Note on the Born series

~T ~

~ Ph~sács 3$ ~1g82j 553~--580 ; ~ í~iowth-Holland Publáshin~ Co., A~raste~"dam tA be u~ed by photoprint o~ micaofilni withovt written permission

from the pubüsher

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J. E. RUSSELL Phys ~ Lleportment, Cornegie Institute of Technology, Pittsburgh 13, Pennsylvania t

A

Received 2? April tQ62

urn ` for the non-retativistic elastic scattering amplitude is investigated for the tary type of non-central interaction, a potential between two spin one-half particles mcet for which the departure from spherical symmetry is due to the tensor interaction. It is shown that ' t conditions for convergence at zero energy are that the potential be static, bounded, of $nits , attractive wherever it does not vanish, and such that there is no resonance at zero energy and no bound state at any energy. Convergence is also obtained if the sign of the potential ïa rev ,

1. Introdaction Devies i ) has proved that the Born series for the non-relativistic elastic scattering amplitude,f~k) converges for all k if the potential Y(r) is {i) spherically symmetric, (ü) static, (iii) bounded, {iv) of finite range, ar~d (v) such that - ~ Y(r)) cannot support a bound state. The pu of this note is to show that the condition (i) can be relayed. The possibility that the convergence of the Born series for an arbitrary nor-central potential upends critically on the absence of a bound state is a complicated problem and is not invest.i ted here . Instead it is shown that there is such a dependence in the of the most elementary type of non-central interaction, a potential ~Y(r, Q, , ea)

_ ~CveCr) + Sab v~(T)~~

~.

>

0,

(1 .1)

between two spin one~half particles a and b, which includes the tensor operator S,b . 3(ea , r~ab - r)r - ~ - (Qa ~ Qa) .

( i ,2)

It is ~~ that v~ and vt individually satisfy conditions {i), {ii}, {iii) and (iv). It is also assumed that the potential ~. w{r, e~, eb) is attractive wherever it does not vanish, which is equivalent to requiring It is shown that the lack of a bound state guarantees convergence at zero energy for both the potentials + íîY and - ~Y if ,~ Y{r, a~, Q b } is such that there is no resonance at zero energy. r central potentials, ' } for devised by Day°ins The Proofi~ similar to the one which was t Supported bt the U. S. Atomic Energy Commission . :í :i3

i.

-~~

N(~rB ~r+N ~

g6

BO$~ SE~ES

{2.9)

0 0l' {x) _

~f, f{x)

Uf, a{x)

~a, f{x)

~a, a{x)

1/x

0

0

a 3 y ~Sx

IIy

0 xaÍSys

û

t

(2.10)

for x > y", for x < y.

The math Ü is diagon~d by the transformation SÜS °' _ -

v~ - 4u~

0

0

uc +2v,

where J

_1

,

~

JZ

{2.12)

J1 ~2/

{2 .13)

Each element of the matrix SnS° ~ is greater than or equal to zero. It follows that each teen in the series (2.8) is positive if the condition (I .3) is satisfied. Therefore, if the Earn series for,~0) converges for the potential + ~:Y, it also converges for the potential -- ~,Y. The convergence of the series for ~f, f is established by relating it to the iterated for3r of integral, ,equations for the radial wave functions of states with total angular momentum J = 1 . á. Ist

F~q~atioms for itadial Wave Functïoas

The cciuple+~ di~%rential equations for the radial wave functions ~Áo(r) and ~a(r) of states with total angulaz momentum J = 1 and respectïve orbital angular moments ~ = f~ and I -= 2 are written in matrix form as . .~r) s E-k 6~- + ? P- ï~C1(r)~~~r)~ (3 .1) a~ ~ where ~{r) = aA~I

~o{r) ~~ir)

(3.2)

~3.3)

,~
~ >

(3.S

ra

(~.s

a

....

(~

~(

~3

~

a

} ~

ra

I'} ""

+

i~

a "

a~}~G~~°a

(3 .7~ It

nn fu

a

a

t na~ nf

a

t .E1

la

1, 1 '

n~~ , t ~ c~}~~"~r, n~e ~f ~ ~cacrs ., ' ï~ fnr u ~f ' ~ina ~`~( , ~~,~

, ~}

(3. )

o~i írï o ut~o~ ~,r~other ~ ~~d si~t ed i G~(k, r} -3í~

k, r, í~}

. (3.1) which vanishes at r = 0 has been written .ble deuil by ~lewton 2). It is defined by ®f

dss -i Go(k, r)pU(s)Q i k ; r, ~)~(S

wh q

~k, s, ~)-3s-iGo(~, r)PU(s)QJ,

k ; r, s}

Go(k= r}~oCk~ ~) ` HoCk~ r)~o{k~ S)expr in the form

~ on ~ .15) ~

rs ~} ® ~

wh

1 ka r - ~

(k, r, ~.) =

h

t

'~k, r)

Ods

~a{k ; r, S) II(S)~(h, S,

{3.15~ (3.16)

I~. ,

~(k, r, ~) - Go(k, r),

(3.17) (3.1S)

ds[l~~{k ; r, s) U(s)G ® (k, s) - 3s-- 1 G o (k, r)pU(s)~J .

(3.19)

t the iterated solution

0

ds K k ; r, s)U(s)~ _ 1 (k, s),

n > 1,

(3.21)

' o y and a lutely for ail k, all í~, and all real values of r >_ 0. i~ ' by the si~il~trly convergent series

c~n~per e~ `fh ore,

R=o

where

(3.23}

It is con~nieut to write ~ in the form ~ccordi~

~

(3 .24) k, r, íl) = Go{k, r)Bß~ , r, ~) + Ho(~, r)c~tk, r, ~~). (3.15), (3.16), and (3.1 S), the matrices B~ and ~~ are given by

0

s} ds[No(k, s} U~(s}G®(k,

3s ~ P U(s}Q ;

3.25

~,~ `;, r .

S O

:t for

t

r ~

r2~

~, i T3 ~`

T4

3 .2~)

rent as r --~ nds ïn enerai on U(rj, ït is a ~~iour of v~ers of r to rc~m e~s. {~. , (3. ), it d not ïnvoive n ra. (~.2?) t and ~ in s. (3.25j and (3.2~) and inerefore, by su ' ing the seri far wer series in ~. in term t ~ the .~ and +C~ are obtained as ch conver far :, , and r > 4. r F®r the tri B d ~R are constant and uai to ( , , ~) and ~vely. If (k, , ~) pr s an inv~rse, the function ,, which ~~~~_ ~a ~~~ re~ ~~ de~ned 1?y k, r, ~.) =

GA

, ~)

~~'

r, ~)+x~(k, r)c,(k' ~' ~),

(3.2 ) (3.2~)

is a solution of e~. (3.1) which vanishes at r =- ® and which a form identical to that of ~. e mott oes ~ and C~ are liven by

"\ .

e matrix B 4 (k, B, ) nay e~ Banded in a canver ent mower rice in ~. if B~(k, R, ~.') has an inverse for all ~~`~ ~ ~,. Therefore, the conver nre of the series (3.9) and (3.1ií) for B~(k, r, ,~) and C~~k r, ~) is determined by the singularities of Balk, R, ~'). In icular, front ecl. (3.1 ), the Born series converges if th+~ determinant of B~(1D, , ~.°) d~s not vanish for f ~.'~ S ~,.

It is îarst sho vanishes . e

that .~' must t ' solution

~s ~,r,~')

r ®

c~s the c®h

s af

real and stove ïf the d~termin~nt of Bß(0, R, ~.' ) (1D, , ~,') of the zero energy radiai egvaation

( , r,

0 1~3 ~~ l

') are individ

0

3ir

ly solu~ions of eq. ( . lj vahïch vanish

~+ t t e ori i ~ '

tri

~ 'ven by 0, r, ~.')d,

d

ö

d

(4.3)

2

d

Iso s~ ~ ~ o%ut~on i

ds to

, n

a

constants. Consideration of the integral

"o T=

rs)-

o

dr{ ' r))~

r) -

t it i It is ible to ch eve ~~ ~h~r~ att ~u~h t t f®l' i R,

3

r~

h re t if and ®n

the if t

dr~ r)C6r-aP-%``U{r)~~a{r) .

di and d~

di Ci . m~- da

(d~ ~~ . ~

i

.z

such that ~~ does not

r

{4.4) vanish

{4.5)

d~ ~~. ~)

a

nts of the rt~atri~ C' (0, , %'). Such a choice is possible tinant o (0, R, w') vanishes. As rl -~ , eq . (4.4) becomes ®

~®

dr~r ®'

dP~d

(4.6)

~, ail the rote ands in eq . (4 .ßa ) are real and greater than or equal r®. h real and positive . fo , if ~! is not eve here equal to zero, ~.' must Final , it i s own that the det~rnlinant of Bß(0, R, ï') does not vanish for real, itivc ~ i of ~' ~. the pr f involves the use of the matrix FC {k, ï~ ' ), the Pro rt.` of r$i~h have n di~uss+ed by ev~ton 2). For Im (k) <_ 0 it is a cont"_'nuous function of k which, for the fini+e range potential being considered here, ritt~n Y lf ~~~

'o

e

ds[F~, ®(k, s)U(s)G®(~, s) - 3s ® 1 P U(s)Q]

(4.7)

here ..~ ~[n~(ks)~-i~o(~s)~

(4.s)

It is noted that In order to prove that the determinant of Bs(O, i~, a') d~s not vanish, it is su c'ent ra (k) Q. to show that Fe (k, ~') does not have a singularity fox and Sri For Im(k) <_ 0 Newton has proved that ~a) the determrnal~t of .~eCk, ~;') n vanisl, only if Re(k) = 0, (b) there can be a singut~rity of ~Ck, A') at fit® 0 only if the J ~ and energy potential - ~,'tI has a bound state with total an~utar tno a bound kó, and (c) there can be a singularity at k = 0 only f`, for J ~ t~ ---,~'-U state or a resonance, or both, at zero energy. e~ for mad su tituted i n Since the absolutely and uniformly conver e~nt eq. (4 .7) and integrated term by term to abtain a convergent powee series in ~' for to unity F~(k, ~,'), the determinant of F~ is a continuous function of ~' which ; u , the matrix for ~,' = 0. Therefore, there exists a ~,~ > 0 such that, for 0 ~ ~' F~(k, ~,') is non-singular for all Im (k) S 0. The assumption that the potential -- ~ U has no resonance at zero energy and no bound state at any ener~yr implies that â < ~, and, from the preceding arguments, guarantees the conwer nrre of the $orn series.

1) H. Davi~, Nuclear Physics 14 (19Sß/60) 465 2) R. G. Newton, Phys. Rev . 100 (1955) 412 3) J. A+I. Blatt sad V. F. Weiskopf, Theoretical nuclear physies (John Wilay and Sons, New Yc,rk , 1952) p. 99