An investigation of Coulomb scattering amplitudes in the complex angular momentum plane

t° .P c~t

i~

~

~~

119~ 1 6

s~ra

f~s ~~

~~ ~t

~it~~~t

b~ ph~iop~snt or

p

o., ,~`

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~ ~dca~a

~asb~ish~~

~TAAI~E~ 1~i~~~l~tL ~d~,~ornio , of P ys~cr, ~n~~~rsity o~ CeÎi~ortiia ad L~~ ~gele~, â o~ ngel~~

i~eceived 2~3 ~une 1962 " . ~. ~onttDiled apprt)ach suggests b~ eg~e to sttidy~ scattering an1 ÎHtu i , W as i °oar of co$nple~ angular manientunt is applied to the ~c can and frac uations with (:oulomb potentials. It is found that t ai tai ~o enturr~ eigen~alues of and statu are given by the equation fur the ' a~ areaplitu~le at n ' e energies. There are no Regge pole rasons , i.e. poles ïn the first quadrant of the d plan for positive enrgi~ . Somm aid- ~ ~ i~x ~entatrons are gig°en ' all . In the relativistic aises a branch cut ap ~~ th+t ~ pl physical rote tom for ' is di ed.

t has

n su

es

that baryons and

wons are assaciated

ith

e

e pales

of at ' et ants . These poles nZa~e i n the complu an lar gtnentaa plane as a function of energy f ). ey appear in scattering atnp t~des ~b mati from salufions of the ~ er equatian with a ~enn~l function ' h n represented as the sta sitian of a tentials. ® explicit solutit~rts for such tendais ~e bven faund . owever, an explicit salution is d~.no n for the crnl~atnb s~atteri~cg pütude. e ®ulornb fennel ntay n®t be represented a~ a s~~ ositian o~ wa ten ° s so that so e of the results of egge a3 do not. haâd~ t is of interest to co e tine non-relativistic ~:oulotnb st~.ttering p it de ~!~ith scattering pliru c do satisfy the egge con itians . ~n addition e ~ltatdy the properties of t e relay ° tic ~oaxlomb pli de. ts . , , n °n ris ctiveiy the rests ®r the hr ' ~, ein-t~ordon ir~ e ~ tiens. discussion of le tray tories is lnc n

e

tion an litu e na

s te

o

°ne fra

°ts of t e

~3g

n `

an er

if

~

~~ in `~°hieh

tl n

~

an attracti~ra;

Ci)iJ~.o ~ SC

RI~(â A2ISPLIi`t,li.DES

63~

written in the form

r(t + 1-~ ~~~

es

o e ns~der (~C, ~~ for ed ~ as a function of 1. Eq . (3~ shows that negative ob ' ed fr® pure imaginary values of k. Since t:~e poles of the Sn ~ ,~ t® and states occur for positive iz~laginary k we first take ~ see from e . (2~ that the poles of in the -~otnplex 1 plane occur for ~ -1- .-

°rhe

,

jnte r

a

t - -- n~ , f_2 )

~ai=0,1, . . . .

is the ~~a iat quantum number, i .e., it is equal to the number of radial Rer

et~ ~f

Ie

®

pot~~ti~l . or ~~~ Sdhr~d~n~er ~quation with~Coulomb

at~ ~ es t ~ ~ es n ` ual to ~ n®n®ne ti set ° level is ¬~ set of c I~es of i'or . . sing ~ . g we s et~ e . l in hese e sho t eter

i . 2.

intc er oe ~ue~s s

. ~ or ` ~ e ~le ~ a ~

.~~j ole i tions in tû~ co pYex 1plane for ~ fix walue ®f l t to ., ' s with the o ~~ 'te ~ r of l+e~. e uen Ieft. and continues to the 1= ~ ancï 1= ,

line d = -- r~ ® I, -~ - . s ~ increases iro~rn ~- $xa to , ewe toti ly e con ' u®usly increas` sl® an ap r®aches as ~io is sho in . I where e und states are identi ene ~ ~e h ~ir®~en ato ener levels is ap rent in f~ ue is to ass ~ate wIt a ~te r ®f s~i~ o fro Incr ° an co 'n i nite s

~, ole ~

t~c~~~s

e c® of

l 1 1 l~s to

e for a valve of , left of the ° a °

lot the

les

ys

.

ne

e a .

it.

,

le

~

i

sitions in

e

in in ° is

at

~. ~ t

hY

v

l

°~~j °

curr~e rises `th = o. is line y crosses . The . . h ener ~ te e v es, lo

.T m .

1

s the

i

r

t

~ite

uence

le ~

ane .

- ~

t t

lled and

r ,~ht .

CoüLO"

Cy

A

hfl

~LiTi1~?ES

it i a ~~Mtx t ere is an i nits nu r of poles having 1 > . ~ bo nd s f `e les . ~t~e o i the results s own in fig. 3 . All the poles appear sere c ecreases onotonic ly with and c(c~o) = 0. This > is of t .e s e as that w °c r uces the resonances described t , a le lies at a finite int on the real I skis for E ~- 0 and into the u r half plane. In the Coulomb case for E > 0 r tts the Left o t e line el ® -- . high xnay expressed as a superposition of ~ukawa potentials, b~°t~veen the residue of ~ik, Ij (considered as a function c~f k for fixed s to le and Eile square of the no ir~tion constant C is given by

c~r es

a

onali lc

be y

~I~es p~ all t e or tlte relate I) at

e as

In this

totic fo

o the radial wave function u(r) is of the form lim u(r} -~ Ce`~`. .-~

sinus imaginary corresponding to a bound state. n the Coulomb

ln e . ( ~ case princi

oti~ f® of the square of the wave function for a bound state with Btu nurn r is given by 2r ~ . _ .1_ ._ .___ liro u ~ --~ na f~+i-~ l)I' n-I) n

Z~e

- ar,, ",

e a~nay ex nd for e 1 as a function of k in the neighbourhood t k ï/r~, obtaining the residue

siat of the

if we

sider the ~nalogue of e

we see

t

in eq . (7) to ~

. (7~ holds except for the factor (- ? )~.

we inSommerfeld-~~atson integral representation for f(®} write 1 = pe~~ and let p -~ x foc large AIE, real k and ph ~~~~l Q. It' ~'e

art order to write

vesti

(7)

te

~ ,

(9) ~~ -~ e(2i ~~ togs°(~~~'~) _ ~.~ < rp < ~~..

°I"his ~s

,pts~ti~

®f the integral vio~ar assures the con~`ergence ïk c ---?ï sin d

S®

is str~~ ~tt

s~n

I tc

p~it~a . . e cont~sur t ~ ~ntc 1 r~ ntation of t ï~n t~ " t. s~ nto r fra~ 1= - --~ao to 1= -- -~- ë~ and clo on âc~ r~ ht atf plant.

e

e vv °- ce~s (1 - .~ ~~~( ' ( p ~. -~ iv ,

-~ ( iv-~- ij%~

i

(12)

.

c

ete

ctfl l1 ,~ ~,5

~ci t cI

cc

o

c u l~

t_ o tio t e ei - or o e tion t t e e ~ ~ since only t en o t ere e °st e scatteri a li e rt~ay written

('

~

`~tc®~)

(s+ ~ ~) s+ ®~.)

(

s the s e 1 plane cut as s. is cut âs due to the ~ 2 in the ~i ® or on equation . t represents a strong n.

.

old-

stem cant®ur for ¬he

sin-Cordon amplitude.

for f(8). The vertical contour y its a S- representation end of the _ ~ n Y in this case, o `~~°er. ~t must run tw°een the variable v by ~s shown in fib. . ~ can then define a new :he inte al repress cation

rce~

e ~~xnd s te

4

8i~9

ere ~s an

ite t~u

~Ie~ are ob ai

Y7~rs

r of b nd st~:tes and nc~ reso ;t

'~i~ ~r~ iv~l t ad~e

es

a

~~

1 r tote t

~t'fe

~t

~.

~~

~~

~~

oUB.~

~~~

~~

6 .~

~ ~ s t ere a j ïs nti i of a. ~.rïse u e crossïng all ïtive inte al 1 v lues, ~ as ~ °~ . he und state ener ïes ~re the

tri+ on the top shit of the compie~ 1 plane for E > r~c~ for the ordon equation .

lcin-

o~ pote trajectoric$ oc' the sccc~~c~ sheet of the complex 1 glane for E > me=.

r~a

ït

~ï

ndïng to a cre~ssing of a no~~-negatitite integer 1. There is an inwith r, y~ctorïes and an infattite num r of a®'1nd states associated . The the solutions c~f eq . (~7) have complex 1 values ~ïan but with t~ve . n region I~ the values of 1 are also complex three regions ï o~ the phase of the function s~l) in all h~

r~s t

.

~,f lo t s te l ~r the s etches .

t

~ to

on-s ° n ar

s

~1

ed

t e ~,

v~rittßn as a s litu e ~na j, viz. ~ o ent

(s ® 6y)

` . ~~ ~ s ~ r~

t

t ft

t~ ~

.

o~ ~

f'

~ ~~

~s of

1'~s ~ 1 ~- gY

S

ire v is the relativistic ,velocity. n this case the ®f t e irac hydrogen a~to . The equation

lee o ~~ ~

rve the

stags

s-l-l-iy es t e eher ~e seen to de in

lace of d.

f

ei envallues and s o~rs t e it e e er e sin t ener leve6s nd on j lone. e results are t e e t of t. , ith j ~~ o ~;rfeld- atson represen tion is o ore ' e as

-a~

2i cos~ ~~

e slo

s

te

oh t e tra~e~ctory of a egge le in t e ~ ver us .~ tity ne %s a nce in applications ~~. or tent° tter° t ~~ slo evaiuated un siate ener is related to an average radius iated ~ - t t hound t e fo a y

relativistic

o lo h scat

in t e appro ri te re

~o is (22~

l , i the relativistic case, e ave 2

~e

l ~ ta. 2~ i`:

y9~9~

t

t e

~ 3

r®lessor i vers for n~ valuable suggestions. atic~nal fence oundation.

hi, Phys . ~~ . Letters ~ (1962) 4I (1959) 951 hanks (Per~anron Press, London, 195 ) P. ä if 'tz, Q stum ry of atomic collisions (The Glarendon Press, London,

u