success of these expansions
will depend on appropriate choice of expansion functions
f or g. Also, it is now easy to realize that expansion
(2.15) can also be derived from a variational principle of
the Schwinger type.

Next, we consider expansion (2.17), which is really a
separable expansion for the operator T —V rather than
for T. This expansion can be obtained from the usual
Kohn variational form for the t matrix:

( p I
T

I
p' & = & p I

v
I

p' & + & p I
v

I &,'+ '
&

+(x,'-'I v Ip')
—&x'

I
(G —v)

I

xp'+'&, (3.8)

which is stationary with respect to small variations of
7'+' and g' ' and where

P P

ly(+))
I

@(+))
I p ) G Iy(+)) (3 9a)

'
I

—&p I

= &&'

with IXp ') and (Xp '
I

defined by Eq. (3.5). The sta-
tionary property of (3.8) follows from Eq. (3.5). Using
the following variational expressions,

N
Ixp'+') = g ~;(p')

I f, ), (3.9c)

N
(I'' 'I = g b)(p)(f& I

(3.9d)

in (3.8) and, demanding that (3.8) be stationary with
respect to variations of a;(p') and b~(p), one can solve
for a;(p') and b~(p):

N

a;(p') = 2 D J &fj I
v

I
p'&

j =1
N

b, (p)= & &p
I

v if; »,

Substituting (3.10) in Eqs. (3.8) and (3.9), one recovers
(2.17a).

In practical application of the Kohn method, the con-
struction of the D matrix via (3.10c) is a delicate task. '

The matrix to be inverted involves E —H. For positive
energies, E —H has a continuous spectrum. So it can
easily happen that one eigenvalue of the finite matrix is
small, specially if the number of basis functions is large.
Then matrix D ' becomes nearly singular, which causes
special problem with expansion (2.17a). With all other

with D given by (2.17b):

'), - = &f, I
(Go ' —V) If & = &f) I

« —»
I f &

(3.10c)

expansions of Sec. II the matrix to be inverted becomes
singular if VGo has an eigenvalue near 1, which is the
condition for a true bound state or resonance pole. This
is why expansion (2.17a), although very instructive and
simple, is not very useful from a practical point of view.
For a numerical application of (2.17a) special care is
needed for the choice of the expansion functions f and

It is now easy to realize that expansion (2. 18a) can be
obtained from the following stationary expression for the
t matrix:

&pl T Ip'&=&x
I
v

I
p'&+&a

I
v l~',+'&

+ &x'
I
G.v

I

p'&

—&x'
I
(1 —G. v)

I i,'+'&, (3.11)

where
I

f' +, ') and (Xp '
I

are defined by Eqs. (3.5) and
(3.9). Equation (3.11) is again a stationary expression for
T —V, rather than for T. The stationary property fol-
lows from the use of Eqs. (3.5) and (3.9). Using the fol-
lowing variational expressions,

(3.12a)

(x' 'I = g b (p)(g
j=1

(3.12b)

TN ——Q Ig;)D; (g IG V,
ij =1

N
T~= X VGo lg &D, &g) I,

(3.14a)

(3.14b)

which they termed nonvariational. Expansion (3.14b) is
essentially the same as expansion (2.15a). It is now clear
that expansions (2.15a) or (3.14) can be derived from a
Schwinger-type variational principle.

The stationary expression (3.13) for the t matrix gives
essentially a stationary estimate for the operator T —V
and hence is easily related to the Kohn variational form
(3.8). So, variational principle (3.13) should be termed a
Hulthen-Kohn —type variational principle and not a
Schwinger-type variational principle as suggested by
Sloan and Brady. Expansions (2.17)—(2.19) are easily
related with each other if we make special choices for

in (3.11) and, demanding that (3.11) be stationary with
respect to variations of a;(p') and b~(p), one can obtain
expansion (2.18a).

Finally, expansion (2.19a) can be easily shown, in a
similar way, to follow from the following variational es-
timate for the t matrix:

&p I
T

I
p'&=&p

I
v

I
p'&+&p

I
VGo I&p+'&

+&x' IG.vip'&
—(yp '

I
(Go —GovGo)

I
Xp+') . (3.13)

This variational principle was used by Sloan and Brady
to derive expansion (2.19a); as a by-product they also de-
rived two other approximations:



1280 SADHAN K. ADHIKARI AND LAURO TOMIO 36

the expansion functions in (2.17), and all of them essen-
tially follow from the Kohn variational principle. Simi-
larly, expansions (2.9), (2.10), (2.13), and (2.15) can be
easily related with each other if one makes special
choices of expansion functions. All these latter expan-
sions essentially follow from the Schwinger variational
principle, but no such relation can be established be-
tween an expansion of the former type and an expansion
of the latter type.

Now it is not difficult to realize that expansion (2.20)
follows from the following stationary expression for the t
matrix:

&P I
T

I P & = &P I
(v+ VGov)

I P &

+&p I
vG

I g +, ')+ & 4 ' '
I

G v
I

p')

&q( ) I(v 1 G ) I@(+)) (3.15)

where

I

q(+)) vG v
I

q(+ )
P P

and

Variational principle (3.15) is really a variational princi-
ple for the operator T —V —VGO V and hence is neither
one of the Hulthen-Kohn nor one of the Schwinger type.

From the discussion of this section it is obvious that
one can always associate a variational principle with a
degenerate kernel scheme for the t matrix. This will be
true, in general, and one can construct other degenerate
kernel schemes for the t matrix and find out the underly-
ing variational principle from which it is derived.

IV. DISCUSSION

(4.2)

However, they failed to note the relation of this ap-
proach to the Kohn variational principle. With the
Bessel-Weinberg states the numerical problem with the
inversion of matrix D ' was avoided. We do not see
much advantage in their approach, as they have to rely

The methods presented in Secs. II and III not only
give a unified degenerate kernel approach to the solution
of the LS equation, but will have application in more
complex multichannel few-body problems, especially in
making separable expansions for the three-body t matrix.
Many of the approaches have already been used in vari-
ous contexts. Also, many approaches of other authors
can easily be related to the methods presented here with
special choices of expansion functions. In this section
we relate our approaches to those of other authors.

Essentially, the simplest Kohn variational expression
(2.17a) has been recently used by Rawitscher and Delic'
in the special case where f; =f; =(t. ;. They used the en-

ergy dependent Bessel-Weinberg states P; with the prop-
erty"

(4.1)

where (2.17b) simplifies to

on numerically constructed energy dependent functions,
in general, which did not yield impressive convergence
for the Woods-Saxon model studied by them —for the 5
wave case 15 expansion functions were needed to obtain
results correct to three significant figures. (The Woods-
Saxon potential studied by them is expected to converge
faster than the soft core Reid potential studied by other
authors. }

The method proposed by Revai, Sotona, and Zofka'
bears some similarity with the method of Eqs. (2.10).
Specifically, they employ g;=f, in (2.10) and approxi-
mate the coefficients & fj I f() by certain weight func-
tions:

The functions f are the harmonic oscillator basis func-
tions. The convergence obtained by them is rather slow
in the case of the Reid 'So potential —some 20 expan-
sion terms are needed to get a result accurate to two
significant figures. In the case of the Woods-Saxon po-
tential, which is usually expected to show better conver-
gence properties, their convergence is reasonably impres-
sive. From this it is expected that the method of
Rawitscher and Delic, ' if applied to the case of the
Reid 'So potential, will show poorer convergence prop-
erties than those obtained in their treatment of the
Woods-Saxon potential. In our numerical study with ex-
pansion (2.10), we shall see that if the expansion func-
tions are properly chosen, it can show good convergence
properties, and the poor convergence obtained by Revai
et aI. ' is related to their use of inappropriate expansion
functions.

Sloan and Brady studied expansion (2.15a) and did
not obtain satisfactory convergence. We believe an inap-
propriate choice of expansion function was responsible
for the unsatisfactory convergence obtained by them.

Much effort has been recently made to relate
Hulthen-Kohn — and Schwinger-type variational princi-
ples, and also to find out which of these two types of
variational principles converges faster. Though answers
were given to this question in limited contexts by
different workers, we do not believe that an absolute
answer to this question even should exist. This is be-
cause we have seen that these two types of variational
principles are really stationary expressions for two
different operators —T and T —V for the Schwinger-
and the Hulthen-Kohn —type variational principles, re-
spectively. So, in a general context one should not try to
relate the two types of variational principles. Also, the
success of these separable expansions should depend on
an appropriate choice of expansion functions. As we are
expanding two different functions in these two ap-
proaches, it is expected that the expansion functions
should be different in these two cases. The expansion
functions, which give good results in the context of the
Schwin ger variational principle, should and may not
yield good result in the case of the Kohn variational
principle. Hence the calculation of Takatsuka, Luc-
chese, and McKoy, who demonstrate the better conver-
gence properties of the Schwinger variational principle
over the Kohn variational principle while using same ex-



36 UNIFIED FORMULATION OF VARIATIONAL APPROACHES. . . 1281

pansion functions in both approaches in a simple model
potential, do not and should not give any general answer
to the question of the superiority of one type of varia-
tional principle over another, because the conclusions of
Ref. 6 are expected to be model dependent.

It is interesting to note at this stage that, with a spe-
cial choice of expansion function, expansion (2.9) yields
most of the commonly used separable expansions. The
Weinberg series is readily obtained if we take

LS equation is schematically written as

T = V(E)+ V(E)Go (F)T, (4.7)

where the energy dependence of T is not explicitly
shown. We show that the GUPE follows with a special
choice of expansion function in Eq. (2.10), which pro-
vides us with a simple way of making a separable ap-
proximation for the t matrix with energy independent
form factors, if the functions g are energy independent.

Here we have expansion functions g satisfying

in Eq. (2.9), where the P's are (energy-dependent) eigen-
functions of the LS kernel, satisfying

A, ; V( —8)Go( 8)
I

—Q;( 8—)) =
I g;( —8) &,

&, (1(J(—8)
I
Go( —8)V( —8)=(qJ( —8)

I

(4.8a)

(4.8b)
~ VGolf &=If &

A,~. ( g, I Go V = ( QJ I

The biorthogonality relation

reduces (2.9) to

TN

(4.3a)

(4.3b)

(4.3c)

(4.4)

and normalized according to

(qJ( —8)
I
Go( —8)

I qg( —8))= —
&gj .

Now, in Eq. (2.10), if we take

I fi & =Go( —8)
I 4( —8) &

the use of (4.8) reduces (2.10) to

(4.8c)

(4.9a)

(4.9b)

Weinberg series (4.4) uses energy dependent expansion
functions P.

The unitary pole expansion (UPE) uses the eigenfunc-
tions g of Eq. (4.3) at a bound state energy —8—and
then Eq. (2.9) reduces to

where

(4.10a)

(4.5 )

(D '),; =8,; —(f, ( —8)
I
Go(E)

I P;( —8) ) (4.10b)

(D ')J; ——k(( —8)5); —(Q, ( —8)
I

Go
I Q;( 8)) . (4.5—b)

With appropriate choices of expansion functions Eq.
(2.9) has been shown " to yield the expansions' of
Ernst, Shakin, and Thaler, and of Bateman.

The UPE (4.5) is very useful because there the expan-
sion is done via energy independent form factors. The
UPE of (4.5) is not readily generalized to the case of the
t matrix involving more than two particles.

The main problem with the three particle system is
that though in this case one has a LS-type equation, the
potential appearing in this equation is energy dependent.
It is easy to realize now that T of Eq. (4.5a) can be ob-
tained if we solve the LS equation with the following se-
parable expansion for V:

(4.6)

As V is energy dependent in the case of the three-
particle system, the energy independent approximation
(4.6) must fail at energies E&—B. Hence a modification
of the UPE is necessary. One modification of the UPE
has been proposed by Casel, Haberzettl, and Sandhas, '

known as the generalized UPE (GUPE). In this case the

(8 ')J, = (QJ( —8)
I Go( —8)V(E)Go( —8)

I 1(;(—8) ) .

(4.10c)

Equation (4.10) constitutes the GUPE of Casel, Haber-
zettl, and Sandhas. ' If V is energy independent, Eq.
(4.10) reduces to the usual UPE (4.5). The present
derivation of the GUPE explicitly demonstrates the fact
that it follows from a special choice of expansion func-
tion in a more general approximation scheme —Eqs.
(2.10)—considered in this paper. The relation of (2.10)
to the Schwinger variational principle as demonstrated
in Sec. III also explicitly shows that the GUPE should
follow from a special choice of expansion function in a
Schwinger-type variational principle —Eq. (3.4)—for the
t matrix.

So the formalism presented in this paper has interest-
ing application in the case of the three-body problem.
Another alternative to the GUPE in this case is the en-
ergy dependent pole expansion (EDPE), which was con-
structed by Sofianos, McGurk, and Fiedeldey' using ex-
pansion (2.9). Now it is also easy to realize that two
more expansions proposed by Fonseca, Haberzettl, and
Cravo' follow also from expansion (2.9) with a special
choice of expansion function, which has never been ex-
plicitly pointed out in the literature. All these demon-
strate the usefulness and generality of the formalism
presented in this paper.



1282 SADHAN K. ADHIKARI AND LAURO TOMIO 36

V. NUMERICAL CALCULATIONS

In order to see how the methods presented in this pa-
per work in practice, we have performed numerical cal-
culations with the Reid' 'S0 soft core potential, which
is highly repulsive at short distances, and this makes it
particularly difficult to obtain good convergence with
this potential. So, if we can demonstrate good conver-
gence with this potential, it is expected that the method
will converge well with other potentials. This potential
is constructed as a linear superposition of various Yu-
kawas and in momentum space it is '

(p +p')'+P, '
V, (p,p')=, g v,ln, , (5.1)

4 up', =, (p p')'+—P,'
where P& ——0.7 fm, P2 ——4P~, P3 ——7P&, v

&

———10.463 MeV,
v2 ———1650.6 MeV, and v3 ——6484. 2 MeV.

We chose to study only four approximation schemes
presented in this paper, namely Eqs. (2.9), (2.10), (2.13),
and (2.18). We choose to study these expansions numeri-
cally, not because we think them to be superior to the
other expansions in a general context, but because they
are reasonably simple and will allow us to make some
general conclusions about the convergence of these
methods. Specifically, Eq. (2.9) was studied before and
was shown to produce good convergence. We shall
demonstrate that if the expansion functions are ap-
propriately chosen one can significantly improve on the
convergence rate obtained before. Equation (2.10) forms
the basis of the GUPE. A previous application of Eq.
(2.10) in the context of the Reid 'So potential produced
poor convergence. ' We show that though numerical
application of Eq. (2.10) is more complicated, in general,
because it involves inversion of two matrices, good con-
vergence can be obtained with the proper choice of ex-
pansion function. Equations (2.13) and (2.18) use non-
symmetric approximations for the t matrix. We shall
demonstrate that numerically this does not cause any
special problem and rapid convergence can be obtained
with an appropriate choice of expansion function.

We shall use simple analytic normalizable L func-
tions as expansion functions in these cases. As we do a
momentum space calculation for the t matrix, we would
like to build in appropriate asymptotic properties in
these expansion functions in the momentum space. We
would like to construct the expansion functions using
the following functions, '

2 2

+.(P)=, , C„', », n =1,2, . . . , N (5.2)
p"+0' p '+0'

which have been successfully used in similar calcula-
tions. ' In Eq. (5.2) C„', is the Gegenbauer polynomi-
al ' of degree n —1, where )33 is an arbitrary parameter.
The set (5.2) is formally equivalent to the set of functions
(p +P ) ", n =1, . . . , 1V. The functions defined by (5.2)
are actually Sturmian functions for the Coulomb poten-
tial and hence are orthogonal with respect to certain
weight functions. Because of this orthogonality proper-
ty the matrices to be inverted in the present calculations
become well conditioned.

1/2

f Jo(pr)f, (p)p'dp' (5.3b)

with fj(p) = ( f~ I p ). It is straightforward to verify, by
using Eq. (5.3), that if we take

f (p)=p (p +P ) ", n=1, . . . , E (5.4)

then ( f~ I

V
I p ) has the desired p dependence: In the

case of a Yukawa potential, we note the following.
(i) As p ~0, (f, I

V
I p ) tends to a constant.

(ii) As p —+ oo, (f, I

V
I p ) behaves like a polynomial

in p with the leading term -p . Recalling that the
functions of Eq. (5.2) are formally equivalent to the func-
tions (p +P ) ", n =1, . . . , N, we use the following set
of functions in our numerical study of (2.9) in place of
the set (5.4):

f.(p) =p'J'. (p»)—

p' p'+P'
p'

2 2, n=1, . . . , N .
p +/3

(5.5)

The functions (5.5) are orthogonal with respect to cer-
tain weight functions and hence have the advantage of
leading to a well conditioned matrix D ' of Eq. (2.9),
which is to be inverted numerically.

Next we consider expansion (2.10). Here we have two
sets of expansion functions f and g. It is easy to realize
that the functions g should be so chosen that ( g; I p )
represents well the p dependence of ( q I

T
I p ) or,

equivalently, of (q I
V

I p ), namely, (g; I p ) should tend
to a constant as p~0 and should behave like a polyno-
mial in p with the leading term p as p~ ao with V
given by (5.1). It is easy to realize that functions (5.2)
satisfy this property. Hence we take

g;(p)= &g
I
p)=F (p—) (5.6)

where F; (p) is given by Eq. (5.2). In Eqs. (2.10) f 's

enter in the following expansion,

with

l, m =1
I fi &ci &fm I

(5.7a)

First, let us consider expansion (2.9). It is easy to
realize that this expansion will converge well if the p
dependence of (q I

T
I p ) is well represented by the p

dependence of (fJ I
V

I p ), or equivalently, if the p
dependence of (q I

V
I p ) is well represented by the p

dependence of (fj I

V
I p ). For small p, (q

I

V
I p )

tends to a constant independent of p and, as p~~,
(q I

V Ip) behaves like a polynomial in p
~ with the

leading term -p, when the potential V is given by
(5.1). The p dependence of ( f~ I

V
I p ) can be obtained

from the following explicit forms,

(fj I
V

I p ) = j j 0(pr)V(r)b~(r)r dr, (5.3a)
0

where



36 UNIFIED FORMULATION OF VARIATIONAL APPROACHES. . . 1283

') i=&f
I

v lf~& . (5.7b)

In the limit N~ oo, V&
' of (5.7a) ~V ' in the space

spanned by functions f; and in this limit, by multiplying
Eq. (5.7a) by V from the right and from the left, one ar-
rives at

the first function(s) in a set following Refs. 7 and 11, and
this will significantly improve the convergence rate for
small N.

In the case of expansion (2.9), the new basis functions
f; are defined by

Vw= g V
I fi &Cim &f

I, m =1
(5.8)

V If i & =T
I p &,

If; &= If; &,

(5.9a)

(5.9b)

As Eqs. (5.7a) and (5.8) are equivalent, the functions f,
which are good in the context of expansion (5.8), will
also be good in the context of expansion (5.7a). As ex-
pansion (2.9) results if we solve the LS equation with the
separable expansion (5.8) for the potential, it is easy to
realize that functions which were appropriate for expan-
sion (2.9) will also be good in the case of expansions
(5.7a) and (5.8). Hence in expansion (2.10) we choose the
same functions f used in expansion (2.9), namely fj(p)
given by Eq. (5.5).

Finally, we consider approximations (2.13) and (2.18).
In Eq. (2.13), (q I

V
I f; ) and (g, Ip ) should be good

functions to represent the q and p dependences of
(q I

V
I p ), respectively. This can be easily achieved if

we choose f and g to be given by (5.5) and (5.6), respec-
tively. Recalling that Eq. (2.18) results if we substitute
the approximate t matrix given by (2.13a) on the right-
hand side of the LS equation T = V+ TGo V, it is easy to
realize that f and g given by (5.5) and (5.6) are ap-
propriate choices in (2.18).

Though we shall not present in this paper numerical
results for other degenerate kernel schemes presented in
Sec. II it is now easy to choose appropriate functions for
other methods. For example, approximations (2.15) and
(2.19) will present good convergence if we take all the
expansion functions g in these methods to be given by
(5.6).

After having decided about the expansion functions,
we are now prepared to perform numerical calculations
with approximations (2.9), (2.10), (2.13), and (2.18).
Though we have chosen the functional form of the ex-
pansion functions, there is still some arbitrariness left in
the choice of the parameter P in (5.2) and (5.5). This ar-
bitrariness is turned to good advantage —we vary P nu-

merically to obtain rapid convergence. Though in the
context of each of the above approximations a new P
would lead to best convergence, we prefer to choose the
same P in all approximations for the same function.
However, P is taken to be different if the functions are
different; specifically, we choose one P for Eq. (5.6) and
another P for Eq. (5.5) in order to obtain best conver-
gence.

We performed numerical calculation in momentum
space in double precision —correct up to 14 significant
figures. An explicit momentum space representation of
the quantities involved in numerical calculation can be
obtained in Refs. 7 and 11. After a little experimenta-
tion we fixed P=3.5 fm ' in Eq. (5.6) and @=2.5 fm
in Eq. (5.5). In all four cases final convergence was im-
pressive, but the results were not very good for small N.
By making a change of basis functions we can make cer-
tain rank-1 (N = 1) t matrix elements exact if we modify

This new set of expansion functions when used in expan-
sion (2.9) will lead to a rank-N t matrix T~ with the
properties:

T~ Ip)=T Ip), N=1, 2, . . .

&p
I
T~=&p

I
T, N=1, 2, . . .

(S.loa)

(5.10b)

I
f()= g f;)C;(k), (5.11a)

with

C;(k)= g DJ(f) I
VIk), (5.11b)

where all the calculations in (5.11) are performed at zero
energy. Thus the new basis function f ~

is available as a
linear combination of the old functions. In practice, we
used N'= 10.

The same thing can be easily done in the case of Eq.
(2.13). It is easily seen that in order to satisfy property
(5.10) in this case, one must use new basis functions
defined by

If &= If; &, &g;
I

=&g; I, t=2, 3, . . . .

(5.12a)

(5.12b)

(5.12c)

Again, in practical implementation we take
p =k =0 fm, the on shell momentum at zero energy.

—1

Using (2.13) and (5.12), one easily obtains
N'

If, &= y I f, )c,(k),
i =1

&gl I

= y &g,
I

(k»

(5.13a)

(5.13b)

together with Eq. (5.12c). In Eq. (5.13),

where, of course, we use the modified set f; for both
I f; ) and its transpose (f; I

. If we take p =k, the on-
shell momentum at an energy, the half-on-shell t matrix
of any rank will be exact.

The practical implementation of (5.9) is easy after we
have performed calculations with the old set at a partic-
ular energy. In fact, we replace T of (5.9a) by Ttv of
(2.9), where N' is sufficiently large that Tz has essential-
ly converged. Then, (5.9a) becomes, for p =k =0 fm
the on-shell momentum at zero energy,

N'

VIf, )= g VIf;)D;, (f, I
vIk),

i j =1
so that

N'



1284 SADHAN K. ADHIKARI AND LAURO TOMIO 36

(5.14a)
TABLE I. The coefficients C, C, and 0 * defining the new

functions at energy E=k =0. These coefficients are defined
by Eqs. (5.11b) and (5.14).

(5.14b)

and

I'~ fi &=(T—&)
~

k & (5.15a)

Again, the new basis functions are obtained as a linear
combination of the old functions. In practice, we em-
ployed N'= 10.

Finally, it is easy to verify that in order to have the
property (5.10) in the case of expansion (2.18), one
should choose, for p =k =0 ' fm,

1

2
3
4
5
6
7
8
9

10

C;(k)

—36.5758
—56.0848

31.1240
28.8955
11.8785

—0.7737
—6.9354
—9.9163

0.2502
—0.1465

C;(k)

—36.3284
—56.5445

30.1631
28.7684
12.8540

—0.7223
—4.7506
—7.2262
—2.9682
—3.1923

C;(k)

70.9539
36.0544

—81.7214
—10.0807

10.5718
12.9460
5.8926
2.4709
0.6873
0.6361

(5.15b)

together with (5.12c). The implementation of (5.15)
again is not difficult. However, in this case we used the
expansion functions given by (5.12c) and (5.13). With
this choice —given by (5.12)—it is easy to verify that
only Eq. (5.10b) is satisfied and Eq. (5.10a) is not
satisfied. In this case we have a one-sided Kowalski-
Noyes property.

In case of approximation (2.10) we do not perform any

change of expansion functions. This case is particularly
more complicated than other examples studied, in that
here we need two matrix inversions, whereas in other ex-
amples we need only one matrix inversion. Even then
we obtain good convergence in this case.

In Table I we present the coefficients C; of (5.11), and
C; and C of (5.14). The values of P used are /3=2. 5
fm ' in Eq. (5.5) and P=3.5 fm ' in Eq. (5.6). In Table
II we present the on shell phase shifts at different ener-

TABLE II. Phase shifts for the Reid 'So potential for different N calculated by using the following
degenerate kernel schemes: A—Eq. (2.9), B—Eq. (2.13), C—Eq. (2.18), and D—Eq. (2.10). The en-
tries at E, =0 are the scattering lengths in fm. The results for N =16 have all converged to four
significant figures. The numbers for schemes A —C for N & 10 have been calculated using a
modification of the basis function as described in the text.

E,
(MeV)

B
C
D

—17.15
—17.10
—17.12

0.52

—17.1S
—17.10
—17.12
—14.02

—17.15
—17.10
—17.12
—15.96

Results for N =
8

—17.15
—17.10
—17.12
—17.15

10

—17.15
—17.10
—17.12
—17.12

—17.1S
—17.14
—17.14
—17.14

16

—17.15
—17.15
—17.15
—17.15

48

72

104

A

C
D

A
B
C
D

A
B
C
D

A
B
C
D

A
B
C
D

0.8191
0.8188
0.5796

—0.2776

0.3705
0.3707

—0.7158
—0.5412

0.2053
0.2059

—1.0582
—0.6522

0.0672
0.0672

—1.2694
—0.7678

0.0299
0.0303

—1.4458
—0.9561

0.8500
0.8279
0.8367
0.8664

0.4535
0.4174
0.4411
0.4330

0.2783
0.2596
0.2624
0.2558

0.0859
0.0925
0.0730
0.0717

—0.2499
—0.2019
—0.2212
—0.2549

0.8554
0.8467
0.8545
0.8378

0.4363
0.4448
0.4444
0.4356

0.2577
0.2702
0.2636
0.2691

0.0758
0.0831
0.0795
0.0911

—0.2217
—0.2289
—0.2148
—0.2048

0.8604
0.8576
0.8603
0.8573

0.4398
0.4417
0.4403
0.4403

0.2620
0.2628
0.2621
0.2636

0.0781
0.0749
0.0799
0.0815

—0.2167
—0.2240
—0.2183
—0.2253

0.8604
0.8605
0.8606
0.8608

0.4398
0.4398
0.4402
0.4403

0.2624
0.2626
0.2628
0.2636

0.0798
0.0801
0.0802
0.0803

—0.2180
—0.2180
—0.2166
—0.2177

0.8605
0.8606
0.8606
0.8606

0.4401
0.4402
0.4402
0.4402

0.2627
0.2629
0.2630
0.2630

0.0802
0.0803
0.0803
0.0805

—0.2167
—0.2164
—0.2164
—0.2160

0.8606
0.8606
0.8606
0.8606

0.4402
0.4402
0.4402
0.4402

0.2630
0.2630
0.2630
0.2630

0.0803
0.0803
0.0803
0.0803

—0.2164
—0.2164
—0.2164
—0.2164



36 UNIFIED FORMULATION OF VARIATIONAL APPROACHES . . ~ 1285

gies and scattering lengths in these four cases. It is easy
to realize that the convergence is much better than that
previously obtained in any degenerate kernel method
using the Reid So potential. For example, it is better
than that obtained in Refs. 7, 9, 11, 12, and 14. The
off-shell matrix elements also converge equally well, in
general.

Unfortunately, we should not compare results of
different approximation schemes shown in Table II and
claim superiority of one scheme over another because we
are using basically the same expansion function in all the
cases studied. In order to obtain the best convergence
the expansion functions in each scheme are supposed to
be different. Specifically, though with a particular
choice of expansion function and for a particular
potential —the Reid 'So potential —one approximation
scheme converges faster than another, the situation may
change if the expansion functions and the potential are
changed. The schemes (2.9), (2.10), and (2.13) can all be
derived from Schwinger-type variational principles and
wi11 yield identical results if the expansion functions used
in the scheme are related in a certain way. For example,
Eqs. (2.9a) and (2.13a) will yield identical results if ( fj.

~

of (2.9a) is related to (gj ~

of (2.13a) by (f, ~
V=(gj ~.

In other words, expansions (2.9), (2.10), and (2.13) are all
equivalent. Expansion (2.18) can be derived from a vari-
ational principle of the Hulthen-Kohn type and is not
related to a variational principle of the Schwinger type.
Also, as has been pointed out before, one should not
claim the superiority of one type of variation principle
over another, because such a claim will be limited, in
general, to a special case and will depend on the expan-
sion functions and the potential used.

VI. SUMMARY

We have presented a unified formulation of the degen-
erate kernel scheme and the method of separable expan-
sions for solving LS-type equations, and have established
a relation between them. We have shown that all degen-
erate kernel schemes can be derived from a variational
principle. We have proposed various practical methods
for solving LS-type equations. We have performed nu-
merical calculations —for some of the schemes —in the
case of the Reid So potential. It is pointed out that if
the expansion functions are appropriately chosen, good
convergence can be obtained in each of these cases. The
numerical convergence obtained in the present case is
better than that obtained previously in the case of the
Reid 'So potential. A correct and appropriate guess of
expansion functions is crucial for a quick convergence.

Though we have considered the application of the
present approach to the solution of the LS equation with
the Reid potential, the present approach should find in-
teresting application in other areas of physics in the
solution of Fredholm integral equations of the second
kind. Interesting applications of similar approaches in
the area of atomic physics are to be found in Refs. 3, 6,
and 22. The last article is particularly interesting,
wherein the authors solve, for the first time, four-particle
scattering equations, which are integral equations of the
Fredholm type, for the atom-triatom system using the
separable expansion technique of Ref. 18.

This work was supported in part by the Financiadora
de Estudos e Projetos (FINEP) and the Conselho Na-
cional de Desenvolvimento Cientifico e Tecnologico
(CNPq) of Brazil.

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