IL NUOVO CIMENTO u XLVI A, N. 2 21 Novembre 1966

The Born Series for Nonlocal Potentials (S-Wave) (').

M. BEI~TERO

Ins t i tu t ]iir Theoretische K e r n p h y s i k der Universi tat - B o n ~

G. TALENTI

l s t i tu to Matematico dell' Universit~ - Genova

G. A. V~A~'O

Is t i tu to di Fis ica dell 'Universit& - Genova

(ricevuto il 24 Giugno 1966)

Summary. - - A sufficient condition for the convergence of the Born series for nonlocal potentials is derived. The analysis is restricted to the S-wave SchrSdinger equation. In order to have some information on the general structure of the scattering solution as a function of the potential strength g and of the linear momentum k, the separable potentials are reconsidered.

1 . - I n t r o d u c t i o n .

The use of nonlocul po ten t i a l s is r equ i red in m a n y phys ica l p rob lems inc luding the t h e o r y of nuc lear m a t t e r (~) and low-energy nuc leon-nuc leus sca t te r ing (~).

F u r t h e r m o r e the so lu t ion of the Schr6d inger e q u a t i o n for a t w o - b o d y

sys t em in t e r ac t i ng via a nonloca l po ten t i a l is a non t r iv ia l m a t h e m a t i c a l p rob lem of nonre la t iv i s t i c q u a n t m n mechanics .

The t h e o r y of non loca l po ten t i a l s is far f r o m comple te . F o r the m o s t pur r

the k n o w n resul ts conce rn the so-called separable potent ia l s , or l inear combina-

t ions of separable potent ia ls . Of course, in this case, one floes no t rea l ly hand le

(') Work partially performed within the program of the Gruppo di Ricerca Mate- matica no. 23 del CNR.

(1) tt. A. BETH,:: Phys. Re*,., 103, 1353 (1956). (~) F. I'EiCE'( ~nd B. B~!CK: N~cl. Phys . , 32, 353 (1962).

338 M. B E R T E R O , G. TALENT][ and G. A. V I A N O

an integrodifferential equat ion bu t simply a linear, inhomogeneous differential equation, coupled to a linear algebraic system. I t is then possible to reduce the solution to quadratures.

In this paper we restr ict ourselves to the S-wave Schr6dinger equation, i.e. in units h 2 / 2 # : 1 :

§ t *

(1.]) y"(r) § k:y(r) = g | V(r, s)y(s) ds ,

0

where V(r, s) is supposed to be a real and symmetr ic function

(1.2) V(r, s) -~ V(s, r) .

The problem is to discuss existence an4 uniqueness of the scattering solution and its general behaviour as a funct ion of the linear momentum k and of the

potent ia l s t rength g. We have made a first step in this direction; we have found a lower bound

to the radius of convergence of the Born expansion (i.e. the representa t ion of the scat ter ing solution as a power series in g which can be obtained by i terat ing

the scat ter ing integral equation). As is well known (3), for a local potent ia l satisfying the Bargmann condition,

a sufficient condition for the convergence of the Born expansion is

(1.3)

-rco

Igl f rl V(r)Idr < ] .

0

For every value of g satisfying (1.3), the Born expansion is convergent and

holomorphie i~ the half-plane Im k > 0 . For nonlocal potentials we have found a sufficient condition very similar

to (1.3) : +co +co

I].4t Igl|dr| lv/r, 1.

0 0

In fact we prove that , if (1.4) is true, the scat ter ing solution exists and is unique (in a class of functions tha t will be specified later); as a by-product we have tha t this solution can be obtained by i terat ing the scattering integral equat ion and therefore is represented as a power series in g.

The new feature is t ha t now the convergence domain in the k-variable is

restr ic ted to the real axis.

(8) R. JosT and A. PAIS: Phys. Rev., 82, 840 (1951).

T I I E BORN S E R I E S F O R N O N L O C A L P O T E N T I A L S (~ -~VAVE) 3:]9

We demons t ra te convergence and ~mMyticity of the Born series in the strip lira k] < ~, if

dco 4-co

( | .~ ) ~o~'r(li.,'6~~ W(r, .,') ,d.,' ~ @ (x)

(1 o

a~(t k[ g is res t r ic ted by

(I.~;)

+co -t-oo

o o

i n order to tmders tand wha t has to be expec ted whe~ conditions (1.4) or (1.6)

on g are r emoved we have reconsidered the separable potent |Ms.

We use a me thod which is ve ry close to the usual one (4) for local potent ials sat isfyiug the Bargm,~rm condition. Many of the obta ined results are a lready

knowu, but wha t we need here is a general a~d unified discuss|ore

The solutions of the appropr ia t e bounda ry conditions and i~titial values problems are discussed. We find t ha t the Fredho lm deternfi~unt of the scat- ter ing i~tegral equa t ion is no longer equal to the Jos t fuuctioll defined i~ te rms of the regular and Jos t solutions; however the (( ~o~tlocal ), F redho lm deter- minan t has proper t ies ve ry similar to the (( local )) Jo s t furmtion. I n this way we are ~ble to t re~t the bomtd-s ta tes problem aIld to rederive, wi th the help of the origizlal teelmique of LEVINSON (a) the completeness of the wt~ve functions sys t em artd the Levil~son theorem (we reobtah~ the result of MARTIN (s)~ O~ course wi thou t using J auch me thod (7)).

[n Sect. 2 basic definitions and nota t ions are given; in Sect. 3 we discuss the convergence of the Born series for the scattering, regular altd Jos t solutions

aild for the ,%matrix. The proofs of some L e m m a s are given in Appendix. The ~'enerM discussion nbout sel)a,r~ble potent ia ls is repor ted in Sect. 4.

2. - Bas i c de f in i t ions and no ta t ions .

We recM1 the problems tha t mus t be studied in comlee t ionwi theq . (t.1).

a) A boundary values problem de]i~dng the ,~.cattering solution ~(k, r)

(2.1) ~o(k, r) ~ sinkr t r r ) ,

(,1) R. G. Nl,'WT()X: Jour~. Math. Phys.. l, 319 (1960). (r,) N. |,EVIN$ON: Da/tsl,'e Math. Fys. Medd., 25, No. 9 (1949). (~;) A . M A I t T I N : N?[,OVO Cimeuto, 7, 6 0 7 (1958). (7) j . JAUCH: Hehu l'hy,~'..|eta. 30, 143 (l.(}57).

340 M. BERTERO, G. TALENTI and G. _~. VIANO

where ~5(k~ r) is a solution of the integrodifferential inhomogeneous equa t ion

(2.2) Q-~ -}-co

qS"(k,r) + k2qS(k,r)- g/r(r, .~) r .~)ds=gf V(r,.~)sinksds, 0 0

satisfying the boundary conditions

(2.3) r 0) = 0 , l i m [qS'(k,r)-- ikqS(k,r)]=O. r--~'+ co

The me thod of var ia t ion of constants suggests t h a t we wri te the follow- ing integral equat ion

-~co +co

(2.4) ~v(k,r) sinl~r + d dtG(k; r,s)V(s,t)~v(k,t), 0 0

where G(k; r, s) is

(2.5) G(k; r, 8) =

1 -- ~ e ~k' sin ks

1 - - -~ e ik~ sin kr

r>8~

r < s .

b) An eigenvalue problem de/ining the bound states solutions g(r)

(2.6) Z(0) --~O

.~co

f lz(r)l~dr = 1. o

I t mus t be observed t h a t this p rob lem is not exac t ly a Sturm-Liouvi l le p rob lem bu t it can be reduced to a p rob lem of this type a t least in the local

case for potent ia ls sat isfying the B a r g m a n n condit ion (*).

As a consequence the solutions of p rob lem b) can be found as the solutions

of the homogeneous integral equa t ion re la ted to the inhomogeneous one of

p rob lem a). This connection be tween p rob lem a) and p rob lem b) is basic for

including the bound s tates in the f r amework of the S -mat r ix ; however in the

nonlocal case this connect ion does not seem self-evident.

(*) In fact, if k is complex, it is possible to find two linearly independent solutions, one growing exponentially (for r ~ ?- co) and the other decreasing exponentially. The request that z(r) belong to L 2 is then equivalent to choosing the decreasing solution. So the problem can be reformulated by stating that we want a solution which is zero at r = O and r = + oo.

T H E B O R N S E R I E S FOIr N O N L O C A L P O T E N T I A L S ( S - W A V E ) 3 4 1

Problems a) and b) are the mos t i m p o r t a n t f rom the physical point of view. However i t is interest ing to know w h a t happens to the usual Jo s t formal ism and to have more insight on the general solution; so we have considered these problems, too, i.e.

e) A n init ial values problem at r = O, de]ining the regular solution q~(k, r)

(2.7) ~(k, 0) = 0 , ~ '(k, 0) = I ,

for which we can wri te the following integral equa t ion

r ~-r

(2.8) 9c(k, r) s i n k r /'~ i ' _ . s i n k ( r - .~) --~ k ~-g. j jcls ct~ ...... ~ . . . . V,s,t)q~(k,t)

0 0

and

d) A n initial values problem at r = + o% de/ ining the Jos t solution ](k, r)

(2 .9 ) l im exp [--ikr]](k, r) = 1 , r ~ + co

with the corresponding integral equa t ion

(2. ,0)

-boo ~r

v 0

sin k(s-- r)

When one has these solutions for a local potent ia l , i t is possible to defi~e the so-called J o s t funct ion as tile Wronsk ian of the Jos t solution with the regular solution.

In our case this definition has no mean ing because the Wronsk ian is not a cons tant (with respect to r) ; but , if condit ion (1.2) is satisfied, this Wronsk ian takes the same value at r = O a~d a t r - - - ? c ~ . This va lue can be t aken as a definition of the Jos t function.

I t is ve ry easy to recognize t ha t this definition is equivalent to defining I(~) as

(2.11) /(k) = l im/ (k , r)

or, in te rms of tile coefficients of the decomposi t ion of the regular solution in Jos t solutions

(2.12) 1

~(t:,r} : Z~:[](-- ],')/(L', r ) - ](k)/(-- t', r ) ] .

342 ~r. B~RTERO, G. TALENTI and 6. A. VIANO

We recall t ha t if ](k, r) exists, then ](--k, r) also exists and they are l inearly i~dependent in bo th the local an4 the nonlocal case. Of course (2.12) has a mea.ning only when the unici ty of the solution of un initial values problem has been demonstrated. This theorem would imply also tha t ~p(k, r) and ~(k, r) are proport ional .

F r o m (2.10) and (2.11) we get

(2.13)

q-co 4-co

1" /" sin ks j j , g ds d t , . ~ - - V(s,t)J(k,t) o 0

and (2.12), (2.10), (2.8), after some algebraic manipulations, give

(2.14)

0 0

Analogously, from the i~tegral equations (2.8) and (2.4) we get

(2.15) /(k) = 7!r r ) . ~(k, r)

We recall tha t , for a local potent ia l satisfying the Bargmann condition, ](k) is also the Fredholm determinant of the scattering integral equation.

F o r a separable nonlocal potent ial we shall fiud t h a t l(k) is the rat io of the Fredholm determinant of eq. (2.4) to the Fredho lm determinant of eq. (2.8).

This fact is probably t rue also for a wider class of nonlocal potentials (8). F rom (2.14) and (2.15) it follows also tha t

(2.16) /(k) :

Jrr ~-co

o o

The S-matr ix is defined in the usual way

(2.17) S(k) ----/(-- k)//(k)

and satisfies the un i ta r i ty condition. Wi th some more manipulat ions o n th e i~tegral equat ions it is possible to have a representa t ion of S(k) directly in te rms

(s) L. BROWN, D. I. FIVe, L, B. W. LE~ and R. F. SAWYEa: Ann. of Phys., 23, 187 (1963).

THE B O R N S E R I E S F O R N O N L O C A L P O T E N T I A L S ( S - W A V E ) 3 4 3

of the scatterillg solution:

§ + c o

(2.,s) , ' ]1' ~

o 0

We wish to observe t h a t all the integral representat ions we have derived for the Jos t f lmetion and the S-matr ix are, at this stage, purely formal.

3. - A suff ic ient condit ion for the convergence of the Born series.

Let us suppose tha t V(r, s) is a measurable funct ion of bo th variables, O < r < - ~ cr O < s < + - ~ , and tha t for every vMue of ~ such t h a t O < ~ < m (m>0)

§ § ft ga

0 0

As was said in the In t roduct ion , for ~ fixed, we shM1 e(msider only the values of g such tha t

< 1 .

o o

We shall give the derivations only for the scat ter ing solution. Some non- essential modifications are ne('essary for the regular and Jos t solutions; these modifications will be indicated at the end of the proofs.

Let §162

~,(k, r) = g~l~(r, .~')~0(k, s) ds . fD

(:~.3) o

(3.4)

whe re

(:~..~)

F rom the intetzral equat ion (2.4) it follows by inversion of the order of inte- grat ion

+ c o

r(/,:, r) : r,,(/,', r) +f_hr(k: r, .s')t,(k, s)d.~' ,

o

i co ~-co

vo(k,r) .qf,T(,., . , . ) N(l,';r,s)=:gfV(r,t),~(k 0 0

; t, s)dt:

344 M. BERTERO, G. TALENTI and G. A. VIAN0

G(k; t, s) is given by (2.5). From the inequali ty

(3.6) Jsin z I < Izl exp [J lmz]] ,

bounds on the inhomogeneous term Vo(k, r) and on the kernel iV(k; r, s) are easily obtained:

+ m

(3.7a) }%(k,r)J< ]gk] fse~slV(r , s)Jds , l I m k l < ~ ,

0

(3.7b) liV(k; r, s) l< ]gift exp[~(s + t)]JV(r, t)Idt, Imk>~ - :r 0

The integrals are certainly convergent almost everywhere for r:> 02 as a consequeace of condition (3.1).

Lemma 1 . - For every value o] k in [Im k l < a a solution o] the integral equatinn (3.4) exists and is unique in the class o] the ]unctions such that

(3.8) f e~]v(k, r)]dr < ~- ~ .

0

Furthermore, for almost all r > O, v(k~ r) is continuous with respevt to k in Jim k] < :r

Let

0

it is s t raightforward to show tha t X is complete (i.e. it is a Banach space). F rom the inequal i ty (3.7a) it follows tha t Vo(k, r) belongs to X for every k in ] ImkJ<~ :

(3.9)

-}-co

j]vo(k, ")II-~fe~]vo(k, r)]dr< Jk l C. 0

(3.10)

Furthermorei let us write

Z(k)x(r) -~fN(k; 0

r, s)x(s) ds ,

T I f E B O R N S E R I E S F O R N O N L O C A L P O T E N T I A L S ( ~ - W A V E ) 3 4 5

then Z(k) is a line~r bounded operator in X with ]IL(k)ll(1 , I m k > - - ~ . In fact, as it follows from the botmd (3.7b),

(3.11) ,(k,xrj=Je f (k; o o

r~s)x(s)ds dr<C[lx]l ,

C ~ 1 implies our s ta tement . I t is then clear t ha t the resolvent operator R(k) : [ 1 - - L ( k ) ] -~ exists and the equat io~

(3.12) v(k, ~) ---- vo(k, r) ~- L(k)v(k, r)

has a unique solution in X when k is in I I m k l < a . This solution can be compute4 by i tera t iom The uniform convergence with respect to k IIm k I<:( of the i t e ra ted series, implies the cont inui ty of v(k, r).

A bound, which will be usefal later, is eusily obtained:

(3.~3)

i~

(3.1~)

and rememberin~ (3.9)

(3.1,~)

]iv(k, -)IT < ![Vo(k,- )[l+ ][L(k)v(k, .)][ <

< ][vo(k," )ll~-]! L(k)il" ]Iv(k, ")l[ < Hvo(k, ")II+ CIlv(k, �9 )I[,

fir(z,:, ")][ ~ : i - c Ilvo(/,', ")l',,

IklC' i im~.l<~. []v(k'}fl< 1 - ( : '

THEOREM 1. - For every value o[ k in Jim k] ~ o~ a ]unction of r~O exists which :

i) has a ]irst derivative absolutely continuous;

ii) satisfies the integrodi//ercntial equation (1.1) almost averywhere for r ~ 0 and the boundary conditions of problem a) (Sect. 2).

This solution is unique in the class o] the ]unctions such that

(3.16) r) T k2~(k, r ) I d r < + c~.

0

1) Existence proo]. Once the funct ion v(k, r) has been computed, eq. (2.4) suggests t ha t we write

(3.17) ~(k, r) - - sin kr + f G ( k ; r, s)v(k, s) d s .

o

346 M . B E R T E R O , G . T A L : E I ~ T I and G A . V I A N O

By a first differentiation it is possible to check tha t y/(k, r) is absolutely con- t i~uous; a second differentiation shows tha t F(k, r) is a solution of the inhom- ogeneous differential equat ion

(3.18) ~"(k, r) + k:~f(k, r) = v(k, r) .

Fur thermore from the integral eq. (3.4), by an allowed inversion of the order of integration (the integrals are absolutely convergent) one has

(3.19)

+ v +co +co

v(k,r) =gf v(r,t)[sinkt +fG(k; t,s)v(k,s)ds]dt=gf V(r,t)~,(k,t)dt . 0 0 0

So it is clear tha t yJ(k, r) is a solution of the integrodifferential equat ion (1.1) and, of course, of the integral equat ion (2.4)~ too ; (3.18) and (3.8) imply (3.16).

We have ye t to check t ha t the boundary conditions are satisfied. Writ ing explicitly the obtained representat ion of ~b(k, r)

r +co

~ ~" sin ks sin kr F .~s (3.20) ~b(k, r) = - - e j ~ - - v(k, s)ds ~ Je v(k, s) ds,

0 r

it is clear tha t q)(k, 0) is zero and, computing the first derivative, tha t

(3.21)

+co

Iq3'(k, r) - ikqb(k, r) l = fexp[ik(s- r)]v(k, s) ds .<. r

~-co

<fe"'Iv(k, s)]ds -~ 0 T

(r-~ + c~, IImkl<a).

The existence is then proved.

2) Uniqueness pro@ Let ~vl(k, r) and ~(k , r) be two solutions of i) and ii) satisfying (3.16); their difference y(k, r ) = ~pl(k, r ) - -yh(k , r) is such tha t

(3.22)

-~oo

y"(k, r) + k2y( k, r) = g f VIr, s)y(k, s) 0

y(k, O) = 0 ,

~-o~

f e"r]y"(l:, r) o

d 8 ,

lim [y'(k, r) -- iky(k, r)] = 0 , r - - > + co

+ k~y(k, r)Idr< + or

T I I E B O R N S E R I E S F O R N O N L O C A L P O T E N T I A L S ( S - W A Y E ) 347

Let us wri te

(3.23)

+co

r(k, r) = y"(k, r) -~- k:y(k, r) = g f VIr, s)y(k, s) ds , o

+co

f c~]v(k, r)]dr < -f- c~ ,

0

then~ wi th the usual technique, we h~ve

+co +co

(3.24) ~;/k, r) = g f d s f dt V (r, s)G( k; r, s)v( k, t) . o o

I t is s t ra igh t forward to ver i fy t h a t the supposed inequal i ty (3.23) together wi th

inequal i ty (3.6) and condit ion (3.1,), implies t h a t the double integral a t the r ight -hand side of (3.24) is absolute ly convergent in lira k I ~<~. So i t is possible to inver t the order of in tegra t ion and we have

+co

(3.25) v(k, r) --f~V(k; r, s)v(k, s) ds, i.e. v(k, r) = L(k)~;(k, r). o

I t follows theft

(3.26) Iv(k,- )11 < IlL(k)II llv(k,- )]l < Oily(k,- )[] ;

C < 1 gives v(k, ")U= 0, i.e. for every k in lImk[ <~a, v(k, r) is zero for a lmost all r > 0 . Therefore y(k, r) is zero, too.

Theorem 1 is proved, f f we subs t i tu te in eq. (3.17) the i t e ra ted series for v(k, r) and we in tegra te t e r m by t e r m we obta in the Born expans ion of the scat ter ing solution.

Let us suppose ~ r 0.

Lemma 2. - The inhomogeneous term vo(k, r) and the kernel N(k; r, s) ]or s>~ 0 are analytic in the strip ] I m k l < ~ (*) for almost all r > 0 and ]urthermore

lhJ(Ikl +~) (3.27a) ]~'0(k + h, r) - ~,o(k, r) l< ~_ lira k] ~ ]~ C ( r ) ,

(3.27b) ]~o(l,,,r)l < Ikl + ~ ~ - iimk I C(r),

(3.27c) ",'o(k + h , r ) - vo(k,r) ;'o(k,r) ~ Ihl([k I+~)C(r )

(') We indict~te with an upper dot the derivative with respect to k,

~ 4 ~ iV[. B E R T E R O , G , T A L E N T I 2 . r i d G . A . V I A N O

(3.28a)

(3.28b)

(3.28c)

Ihl i~r(k + h; r, s~- 2~(k; r , s ) I < l lmk[_ ]hi e~C(r) '

1 I~(k; r , s ) l< _ iim~ [e~*C(r) ,

l ~ ( k + h ; r ,s ) - .5] ' (k; r ,s) _ ~(k; r ,s) < h

Ihl e~,C(rj , < ( ~ - I l m k l - I h l ) ( ~ - [ I l n k l )

where +o~ +00

0 0

The prool is given i~ Appendix A.

_Lemma 3. - k--->v(k, .) is an analytic function in the strip I I m k l < ~ ~ with

values in the Banach space X . The proof is given i~ Appen4ix B ; we recall here Olfly the definition of an

analytic vector-value4 function. Let (~ be an open region in the complex k-platte an4 X a complex Banach

space. A function v(k) define4 on G an4 with values in X is sai4 to be analyt ic if a f lmction ~(k) on G to X exists such tha t

vik + h ) - v ( k ) _ ~,(k)/-+0, lhl-->0, (3.30) h /

for every k i~ G. We know tha t v(k,-), solution of eq. (3.12), is a vector in the Banach

space X defined above for every k in l I m k l < a, so it is enough to show tha t fu~ctiorr ~(k,-) beiongiag to X exists such tha t (3.30) is true.

I t is clear tha t ix all the proofs the bounds (3.7a) and (3.7b) are the mahl point.

For the regular ~n4 Jost solutions we can introduce the flmctions

+r +co t ~ #

r) = s )V(k,s )ds , r) = g j r ( r , 8)t(]c~ 8) ds

O 0

and the integral equation (3.4) is replaced by

+ m

(3.32a) vx(k, r) : v~.o( k, r) +|iV~(k; r, s)v~(k, s) ds ,

0

+on t a

(3.32b) v~(k, r) = v2.o(k, r) + f ~V~(k; r, s)v~(k, s) 4 s ,

0

T HE BORN S E R I E S FOR NONLOCAL P O T E N T I A L S ( S - W A V E ) 349

where

(3.33a)

+ ~ +co

'v,.,,(k, r) V(r, s) - - k - ds , --~l(k; r, s) = . V(r, t) sink(t--k s)

0 s

(3.33b)

with the botmds

(3.34a)

(3.34b)

(3.35a)

(3.35b)

dt

+oa

v2.,,(I," , r) : : g f V(r, s) e ~ d s , 0

$

.N'~(k; r,s) -- g f s ink ( s - - . . . . . . . . t) V(r , t )d t k

o

+co

lVl.0(k, r) l< Iglfs ,l V(r, s)]ds, l ira k I < a, 0

+ m

I v~,,,(k, r) l < Iglf ,l V(r, s)Id,~, l i ra ~'1 :~ - - ~, 9

IN, (k; r, s)l < Ig Ift e~'l V(r, t l ld t , lira k I< ~, o

+ca

1~%(/,; r, s) l< Ig lse~ V(r, t)Idr, lira k I< ~, 0

For the regular solution the space X defined in Lemma 1 can be used; for the Jost solution it is necessary to introduce the space

+r

0

2]Zl is also '~ Banach space. The proofs go along similar lines and tile results are the following: ~v(k, r)

and /(k, r) exist in the strip [Imk t < ~ and there they are unique respectively in the class of the functions such tha t

(3.36)

+r +cv

fe"l~'(k,r) + k2q~(k,r)Idr<-+-oo fr~"ll"(k,r) + k'l(/~,,)!d,< § o 0

l~urthermore vl(k, r) and vdk, r) are analytic vector-valued functions in the strip I rmk l< a, derived:

(3.37)

continuous on ]Imk I ~<~. The following bounds are easily

C l!',,,(k, ")ll-<.y~-C (i = 1 ,2) , l l m k l < ~ .

2 3 - l l N a o v o C i o t e n t o A .

350 M. B]~RTER0~ G. TAL~I~ITI and G A. VIANO

I t must be observed that all the formal algebraic manipulations made in Sect. 2 on the integral equations are now completely justified.

We can write

f sin ks S(k) = 1 - 2i - -~-- . v(k, s) d,~

o

(3.38)

and

(3.39)

From the inequality

(3.4o)

-~-co

](k) = 1 § ,~) d s .

0

[si.nz] < exp [Jim zl]

and inequality (3.15), it follows that

2 11___+ C ]imki<~. (3.41) lS(k)[~<l§ ~ I[~( k, )11< _ c '

Furthermore~ (3.6) an4 (3.37) imply

1 (3.42) Ill(k) It< 1 + ]lvl(k, .)J]<~- C ' lira kl < ~"

So it is quite clear that S(k) and ](k) are finite and continuous in [Imkl~<a; eq. (2.17) implies that ](k) cannot be zero in the strip l Imkl<~.

Introducing the T-matrix

(3.43) T(k) = ~ [ S ( k ) - 1],

it is straightforward to verify that

(3.44) ]exp [-- ikr] q~(k, r) - - T(k)] -+ 0 (r-~ + ~ , I I m k l < a ) .

T~EORE~ 2. -- ](k) and S(k) are analytiv 1unctions, regular in the strip

I Im ~1 < a- We want to observe that it is possible to give a direct proof of the analyticity

of S(k), if one does not want to work with the gost functions: one must only take care to use appropriate bounds on the function (sin kr)lk and its derivative with respect to k.

Of eourse~ the an~lyticity of ](k) implies the analyticity of S(k); so~ with the results tha t we have at our 4isposat~ the less intrie, at~ proof of the theorem is perhaps the following.

TIIE BORYl SERIES FOR I~0NLOCAL POTENTIALS (S-WAVE) 351

The start ing point is the integral representation (3.39). Let

(345) ](k) = .iieikrvl(k, r) dr q-f eik" vl(k, r) dr. o 0

As i~ Lemma 2 one ca~ prove tha t irL the strip l I m k t < a

(3.46)

So

(3.47a)

(3.17b)

ea ~

]#e~- l - - ' e ~ < - ~ i ~ , . ~ I~nal [

e ~r Ivy(k, r)],

a~d the inr in (3.45) are absolutely convergeat. 5Tow

(3.48) ](~ -~ h)--~ ](k) -- ~(]g) < jet ikrF[ ~hVl( k,h . . . . r) ~)1(~, r ) ] d r [ -~-

o +Qo -t-r

4- f~ladk~h~-'~k'r!dr]Zalf[~h~k~--irFk~]v~(k,r)dr] 0 o

As in Lemma 2 we have

(3.49) 1 A a e ~

j h

It follows tha t

~-Ilmkl-I~'1

- - - - ire'k~ j < (~ - - l i r a k])(~----]Imk]Z ]h~} e ar .

(3.5o) ) + h

]

From Lemma 3 we have

I ](k d- h)-- /(k) ](k) ]___>O (3.5l) [ h . . . . . . . . I

(Ihl-+ 0, I I m k t < a ) ,

and therefore Theorem 2 is completely proved.

352 1~. Bli~RTERO, G. T A L E N T I a l l (1 ~ A. VIANO

4. - Separable potentials .

We now analyse a nonJocal potential such that

(4.1) V(r, s)=v(r)v(s).

without restriction (3.2) on g. The hypothesis (3.1) made in Sect. 3 on V(r, s) implies that v(r) is a real measurable function for r > 0, satisfying the fol- lowing conditions:

+r +r

(6.2) f[v(r)[e'~dr< + o o , fr[v(r)[e~'dr< + c ~ , O < a < m , m>~O. 0 0

Let us call ~'(k) and ~(k) the Fourier sine and cosine transforms of v(r) respec- tively:

(4.3) 9(k) =fv(r) sin kr dr, =fv(r) cos kr dr. 0 0

I t is straightforward to write down explicitly the scattering, regular and Jost solutions:

~(k) (4.4a) y)(k, r) = sin kr + g ~ ~(k, r) ,

sin kr "~(k) ~ (4.4b) ~(k , r ) - - ~ : ! + g ~ ( ~ e f ( k , r ) ,

](k, r) = exp[ikr] + g~U(k) iN(k) A(k) + 7(k, r) , (4.4e)

where

(45a) r) =fG(k; r, s)v(s) d s ,

0

f

0

(4.5e) ~k, r) = f sink(sk -- r) v(s) ds

r

T I I ~ BORN S E R I E S FOR 2~'ONLOCAI, POTlgNTIALS (S-WAV]~) 353

and D(k) and /l(k) are the Fredholm determinants of eq. (2.4) and eq. (2.8) (or eq. (2.10)) r(;spectively:

(4.6a) D(a) : l-r.]fv(r)drf?;(.~)e;(k;r,.~)d.9: , - .qfv(r)~(k,r)dr, 0 0 0

+ c o r +r

,,f r f g f - (4.6b) /l(k) = 1 - v(r)d v(s) sin k(r- s) d.s. = I - v(r)~0(k, r) d r .

k 0 0 0

The solution of M1 the problems has been reduced to quadratures . All the integrands are anMytic functions of k; therefore, in order to derive analy- t ic i ty properties, it is enough to show the tmiform convergence of the integrals. I t is very easy to do it, using inequal i ty (3.6) ~nd conditions (4.2).

The result is tha t , for every value of r, 0 < r < + c~, the scattering, regular a~d Jos t solutions are ~mdytic funct ions of k ill the strip I I m k l < m, excep t for a set of points where the solutions have poles.

The zero of D(k) and A(k) are the source of these poles. (Of course, if m = 0, the solutions are defined only for real values of k and are there con- tinuous, except in the neighbourhoods of the zeros of D(k) or A(k).)

As a consequence, the Jos t funct ion is meromorphic in the strip IIm k[ < m ; using the iden t i ty

definition (2.11.) gives

D(k) (~.8) /(k) ......

A(k) "

I t mus t be observed t ha t D(k) is holomorphic in the half-plane Im k > -- m, whereas A(k) is holomorphic in the strip l I m k [ < m ; so ](k) is meromorphic in this last strip and the zeros of A(k) are the source of the singularities of ](k). However these zeros do not appear as zeros or poles of t h e S-mat r ix ;owing to the symmet ry proper ty

(4.9)

one has

(4.1t))

A(k) = A( - -k ) ,

i t - k ) D ( - k ) S(/,:) . . . . . /(k) D(k)

I t is interest ing that , even if the Jos t ftmctions are defined only in a strip, it is however possible to write the S-matr ix as the ratio of two ftmctions which exist in half-planes, just as in the local case.

354 ~ . B E R T E R O , G. TAL]~NTI ~ I l d G. A, VIh l~O

The nex t step is to discuss the zeros of D(k) and to relate these zeros to the solutions of problem b).

I t is quite clear t ha t if D(k) is zero at k = ko, the~ for this value of k a solution of (2.4), wi thout the inhomogeneous term, exists; it is s t raightforward to recognize t ha t such a solution is v~(ko, r).

Therefore we have to s tudy the behaviour of ~2(ko, r) as a f tmction of r. As regards the behaviour for r --> 0, if Im ko> 0, f rom inequal i ty (3.6) we get

(4.11) d-co

I~(ko, r)]< rfsIv(s)Id,; 0

az~d the condition ~(ko, 0)----0 is satisfied. As regards the behaviour for r - ~ d= c~, we discuss first the case I m k o > 0

and t h e n the ease I m ko ~ 0. I n all the subsequent discussion we shall sup- pose m ~ 0; a t t ke cud of the Sect ion we shall discuss briefly the ease m -~ 0.

a) I m k o > 0. F r o m the inequal i ty (3.6) we get for every value of a,

0 ~ m + ~

(4.12a) I~(ko, r)l ~ e x p [ - rImkoJjse*~'lv(s)lds, 0 < I m k o ~ < m ,

0

d-co

125) I~(ko, r) l<~ e-a~tsc~'lv(s)Id s , [m ko > ~. (~. e g

0

As a consequence~ ~(ko~ r) is a solution of problem b). I t mus t be observed the character is t ic fac t t h a t for ~ no , loca l separable

potent ia l such t ha t v(r)~.co e x p [ - - m r ] , the bound s ta te wave funct ion cau- not decrease faster t h a n the potent ia l for r - + d- cr

I t can be proved by s tandard methods (~) t h a t k~ must be real; fur thermore ,

using the spectral representa t ion of the Green hmc t ioa (7(k; r~ s)

2 f sinqrsinqs__ I m k > 0 ~ (4.13) G(k; r, s) =-~ k2_ q2 dq,

0

it is s t ra ightforward to show, as is well known, tha t D(k) has a t most one

zero on the imagLaary axis in I m k > 0.

b) Ymko = O, ko:/: O. l~or loeM potent ia ls satisfsrLag the Bargmazm con- di t ion i t was impossible for ](k) to be zero at a real value of k; i t is not so

for D(k) (~).

T H E B O R N S E R I E S 1)'OR N O N L O C A L P O T E N T I A L S (S -W~kVE) 3 5 5

The iden t i ty (4.7) implies t h a t i t is possible to have such a zero Lf there is

confluence be tween a zero of v(k) an4 a zero of A(k). Owing to the fact t ha t the zeros of A(k) are continuous funct ions of g, i t

is clear t h a t D(k) can h~ve zeros for reM values of k only when the s t rength g takes some critical vMues. I n other words these s ta tes are instable with respect

to small cha~ges of t im poten t ia l (~). As regards the behaviour for r - + ~- ~ , using ~(ko)= 0, one has

q - 0 o % 0 o

r f sin kos sin kor f v(s) cos kosd s (4.14) ~(k0,r) =-cosk 0 v(s) k0 d s - k0 r r

an4 so for every a, O ~ a < m,

(4.~a)

-{-cP ~co

l~(z,,o,~)l<l. ~ I.,,(.~)ld.~ 1~:2ic -~ ~lv(.~')ld.~< r r

+co +oo

O r

This bo tmd is e~ough i~l order to conclude t h a t ~(ko, r) is a solution of

problem b). YaequMity (3.40) implies in [ m k ~ 0

(4.1(;) q - ~ o

If 0

i.e. D(k)-§ Ik]-~ ~- c~o. I t follows t h a t D(k) has ortly a firdte n u m b e r of zeros in frak~O, because it is holomorphic in I m k > - - ~ n .

All these zeros, except possibly t h a t a t k = 0, are simple.

In fact , using the spectrM represen ta t ion of the Green ~unctio~ G(k; r, s), i t is possible to show t h a t for every k in i m k > 0 the first der iva t ive of D(k) can be wri t teu

+r

(4.17) l')(k) = 2kg(~p'~(k, r)dr. 0

H there is a negat ive-energy bound s ta te a t ko = ib, b > O, ~(ib, r) is real ; therefore l)(k0) is cer ta inly different f rom zero.

When k is real, the in tegral a t the 1.h.s. of eq. (4.17) general ly is not convergent except if k = ko is a real zero of D(k) ; in fac t we know t h a t in this case ~(ko, r)

356 M . B E R T E R O , G . T A L E N T I and G . A . ~ r I A N O

belongs to Z ~. In order to conclude tha t for these special values of k, eq. (4.17) is ye t true, i t is enough to show tha t the integral a t the 1.h.s. is nniformly con- vergent on the closed interval k = ko + is, 0 < s < ~ (~ < m and fixed). Remem- bering tha t v(k0)= 0, it follows t h a t

(4.18) I~(ko ~- is)[ < B s .

Equations (4.5a) can be wri t ten as

+0o; ,

(4.19) ~o(k,r) =- - -~e~k~'~(k) 4-~ v(s) sink(e-- r) ds .

The second te rm is uniformly bounded by

(4.20)

-boo ~-r

f ( )sink(8--r)dsl<e- fsIv(8)le 'd8 r 0

and the square of the first one by

(4.21) k 2 B ~

e 'k' 9(k) < jko[- exp:[-- 2sr] --

B 2 e28 A f e ~ -- ik01(l§ - 2 s ( l + r ) ] ~ < i k o , ( l t L ~ _ , r r / I §

Therefore the uniform convergence is guaranteed. The point k = 0 needs a little different analysis. If

q-o~

(4.22) fry(r) dr =/= 0 ,

O

the zero a~ k = 0 can be a t most simple; the corresponding solution of the ho-

mogeneous equation is r -{-co

(4.23) ~(O,r) : - f s v ( s ) d s - r fv(s)ds . 0 7'

+ ~

On the other hand, if frv(r)dr = O, (4.23) can be wri t ten o

+ c a

(4.24 / ~(0, r) = f ( s - - r)v(s) ds . r

THE BORN S:E1RIES FOR NO!ELOCAL P O T E N T I A L S ( S - W A V ~ ) 3 5 7

I t follows t ha t

r 0

i.e. a zero-energy bound state. I t is ve ry easy to check tha t in this case, on the closed intervM k = is, 0 ~ e ~ a ,

(4.26) l;(k)] < B e 3 .

As ~ consequence, eq. (4.17) is t rue for k = 0, i.e / ) ( 0 ) = 0; by differen- t ia t ion one has

(4.27) i)(0) 2g I-~2(0 , r) dr.

0

Therefore in this c~se the zero ~t k = 0, if i t exists, is u second-order zero; no higher-order zero is allowed.

I t is also very easy to write doom the full Green funct ion and the result is

(4.2s) ~ ( k ; r , s ) ( , ( t ; r , s ) g ~ r)~f(k,s) = t-D(k) W(],:, .

By s tandard methods (4.~) it is then possible to prove the completeness of the system of the scat ter ing solutions (for real k) and of all the solutions of prob- lem b) ; ( the in tegrat ion pa th in the k-plane, which has r be used in the proof, has to go around She zeros of D(k) on the real axis).

Equat ions (4.17) and (4.27) have to be used. Remembering t ha t on the real k-axis:

(4.29) D*(k) = D ( - - k ) , ~V(k) = exp [2 i t ( k ) ] ,

with a(k) a real funct ion of k, if we write

(4.30) D(k) = ]D(k)Jexp I -- ia(k)],

we must observe that , whereas ~(k) is continuous in the neighbourhood of zero of D(k) (in fact a real zero of D(k) does not manifest itself as a pole of S(k) ;

if D(ko) is zero, D(--ko) is zero~ too), a(k) has a discontinui ty equal to z . Therefore we can say t ha t a(k) is equal to ~(k) plus ~n~ where n~ is the number of zeros of D(k) encountered going f rom oo to k. Then, computing

(4.31) :, f b(k) 2~i ~ dk, ff

~ 5 8 M. B E R T E R O , G. T A L E N T I ~t,Ild (~ )~. V I A ~ O

where the pa th C is the same used for the completeness demonstrat ion, i t is possible to derive the Levinson theorem:

(4.32) ~(o) -- ~ (+ ~ ) = (n + �89

where n is the to ta l number of states belonging to Z 2 and q = 0 if the point k = 0 is not a zero of D(k) or if it is a double zero, q = 1 if t h e p o i n t k = 0 is a simple zero.

I n the case r n = 0, our proofs tha t the solutions of the homogeneous equat ion associated wi th (2.4) belong to Z 2 and the zeros of D(k) on the real axis are simple, break down. In fact f rom the conditions

+a~ A-co

fiv(r)ldr< +~o, frlv(r)ldr < +o~, 0 o

i t is possible to deduce t ha t l~(ko, r)l-+O , r - ~ § c~, bu t no more. I t is clear t ha t the hypotheses made on the potent ials have to be s t renghtened

in this case. We do no t want to analyse the problem of finding the largest class of non-

local separable potentials such tha t ~(ko, r) belongs to L ~. We are interested only in finding a class which is large enough from the physical point of view.

I f we suppose tha t

A (4.33) [v(r)l<r~(1 § r)~' 0 < r < § c~,

wi th a < 1, f l > 2 - - a , then it is ve ry easy to recognize t h a t ~(ko, r) (where ko is a zero of D(k) in Imk~>0) decreases a s a power for r - + § oo and tha t

i t belongs to Z *. For the class of potentials (4.33) eq. (4.17) (on the real axis) and (4.27)

are s e t true. We sketch the proof. Let us observe tha t

+co f o

(4.34) i)(k) =- -- gl v(r) ~(k, r) d r ,

0

where ~(k, r) is a solution of the inhomogeneous differential equation

(4.35) ~'(k, r) + k ~ ( k , r) = - - 2 ~ ~(k, r),

which is zero at r = 0. Therefore we can certainly write

" k ~. /" sin (r-- s) (4.36) ~(k,r) = A(k) s i nk r - zJr ~ ~(k,s)ds,

0

THE BORN SEi~IES l"OR N( )NLOCIL POTENTIALS (*~-WAV~) 359

where A(k) is an unknown funct ion of k. Remember ing t h a t ~(ko)= 0 and observing tha t

_s) : I (#~ sin kr, (-1.37) 'g)(r .~.)siTt/,:(r/,: . (J(k; r,.~.) t-/-:

f rom eq. (4.34) we reobtMn eq. (4.t7) (we recall t ha t ,~(x) ~ ] , x ) 0; O(x) = 0, x < 0 ) . In an analogous way eq. (4.27) can be deduced.

In order to conclude tha t the number of zeros of D(k) on the real axis is finite it is enough tha t the first and second der ivat ive of D(k) be continuous. A sufficient condition is t ha t

(4.3s)

~-co

f r~]t;(r)]dr < -~ oo, o

or, for potent ials belonging to the class (4.33), t ha t f l > 3 - - a . For this class of potent ia ls the completeness and the Levinson theorem c~n be proved.

5 . - C o n c l u s i o n s .

The results obtained here for nonlocM separable potentials, are likely true in the general case (i.e. condition (3.1) wi thout condition (3.2) on g).

The proof of the convergence of the Born series (when g is restr icted by (3.2)) given in this p~per is a first step in this direction.

Therefore we can expect, for nonlocal potentials satisfying ~ sttmmability condition, th,~t the solutions of the Schr6dinger equat ion have ~nulyticity regions much smaller th~n those in the local ease for potentials satisfying ~nMogous conditions.

The work for removing condition (3.2) on g in progress.

One of the authors (M. B.) wishes to thank Prof. K. BLEUI~ER for the warm hosl)i tali ty ex tended to him at the Ins t i t u t ffir Theoretische Kerl~physik.

A P P E N D I X A

We prove here Lemm~b: 2. As w~s said, fl'om the hypothesis (3.1) on the po(ellt.iM, i| follows t h~t C(r) ~ -~- c~ for Mmost M1 r ~ 0. Therefore the bounds

360 M. BERT:ERO, G. TAL]~NTI and o. A. VIANO

(3.7a) and (3.7b) show tha t the integrMs defining vo(k, r) and X(k; r, s) are uniformly convergent wi th respect to k in the strip Jim k l < a and the analy- t i e i ty is proved.

We sketch the proof of the inequali ty (3.28c); inequalities (3.27a), (3.27c) and (3.28a) can be proven in an analogous way; (3.27b) and (3.28b) are a consequence of (3.27a) and (3.28a) respectively.

Le t k be a fixed point in [ I m k ] < a ; the Cauchy theorem gives (for Mmost all r > 0)

(A.]) N(lc; r , s ) = 2~i z - - k ~(z; r , s ) d z ,

~ ( k ; r,s) ~- 2 ~ ( z _ k ) 2 N ( z ; r , . ~ ) d z ,

where the integration path is the circle, with centre in k and radius R = ~ - lira k [

(z = l~ + Re ~, 0 < qJ < 2r~, dz =~ R e ~ ida) .

We have

(A.').) N~k § h; r, s) -- s r, s) _ -.~k;

,~ ) r, h

2~zi ~ (z-- k)2(z - k - - h)

2 ~

h f~V(k§ r,s)dq~

0

Therefore the l.h.s, is bounded by

(A.3)

2 ~ 2 ~

h f t ~ ( k + R e ' ~ ; r , s ) f I~(k R e ' ~ ; r , s ) I RIRe i~- hi dq~ <

0 O 2r~

0

Using the bound (3.7b) on N(k; r, s) and the expression of the radius of the circle, we obtain inequMity (3.28c).

I t must be observed tha t when one derives the inequalities for v,(k, r) one finds the integral

(A.4)

2:z

O

Lemma 2 is completely proved.

THE BOICN SERIES FOR, NONLOC&L POTENTISJ .S ( ~ - W A V E ) 3 6 1

A P P E N D I X B

We prove here Lemma 3. I t is enough to prove tha t :

i) v(k~ "1 is bounded in every bounded domain contained in the str ip I m k < ~.

This f~et is ~ consequence of eq. (3.15).

ii) v(k, .) is cont inuous in the str ip }Imk[<ce:

(B.t) C ( lkl+~)[hl llv(k +h, . ) - v (k , . ) l l<(~ C) ' - I I m k l - lhl"

Let its suppose tha t k is such tha t [ I m k [ < a and let us wri te

(B.2)

Ahv(k, r) :--= v(k + h, r) -- v(k, r) ,

d~vo(k, r) =: vo(k 4- h, r) -- vo(k, r) ,

AhN(k; r ,s) = lV(k 4-h; r , s ) - - N(k; r , s ) .

From the iqtegral equ'~tion (3.4) we have

(B.3)

~-oa

dhv(k, r) :--: Al, vo(k, r) § o

r, s)[v(k, .~) + Ahv(k, s)]ds q-

+co

+f2r r, s)A~v(k, s) ds . o

This equat ion (.an be considered ~s an integral equat ion in the unknown A~v(k, r).

Remember ing tha t l]L(k)[!< C < 1, we have

(B .4 ) (~ - C) l ldhv (k , ")J[ < ]i&'~,,(/~, ")[P + I[A,L(~c)[v(k,-) + A~v(k, ")]Jr

and from Lemm:~ 2

(B.5) - lhl

cthl JlAd~(k)Ev(k, �9 ) + Ahv(k, -)3 I'~< ~-- Jim k I - Ih] Ilv(k' ") + A~v(k, ")JI.

3 6 ~ l~I. BEI~TERO; G. T A L E N T I and a A. v i h ~ o

Furthermore , observing tha t

(B.6) Itv(~," ) + A~v(k,. )1] - [Iv(k + h , . )II,

from the inequal i ty (3.15) i t follows tha t

(B.7) ][v(k + h, .) +A~v(k,-) i f< (lkl + [hi) C < !lkl + ~ ) c

~ 1 - C ] - C

(B.4), (B.5) and (B.7) give (BA).

iii) For every k in [ Imk]<~ , ~ function ~)(k, .) exists such tha t (3.30) is true.

Let us consider the integral equat ion

+r +co

(B.8) 0 0

in the unknown ~)(k, r). The term between [ ] is a funct ion of r which belongs to X i f [ Imk[<~ .

In fact from Lemma 2 one has

) (B.9) ][~)~ a-- I m k I

and furthermore, from Lemma 2 and inequal i ty (3.15),

(B.10) C2tkI flL(k)v(k, .)H:< (1- ~)(~- I I m k l ) "

As in Lemma 1 it is then possible to show tha t the integral equat ion (B.8) has a unique solution in X. Tow, from (B.3) and (B.8) we have

(B.11) 1 hAhV(k,r)-i:(k,r) =[-~Aav.(k,r)-~)o(k, r)] +

+ m

+f [hAh~V(k; r,s)- N(k; r,s)Jv(k, s)ds + O

+ m +r

1 +f.~(k; r, s)[~Ahv(k, s) ~(k,s)lds + f ~ Abe(k; r, s)Ahv(k, s) ds 0

and this equat ion can be considered as an integral equat ion in the unknown

1 Ahv(k, r) - ~(k, r) .

THE BORN SERIES FOI~ NONLOCAL POTENTIALS ( ~ - W A V E )

U s i n g a g a i n ILL(/,:):[.~4 C < I :

363

(B.12) ( 1 - zlhv(k, ")-- 'b(k,') < Ahvo(k, ")- 'bo(k, ") +

1 1 [~ JhL(k)- JL(k)]v(k, .) + l-~lhL(k)/Jl, V(Ic, ) .

F r o m L e m m a 2

(B.la)

IhlC(Ikl +~! l h A'~v~ t<(~_ [imk[)(cz_ lImki_ lh,),

IhlCllv(k, )ll "[~-~L(Z~)--~'(k)]v(,~, .) <(~_ IImkl)(~-IImk[-[hi)'

< c][~v(k, -)It ) , mkJ- 1hi"

o (Iht~o) .

Using t h e b o u n d (B . I ) on ][Ahv(k, ")I], we o b t a i n

fl (]3.14) i~ /lhv(k, " )-- i;(k, r),,-->

A n d L e m m a 3 is c o m p l e t e l y p r o v e d .

R I A S S U N T O

Si dimostra una condizione sufficiente per la eonvergenza della serie di Born net caso di potenziali non locali. Si riconsiderano i potenziali separabili allo scopo di estrarne informazioni sulla dipendenza generale della soluzione di scattering dalla costaate d 'ac- coppiamento g e dell ' impulso k.

Pe3IoMe He JIO.rlyqeH.