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« irl "Dl ^ Institute of Mathematical Sciences
.t,^ ^^ .< Division of Electromagnetic Research


On the Eigenvalues Which Give Upper and
Lower Bounds on Scattering Phases



CONTRACT NO. AF 19(122)-463
APRIL, 1956


Institute of Mathematical Sciences

Division of Electromagnetic Research

Research Report No. CX-24


Larry Spruch

Larry Spruch

Morris Kline
Project Director

The research reported in this document has been made possible
through support and sponsorship extended by the Geophysics Re-
search Directorate of the Air Force Cambridge Research Center,
under Contract No. AF 19(l22)-t^3. It is published for tech-
nical information only, and does not necessarily represent
recommendations or conclusions of the sponsoring agency

New York, 1956


The problem of finding upper and lower bounds on the
scattering phases has been reduced to that of finding certain
eigenvalues of an associated eigenvalue problem by T. Kato,
who found circumstances vinder which these eigenvalues can be
determined. We have found additional circurastsinces such that
under specified conditions one can determine these eigenvalues
for an arbitrary angular moraentvim for non-negative potentials,
for non-positive potentials, for potentials which are solvable
beyond the point r ■ a, and for less restricted potentials.

Table of Contents

1. Introduction 1

2, Terminology and notation 3
3« Non-negative "potentials" 5
U. Less restricted potentials lU
$• Potentials which are solvable for r > a 17
6. The choice p -^ br" 20

References 22

- 1 -

1» Introduction

Variational methods have proved extremely fruitful in scattering theory,
but in some cases the variational expressions are rather sensitive to the choice
of the trial function. This is true even though the error involved is of second
order, and even with the improved forms of the variational method in which only
the "inside function", i.e., the difference betv/een the trial function and its
asymptotic form, appears. Further, the inclusion of more free paripieters in the
tidal function does not guarantee improved results. It would therefore be ex-
ceedingly useful to be able to bound the quantities under consideration. Although
this is well known to be very difficult, Kato, in a beautiful treatment L J ^ was
able to reduce the problem of finding upper and lower bounds on the scattering
phases to that of finding certain eigenvalues, labelled a„ and Pqt> of an asso-
ciated eigenvalue problem. Useful though somewhat less general results have also
been obtained by others. To Kikuta» - J and W. Kohn^-* were able to find conditions
under which the Bom approximation could serve as a basis for determining bounds
on the true phase shift. J. Keller' - ' found conditions under which the W.K.B.
phase shift serves as a bound on the true phase shift.

Kato's result, stated more precisely, is the following (see Section 2
for terminology and notation). Consider trial functions u^jir) which vanish at
the origin and which have the asynqstotic form

(1) n^^ -» cos(kr - .^ + e) + cotC^^^- 0) sin(kr - ^ + O) .

(2) ^ . 4 . ^2 - ilWi , «(,,

1- ^ - ^ - -^

and let

- 2 -



wheire p(r) is some weight factor. The eigenvalues a-, and P-^ arise in the
analysis of tiie differential equation

(5) ^pf+Mp^ = 0, 0^r oo,

(7) ^nL^^^ "^ const, sin (kr - -^ + « + nn) .

o-jT is the smallest positive eigenvalue and (-Poj) is the negative eigenvalue
which is smallest in absolute magnitude. It should be noted that a^ and P-.
need not be determined exactlyj even quite crude lower bounds on a^-r and on p^,
will often suffice to give q\iite accurate bounds on the phase shifts.

While there are any number of possible applications of the method, the
present paper is restricted to an attempt to find additional circvunstances for
which a^T smd p_, can be determined. Kato concentrated primarily on potentials
of constant sign and on the case where L = 0» In Section 3, where only non-
negative " potentials" are considered, one of Kato's results is extended to the
case L / Oj also a fairly general condition is found for which an explicit ex-
pression can be given for a^j,, and the use of comparison potentials is formalized.
In Section U, some results are deduced for potentials which are not of constant

- 3 -

sign. Section 5 consists of a number of generalizations of a particularly in-
teresting result of Kato applicable to potentials which are solvable beyond the
point r = a. Section 6 contains a disctission of the possibility and desirability
of choosing as the " weight function" p -^ b r •

2« Terminology and notation
k = wave number.

L ■ angular momentum quantum number.
V(r) " potential.

W(r) - - 2m t"^ V(r) » ■ potential" ,

't. d^ ^ 1,2 ^ ,r( \ L(L-t-l)

cS-. - — «- + k + W(r) - -i-j — •

dr r

6 = normalization. It satisfies < 9 < n but is arbitrary.
A bar over an expression means that the expression is exact.
u_, (r) and u-tCt) are the exact and trial functions respectively.

v^ and '^- are the L-th exact and trial phase shifts respectively.
The total phase is the L-th phase shift less Ln/2.
p(r) > is the " weight function" .
|ip(r) is the additional ■ potential" in the associated eigenvalue problem.
For all values of p., m? must satisfy the conditions imposed upon any
" potential" in order that there be a well defined phase,
j? ,(r) ■ is the eigenfxmction of the associated eigenvalue problem with a
phase shift e + nn, where n » 0, + 1, ..., and with eigenvalue \i^^,
a - is the smallest of the positive eigenvalues |i^»

- u -

(■^gj.) is the negative eigenvalue |i ^ which is smallest in absolute
magnitude •
The values of the a_, and of the p-^ depend upon the choice of p(r).

Any integral is understood to have limits and co unless specifically-
noted otherwise,

P( ^^) - k cot( Yi^- e) - k cot( 7^- G) - J u^L ^ Ugj^ dr.
W (r) is a comparison " potential" .

(- is the L-th phase shift associated with W (r) •

bj{\i) is the L-th phase shift associated with W(r) + |jp(r). It is a
monotonic function of m..

5,(11) is the L-th phase shift associated with Vr(r)(l + |i) when p - W
and W°(r)(l + n) when p = + W°.
s « 0, + 1, + 2, ... is used to specify some crude a priori in-
formation about '^y or ^ •

The term " monotonicity theorem' refers to the theorem which states that
if W^(r) 1.

§ 2. If < W(r) ^ br , and if -i? < Q < n, Kato showed that

(8) Peo - ^ * ^"^^^ " ^® ""'''^ ^^ " ^® "''''^'

This resvilt can be generalized immediately to the case of arbitrary L» Thus,
if < ¥ < br , then, for n < -1,

(9) W + MP - L(L + l)r'^ > (1 + kL)br"^ - L(L + l)r"^ .

For |J. < -1, the term on the right represents a repulsive inverse square law
potential for which it is trivial to find the L-th phase shift or the total
phase, since the total phase associated with a " potential" of the form
-v(v+l)r"" is -^ • The separation of the total phase into an L-th phase
shift and - •=i for the right side of (9) is somewhat arbitrary but per-
fectly legitimate. We seek that value of n, which we call ii , for which
the L-th phase shift for the right side of (9) is e - n or - J (2 - 2en" ),
i.e., for which the total phase is - 2, (l + 2 - 2©n" ). It then follows
that n is to be found from

(10) (1 + [i*)h - L(L + 1) » - (L + 2 - 2©n"-^)(L + 3 - 2en"^) .

(11) Yl, : > ® < " »

then i-^Qj) ^ ti , or

- 6 -

(12) PQL - ^ * b"-'-(2- 2©n"-'-)(3 - 2©n"-'- + 2L) .

When applicable, this resvilt can be a considerable improvement upon the result
given in § 1 if b is sufficiently small and/or L is sufficiently large,

§ 3» We again assume that < W(r) < br" , but this time we seek a lower
bound on a^j rather than on Pqt* Por |i. > -1, sind, in parti ctilar, for M- > 0, it
follows that

(13) W + MP - L(L + l)r"^ i (1 + ti)br"^ - L(L + l)r"^ .

It will not be possible to find a lower bound on a_, unless there exist positive
values of |x for which the phase associated with the " potential" on the right
side of (13) is well defined. A necessary condition on b is then that it satis-
fy the inequality
ilh) b < L(L + 1) .

(The case L = is thereby automatically ruled out.) Let \as denote by '^j the
L-th phase shift associated with the right side of (13) for p. " (see Figure l)#

(15) 1l' T " 'T >

(16) b - L(L + 1) - - v(v + 1) ,
so that

Of course, we have

Now let us assume that

(19) (s - l)n + © < ^^ ,

- 7 -


STT +9


Figure 1
Determination of lower bound on a^, assvuning that < W = p < br





< sn + e. The

that b < L(]>1), that (s-l)n + © < ^^, and ».^^ -^
comparison curve vanishes beyond n = -1 + b L(L+l), which is a
lower bound on a ^ • (If TV < sn + 6 but -k- > sn + e, the com-
parison curve crosses the sn + 6 line^ any valvie of (x less than the
value of n at the point of crossing serves as a lower bound on a.^o
If b > L(L+1), the comparison curve does not even cross the vertical
axis and a„ cannot be determined by this method. These sitiiations
are not shown in the figure.)

- 8 -

where s is an integer. The inequality that follows from equations (17), (18) and

(19) is not a restriction, but will automatically be satisfied. However, there
is the necessary and sufficient condition if we are to find a bound on O/jt* as-
suming that (lii) is satisfied, that

(20) ? L *^ sn + e .

Since the total phase associated with the right side of (13) is < 0, the phase
shift must be < -^ • A distinction must then be made between case (a) for which

(21a) ^ < sn + ©

and case (b), for which

(21b) ^ > sn + e .

(Note that for case (a), equation (20) need not be assumed, but is an immediate
consequence of eqviations (21a) and (l5)») In case (a) there is no value of [i
for which the phase shift associated with the right side of (13) is equal to
sn + 6* The largest permissible value of \i then follows frcm

(1 + n)b - L(L + 1) = ,
so that, for •«- < sn + e,

(22) ttgj^ > -1 + b"-'- L(L + 1) .

On the other hand, for case (b), there is a value of Ji for which the phase shift

is sn + © or - 2. (-2s - 2©n"-^), or for which the total phase is - ^ (L- 2s - 2©n~*'-),

This value of [i is the bound on a^,. We find, for -«- > sn + © ,

(23) Ogj^ > -1 + llb"-'-(s + ©n"-*-) (L + I - s - ©n""^) ,

For L > 0, there will be " potentials" for which one can bound p^, by means of


^ 1 or equation (12) and Og, by means of equation (22) or (23)«

- 9 -

In the explicit evaluation of ^ for a certain potential, Kato
utilized a knowledge of 'O for the Hulthen potential. The use of comparison
potentials, denoted by w (r) , will now be put on a somewhat more formal basis.
The quantities a^. and p_, can be replaced in the various inequalities in which
they appear by lower bounds, but not by upper boimds, on a_- and P^t* ^^ fix
oior attention on a specified L, and let 5j(ti.) and 5^(|j.) represent, in this sec-
tion, the L-th phase shifts associated with W(r) (l+tx) and W (r) (l+p.) respectively.
It can then be readily determined that the possibilities of interest are

(2U t^v) > 5° (ix)

for a negative range of [i, in which case a lower bound on p.j may be obtainable,


(25) 6^i\i) < 5° (ti)

for a positive range of n, in wliich case a lower boimd on a^^ may be obtainable.
The inequalities between 6y(M-) and 6^ (p.) follow if there exist corresponding
inequalities between the potentials from which they arise. It is clear that
(2h) is satisfied if

(26) W(r) > W°(r), f or -1 < ji <
and also if

(27) W(r) < W^(r), for [i < - 1 ,
and that (2$) is satisfied for

(28) W(r) < W^(r), for ^>0 .

We analyze these three cases in S 3 U, 5 and 6 respectively.

§ I4. Assume that < W(r) and that W^(r) :SW(r). The ■ potential" V^{t)
need not be non-negative, and as such 5- {[i) need not be nonotonic. Let '>y -

- 10 -



Figure 2

Determination of lower bound on p„ assuming that < W «= p,
that W° (s-l)n + 6.
Then P-, ^ Pqt > where p% is any number for which
e°(-Pgj^) > (s-l)n + ©.

be the L-th phase shift associated with W°(r). Then ^ ^^, ^j^ ^ Tl »
and, as has already been noted, 5° (ji) ^ 5j^(m.) for n > -1 (see Figure 2). If

(29) 7j^ < sn + © ,

and if

(s-l)n + e < ^j. ,

- 11 -

where s is some integer, and if p^j is any n\amber such that

(31) 6^ (-p^j.) > (s-l)n + 9 ,

(32) Pel 2: PgL •

Since ^ '^-., we nmst have < s. Further, if s ■ 0, it is known from § 1
that Pq, > 1. Since we are here restricted to ji > -1, equation (32) can there-
fore be \iseful only for s > 1.

5 5o Assume that e - n ,


There will exist values of ^^st greater than unity, so that this method can be
an improvement upon the result pg, > 1 given in § 1 for the conditions < W(r)
and y, < e < n, which are satisfied in the present case.

§6. Assume again that ^W(r) a^^ ,

- 12 -

(UO) P( -^t) ^ - i^lr)""- I W - ^(c^, u^,)2 dr .


7. In 3SU>5 and 6, the angular momentum of the comparison potential

was taken to be the same as the angular momentum under consideration. It is also
desirable to consider the case of a comparison potential vdth an arbitrary angular

momentum, L • The case L = is of special interest because in general it is by


far the easiest to handle. Once L < L has been chosen one can hope to determine

only a T . The possiblility of finding P-, would arise only in the generally un-
interesting case for which L > L. The essential point is that for L / L the

nature of the possible inequalities corresponding to eqiiations (2I4) and (2^) is

-2 1,1 X -2
fixed by the relationship between L(L+l)r and L (L + l)r .

Assume that < W < Vtr . 1'hen for p. > -1, and, in partictilar, for |x > 0,

(I4O) (l+ix)W - L(L+l)r"^ < (1+n) W° ,

so that

(Ul) 6j^(ti) - ^ < 5° (ti) .

This inequality will often be too cinide to be useful, HowevBr let us assume
further that W < br" . If b < L(L+1), it may be possible to find a bound on a-y
by means of S 3« Whether or not this condition is satisfied, we can add and
subtract L(L+l)b~'V, and using both upper bovmds on W, we find, for

(U2) H > L(L+l)b"-'- - 1 ,


(U3) W(l+n) - L(L+l)r"^ ^ W^ [1 + H - L(L+l)b"5 .

This inequality, in the domain of |j. where it is applicable, is less crude than
that given by iho) •

- 13 -

A slightly generalized inequality that might be useful if (U2) proves
too restrictive can be deduced by adding and subtracting

UlAl) \m>'^ + (l-A)W°(b*^"^] ,

c c ■*2

where < A < 1, and where W < b r . One finds, for

ikh) Kt > ALCL+Db"-"- - 1 ,


W + pW - L(L+l)r"^ < W^ Jl + n - L(L+l)[(l-A)(b°)"-'- ♦ Ab"-^J I .

If A is decreased, the inequality (U5) becomes cruder, but the restriction (Uli)


on II is eased. If enough is known about the L » phase shift associated with

a " potential " of the form W but of arbitrary strength, a lower bound on a.,
may be sought in a similar way as in some of the previous cases.

It shoiild be specifically noted that in cases ^ h through § 7, W
need not be a solvable " potential" . It may be sufficient to know something
about the phase associated with (l+ii)Vr on the basis of numerical work or some
approximation, for eixaraple. Further, in § 7, where the possible choice for
W° include \^= W, there are potentials for which the L / phase shifts cannot
be found analytically but for which the L = phase shift can; the difference
arises from the absence of the r terra in the basic differential equation in
the L = case. It should also be noted that ^ 2 and ^ 3 are special though

particularly useful instances of some of the formal results just obtained} they

have pec\iliar features arising from the fact that Br is a " potential" only

for B > 0. Cases § 2 and § 3 are not only simple, but have the additional
virtue of giving the same values of a-, and of p^j for all values of k for which
the calculations are valid, as opposed to the other cases where a separate de-
termination of Og, and of p„, is reqtiired for each value of k«

-Hi -

Kato has noted, with regard to ^ 1, the existence of a symmetry between
non-negative and non-positive potentials, in that the determination of a-, in one
case is identical in form with the determination of p^-r in the other. Thus, if
W £ 0, the choice of p = -W leads to a , = + 1 if y^ ^ - ^r ^^'^ to °-Qr i 1 if
^ > - n. A corresponding synsnetry exists in ^^ h, 5 and 6 if one further

replaces VT by -«♦ By virtue of the previously mentioned fact that terras of the

-2 -2

form +Br and -Br behave quite differently from one another, no such siu^jle

symmetry argument can be applied to cases § § 2, 3 or 7 to determine a-, or
Pg, for a " potential" which satisfies -br < W ^ 0.

li* Less restricted potentials

In this section W(r) is not required to be of constant sign, but there
is to be a comparison potential, W (r), which id of constant sign, and we choose

p to be equal to + Vr • VT need not be a solvable potential. 5y(M.) and S-j. (n)

c c c

are nov: to represent the L-th phase shifts associated with W + p, W and W + p, W ,

respectively. 5j(ii) and S^Cjj.) are each monotonic functions of [i* Equations (2li)
and (25) now admit of two interesting cases. Equation (2^ is satisfied for
11 > if
(I46) W(r) s W°(r) ,

while (2I4) is satisfied f or n < if

(li7) W(r) > V^ir) .

(Unlike the case in Section 3 where we choose p " W, the curves associated with
6-(n) and with 5^ (n) do not cross.) These two cases are analyzed in § 8 and
§ 9 respectively.

§ 8. Assume that W < v/^ and that < W*^ « p. Then ^, )



















/ ^L

/ >

(S-I)7r+^/ / _


1 /

1 /

^ /


/ f

/ /

y /







Figure 3

Determination of lower bound on a-, assuming that < W*^« p,
that W < W^, that (s-l)n + < ^j^, and that ^^ < sn + «.
Then a^^^ ^ a^, where a ^^ is any number for which
Sj^(ctgj^) S sn + 0, (Note that the curves do not cross.)

and if

(s-l)n + e < Yl *

^ < sn + © ,

where s is some integer, and if a^^^ is any nuniber such that

(50) 6° (a^j^) < sn + ,


a, but make no assumptions about W(r)
for r < a. Choose p(r) -^ for r > a, and let p(r) - a" for r i a. (The
particular choice of p(r) for r < a is, as was noted above, irrelevant.) If

(63) (s-l)n + © < - ka ,


i6U) ^^< sn + e ,

where s is some integer, and if the trial function satisfies equation (62), then
p__ ->oo. Note that for W(r) non-negative ka must be small if the method is to
be applicable, but that this is not necessarily the case if W(r) is not thus

§ 11. To generalize to L jf' 0, one need merely replace -ka = ^>T^ by

(65) ^ - cot-^ [^L^l^y ^L^^^]


where f?j is the L-th phase shift for an infinitely repulsive square well
potential, and where n, and j^ are the spherical Neumann and spherical Bessel
functions . -tTv has a precise meaning, with no ambiguities regarding ad-
dition of multiples of n. It is a monotonic function of ka, decreasing from
at ka » to -ka + -5- as ka -^00.

# rrt

For definitions of n, and of j, see, for exansple, Morse and Feshbach"^-^, where

values of 5j^(ka) = - ^°? are also included, in Table XV, in degrees rather

than radians.

- 19 -

§12. The essential featvire in the considerations of this secticai is not
that W(r) vanishes for r > a but rather that u_, cam be chosen such that



Online LibraryLarry SpruchOn the eigenvalues which give upper and lower bounds on scattering phases → online text (page 1 of 2)