THE A N A L Y T I C P R O P E R T I E S FOR INTERACTION OF THE SCATTERING V I A NONLOCAL AMPLITUDE POTENTIALS by RONALD STUART B.Sc, University DAVIS of Alberta, 1963 A THESIS SUBMITTED I N P A R T I A L F U L F I L L M E N T THE R E Q U I R E M E N T S FOR MASTER OF in THE D E G R E E OF OF SCIENCE the department of PHYSICS accept this thesis as conforming to the required THE U N I V E R S I T Y OF B R I T I S H April 196^ COLUMBIA standard In the requirements British for extensive be cation without of my Department this that and thesis i n partial by degree at the the of I this Head permission* of The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a Columbia, of the U n i v e r s i t y of f u r t h e r agree that freely per- thesis for scholarly o f my I t i s understood thesis for financial written fulfilment L i b r a r y s h a l l make i t study, copying granted representatives. this advanced agree for reference p u r p o s e s may his f o r an Columbia, I available mission presenting Department that.copying gain shall not or or be by publi- allowed ABSTRACT The d e r i v a t i o n of a partial-wave amplitude a separable, n o n l o c a l p o t e n t i a l given by 29, 4153 (1963) i s reviewed. f o r the amplitude where V ( r ) = r asymptotic a f o r s c a t t e r i n g by M°Millan i n Nuovo Cimento Using h i s r e s u l t s , an exact expression i s d e r i v e d f o r a p o t e n t i a l of the form -g V(r)V(r ), f e""' , and i t s a n a l y t i c p r o p e r t i e s are s t u d i e d . The Ar behaviour of the amplitude usual angular-momentum as \t\ (where t i s the -* » parameter) i s d e r i v e d , and i s shown to permit a Sommerfeld-Watson t r a n s f o r m a t i o n to be performed on s e r i e s expression f o r the t o t a l the p a r t i a l - w a v e amplitudes. s c a t t e r i n g amplitude the i n terms of By means of t h i s t r a n s f o r m a t i o n , a d o u b l e - d i s p e r s i o n r e l a t i o n i s d e r i v e d f o r the t o t a l amplitude both the complex-energy and complex-cos ©• p l a n e s . in E x p l i c i t forms are d e r i v e d f o r the weight f u n c t i o n s , and the convergence of the integrals involved i s studied. In a d d i t i o n to the usual branch cuts along the p o s i t i v e , r e a l energy and cos ©• axes, an e x t r a cut along the negative , r e a l energy a x i s i s found which i s not present f o r the l o c a l case. of two I t s o r i g i n i s t r a c e d to the f a c t t h a t the Wronskiah' s o l u t i o n s of the n o n l o c a l r a d i c a l Schroedinger equation i s not n e c e s s a r i l y a constant, as i t i s i n the p u r e l y l o c a l case; to the c o n d i t i o n s necessary to ensure convergence of the e x t r a i n t e g r a l i n the n o n l o c a l Schroedinger equation. and iv ACKNOWLEDGEMENTS I am i n d e b t e d and t o D r . J , M. M ° M i l l a n f o r s u g g e s t i n g f o r generous a s s i s t a n c e with the N a t i o n a l Research C o u n c i l i t . the problem T h i s work was s u p p o r t e d o f Canada. by iii TABLE OF CONTENTS I Introduction 1.1 II A closed form III The a n a l y t i c p r o p e r t i e s f o r the p a r t i a l - w a v e s c a t t e r i n g amplitude of ^ ( k ) v and v ^(k) fora 2.1 3.1 particular potential IV The a s y m p t o t i c behaviour of a^(k) V The a n a l y t i c p r o p e r t i e s o f as \l\ -* f ( k , c o s ©•) 0 0 i n the 4.1 5.1 c o m p l e x - c o s ©• p l a n e VI The a n a l y t i c p r o p e r t i e s of f ( k , c o s ©•) i n the 6,1 2 complex-k VII The o r i g i n plane of the " e x t r a " c u t i n the k plane V I I I Summary 7,1 8.1 Appendices s I C o n v e r g e n c e o f t h e i n t e g r a l s d e f i n i n g D^(k) and II The a s y m p t o t i c b e h a v i o u r Bibliography. of P^(t) as | t | -» » 1.1 II.1 111,1 1 Chapter In this properties by thesis, of the a particular advantage it of a 1, Introduction detailed partial-wave class of to obtain an amplitude. used here, a m p l i t u d e has is possible amplitude, to derive and to Separable, physical the be reformulated to detailed 310 the Lomon and form the Model for are the for the to potential so that shown of that interaction and Tabakin s e p a r a b l e , has nuclear-nilclear i t relation. (1963) have be that partial- e n t i r e l y devoid potential, potential is scattering nuclear-nuclear separable available not The (1963), total analytic scattering simple form, McMillan scattering can (1964), obtained data up similar to mev. ( 1 9 6 3 ) and Mitra (1963), using have a l s o work goes beyond t h e i r s , h o w e v e r , by for the convergence of o t h e r s by the closed form f o r the potentials. McMillan a relatively those used here, of for p a r t i c u l a r type of potentials nuclear-nuclear Cushing forms amplitude shown by For i n terms of the total i s made o f i t i n a double dispersion nonlocal f i t to as explicit express Boundary Condition assuming the a an significance. and explicit, wave s c a t t e r i n g this investigation separable, nonlocal such p o t e n t i a l s , i s possible 1.1 spectral the applying o r i g i n of amplitude for the proven double-dispersion functions spectral presenting involved, integrals. method of certain potentials analytic general nonlocal Bottino and I t also explicit, by extends potentials. of the an the This closed investigating (1962) to properties relations. work the of investigation scattering Chapter 2 - In for A this c l o s e d Form chapter, the scattering f o r the Partial-Wave the integral wave equations f u n c t i o n and Scattering given Amplitude by M c M i l l a n (1963) the partial-wave amplitude f o r i scattering The V separable, Schroedinger + 2 v i a a k * ( r ) 2 equation f - a l l and the corresponding function *(r) = i s exp(i k«r) + J where nonlocal potential r = / dk integral j dr' = equation J*dr" 0 f o r the G(r,r') i s (2.1) scattering-wave V(r',r ) f ( r " ) (2.2) H ra l la nrde a rl 1 G i s exp(i / k a l l a nonlocal potential dr' V(r\r') real the Green's f u n c t i o n 0(?,?') with are derived. . k - ( r - r )) , ( 2 < 3 ) -k^+ie real k The f i r s t and the second Green's outgoing term waves integral and may given the gives be by by (2.2) represents the effect equation at infinity. equation derived f o r scattering (1961), ( 2 . 2 ) may of the quantities appropriate be A expansions expressed are^. incoming (2.2) i s a given 3.1 as plane to the potential. much more chapter concerned due an (2.3) contains + Equation similarly. McMillan Equation each term function given the is i n equation elegant i n terms of to (1953), yield of page 1077, derivation appendix radial The generalization i n Morse and a i € wave I . equation spherical by expanding harmonics; 2.2 1=0 m=-l <t=0 m=-<I 00 exp(i which the £.?) (2.2) = 4ff V define usual -t the V i 1=0 m ^ t new functions spherical expressions Bessel may a l l be relation (2.4), separated ( k r )T * (k) T ( r ) , -Blatt (1952) Appendix t^( ) and r function). and (2.3) a r e expressed the * j When i n spherical V^(r,r ) • (j^(kr) the integrals polar form of i s equatims b y means of the angular portions of the integrations and evaluated by means of the orthonormality f o r spherical harmonics (Dicke(1961), A page 1 8 9 ) : sphere Equation ^ ( r ) (2,3) thus =.j^(kr) +J leads dr r to a d family of equations G j ^ r ) jT dr r of the form ' V^(r , r ) ^ ( r) (2.6) where G j r . r and 1 a ) = / 0 I = 0 , 1 , The method denotes dk k k 2, ... » J -k +i€ (2.7) (the"physical" values of McMillan the spherical (1961), polar appendix coordinate of I , may angles I.). be used of •* a t o show i n some — » arbitrary coordinate system; a denotes the magnitude of a. 2.3 that G^(r,r') where = - = ^ An a^(k) m a h^ * stituting be (2.8) (r,r'). x mm derived ( k r ^ 1 ' x ' expression f o r the may (2.6) i k j ^ k r j partial-wave from the e x p r e s s i o n (2.8) scattering asymptotic form f o r the Green's (kr) / dr' amplitude of Jr^(r). function = j (kr) - i k h ik ( 1 ) ^ ( k r ) i f r ' io / r '* y' dr' 2 / dr" r " 2 / xV^(r 1 r -* °°, h * the integral ( k r ) -» j second 1"* term ~ i n the last exp(i 1 angular of momentum / \ ( r ) of as (2.9) r » J vanishes °°, (Morse, and, (2.9). using (1953)) becomes ^ E i L i - i i s part 4 ^ ( r " ) ... / d r ' r ' 2 / dr" r " 2 J^kr') V ^ r ' . r " ) ^ ( r " ) as which "v ,r ) * 1 2 term kr) j,(kr') " »2 ( l1 ) dd rr " rr "^ h h^ * ' ( k r ' ) 2 xV^(r',r") the i n equation gives isAr) As Sub- the outgoing, equal to £. spherical, The scattered coefficient of i " r -• wave (2.10) », with exp(i k r ) / r i s 5> 2.4 defined t o be =JT a^(k) The ponding dr dr up expressions using V r equations V(r,r*') or, the partial-wave = r ^ to this j^(kr point f o r t h e more ) V^(r ,r ) ^ ( r ) . may be familiar converted local to the p o t e n t i a l by (2.1l) correswriting V(r) 6(r-r') equation (r,r*) scattering amplitude,i . e . , (2.4), = ^ r «(*-*') since r The that condition i s now i s , that i t may V^(r,r') = where g i s a convenience, exactly, as - be since i s now imposed written g V j r ) constant. Substituting yields 1=0 r under that i n the m=l t h e p o t e n t i a l be separable; form (r') This this form (2.12) i s chosen condition, primarily f o r mathematical equation (2.6) can be solved shown. expression (2.12) f o r V^(r,r ) i n equation (2.6) 2.5 (r) = 3^(kr) - gjf dr' r ' V^r')^ G^(r,r') 2 d r x ^ ( r The double and the integral kernel integral now be Substituting in (2.14) of thus the becomes of may V*"* 2 ... (2.13) single degenerate. integrals Let the second denoted dr r the right-hand 2 ^ V j r V (r) ) ^ ( r ). side of j^tkr) - g A ^ t be solved ^ (2.14) e q u a t i o n (2.13) for ^ ^ + dr r k r O g / Substituting ( 0 this 2 00 dr r algebraically CO (2.13) ) two 00 dr r 1 r ^ ( r ) yields A- to 2 I V (r)m^ dr r t I o 2 I G^r.r . t ^ / dr 0 I O r ) y i e l d > G,(r,r expression f o r the <2.„> t )V ( )V r . second * ( r • ) integral of equation yields dr t^(r) product e q u a t i o n becomes CO A = / which a " " = r G,(r,r )v^(r )J dr r ^ ( r Jj^Ckr ) ^ ( k r ) 1 + g / dr r' V (r ) / t 2 , ( t < ( B d r V G ^ ( r ' ,r" ) V ^ ( r " ) 2 (2.16) 7 2.6 Now new functions are defined: GO V j k ) =J^ dr r V ^ r ) j^(kr), (2.17) and D,(k) = 1 + = ! drV g / 2 £ ^° dq'q + "T^O Equation r (r) 4 (2.16) = ^ ( k r ) - ( k q - then ^ D drV 2 G^(r',r") V ^ r " ) V-^ (q) .2 k may V^(r')^ 2 (2.18) ( 2 # 1 9 ) 2 be w r i t t e n t ^ ) • (2.20) * Then a.(k) V' = / a ~J d r r ' r r 0 = -gj^ V where drV ( k (2.12), Defining N j k ) equations 1-o d r 2 dr r r hj . ( k r ) { j^(kr') V V ^ ( r ' ) ^ ( r , r ) * . ( r ) drV 2 (2.11) V^r") ^(r") (2.21) ) (2.14), (2.17), and (2.19) have been used. now = g V^ (k), (2.19) D, ( k ) = (2.23) 2 and (2.22) 1 - 2 r d c *J0 K L q 2 may V*> be r e w r i t t e n ,2 - k' (2.24) and N.(k) a « t( k ) V i W = " -fe-pjv D,(k) *. (2.25) S 2 . 7 In appendix I i t i s shown t h a t t h e i n t e g r a l s i n equations ( 2 . 1 7 ) and ( 2 . 1 9 ) converge f o r p h y s i c a l v a l u e s o f I i f V (r) = M r -t 5 / 2 ) and as r - 0 (1.3) asr-*°>. (1.4) Chapter 3: The A n a l y t i c P r o p e r t i e s of ^(k) Particular In terms this chapter, of hypergeometric V^(r) where the several = r a a a n d u. a r e r e a l w i l l be seen McMillan one that (1963) with h i s work f will 3 (a where P(a, + y P i s a p 2 p ) ^ / f c = of by Cushing Volume ^ l / , J4±M±L, ' 2 formulae, are derived. a^(k) derived i n case, work, A (1963), I t this being particular and comparisons course. I I , page 29: 2 -v ^ f f + 2 i ; = k K ) (3.2) ( 3 < 3 ) y 2 function", 771—TP c a 2 . a— ( "hypergeometric V + 1; - ^ V 2 ]j a b ; c j z ) = 2-x (a) of these (k) o u t i n due 2 k=0 where case v r(^v) 2 f o rthe special i n the present (1954), " 2V^r(v+l) / F and of the present be pointed QMV). S are derived i n v By means e x p (-.ax) J ( x y ) ( x y ) 2 f o r ^(k) functions properties studied t o Bateman dx x ^ - / 0. explicitly motivations o f (3.1)has been forms (3.l). ^ ( k ) v the general case According u. > of appear o f the main closed u.r), and analytic properties (k) f o ra Potential. and Legendre e x p (- 2 and o f v Z defined by (3.4) * ( a + n ) ( 3 « 5 ) 10 3.2 (See, f o r instance, associated The the Legendre most present useful work P(a,bj and that Using v ( ) k function (EI chapter are i f ff.4), P~ i V i s an 3). of the hypergeometric function i n 1, (3.6) defining |z| < 1. i tconverges Also, there a v a i l a b l e f o r i t , some (3.1), and where that c; 0) = of z continuations ( l ) properties the s e r i e s function E I 2.1.1 (3.2), and and i s an a n a l y t i c are several of which analytic are used i n the following (3.3), F ( ^ ± 2 , S±§±± ; ^ 3 / 2 ; - 4) ( 3 . 7 ) = 2 - ^ V 1 _/lf" V TU+3/2) r(a+£+3) a+5/2 2 k (, 2 k ) 2 + 2 2 2 p--t-l/2 ^ a+3/2 u / V 2 | i ) . (3.8) 2 " The abbreviations EI and E I I a r e used f o r volumes 1 and 2 r e s p e c t i v e l y of "Higher Transcendental Functions" by t h e makers of references (Bateman (1954)). The s u b s e q u e n t n u m e r a l s a r e equation numbers i n those works. e 11 3.3 Equation the (3.7) i sv a l i d region always i n which converges. available V (k) = there < 2 u because 2 are several analytic t h i s is function continuations we now u s e . function given i n equation T(a+-M-3) ( r ( - a - 2 ) i ^ " 1 However, the analytic continuation hypergeometric lk l the series defining the hypergeometric f o ri t w h i c h Using o n l y i n t h e domain a + (3.7) y i e l d s < t + 3 g r ^)r(^i)k 2 l+1 b y E I 2.10 ( 4 ) o nt h e (a+U3,aU+4 . a + 3 2: 2 ( k r { a i p(i=£=a=i=l, ( *4) , _T(a_+2) l4 | ± 3 ) r ( ^ ± t l ) a ( + ,x 3 A -a-1; f 2 )). (3.9) k^ k +» i sv a l i d i n the region i n which < 1? 2 k ) < However, according t o t h e r e s t r i c t i o n s 2 This 2 continuation 2 that is,.Re(k specified the with negative E I 2.1Q: real k 2 ( 4 ) ,t h i s axis region 2 -u. from 2 later that the singularity Similarly, a t ^ ( k ) " 2^V "' + -®. + 3 rU+3/2) U+ 7 ( I t w i l l may sometimes continuation i svalid 2~ J i n the region a+-t+3 ( l + k /\i )~ 2 2 cut duet o t h e f a c t o r be seen ( 2 ' be a pole,) equation (3.7) ' ^' 2 2 ki This along 2 s u b s t i t u t i o n o f E I 2.10 (6) i n t o reveals V t o = —\i k i s c u t i n general 2 ^+\i 2 ). (3.10) 2 Re(k ) > - 2 [i. / 2 . T h e i s outside this 2 region. 12 3.4 Another (u) v V K - ' ~ 2 continuation, based o n E I 2,10 3 /___JsZ _ p(S±kti) p(iz^) rU+l+3) t+l a 2 uk~ " a Y(a+|±3) (2), reveals p/a+£+3 2 r(i=|=i) a^t+2. ' w/a±£+£ 4 2 ' 3. ' ' 2 2 ji . 1„ ' 2' a-4+3- ' 2 that 2 " x k 2 (3.11) This i s valid negative and a n a l y t i c f o r a l l real Thus, k 2 I k i = u. , a n d b o t h assistance, k shall As the properties now be k -• 0 approaches and V j k ) Thus, and V^(]a) a pole [j, 2 except along the both inside and outside the and to the r i g h t of the 2, h a v e of v been ^ ( k ) found. With as a f u n c t i o n of their complex studied. i n equation 1; valid to the left R e ( k ) = - \ij line | > 2- 2 straight 2 axis. analytic continuations 2 circle 2 jk (3.3), function hence, a K as l has a branch there the hypergeometric f o r I k - point 0. at a negative (3.12) k = 0 f o rnon-integral integer; (k) has a -t, branch 2 point 21 at k = 0 i s a negative singularity positive when 21 integer. i s generally real axis to i s n o t an i n t e g e r , The b r a n c h considered infinity. line t o go and a pole corresponding from the origin there to whai this along the 13 3.5 Equation ( 3 . 5 ) shows a branch point a t k 2 = - \i 2 o f t h e form /, 2 2 \ - a -2 (k + u. ) The branch to k 2 line = . - <=° f o rt h i s to disappear infer this hand side with obtained p u./ x _ valid (k)- f o r a > 2 a k 2 = ~u. The b r a n c h point b u t i ti s impossible (3.5) because f o rt h i s case. to the right- Further point 2 information may b e o f E I 3.9.2 ( 1 9 ) , r ( y + 1/2) V v v - u.) f o r Re(v) > |z| as - », z - 1/2. Substituting this into equation = - |i 2. vV-(r) V ' s5 / f a + ^ 1 - 2. a - 5 / 2 r ( a + 2 ) ( l + 4)" " » Thus, as expected, v a 2 ^ ( k f o r integral Further,EI _ , z 2 ( X - r(l-|A) Substituting 3.2 ) h a s no b r a n c h ' ( 2„ )tx/2 z into - 2, b u t r a t h e r ( 2 4 ) a n d E I 3.2 • t l + v a > 1 equation pz-u^v n 2 (3.7) , r-i P f point 1-u-v. 2 ' ' , 1 i_I ^ 2 ?U+l/2) -a- -5/2 E I 3 . 9 . 2 (19), w h i c h of order z 2 ( 3 14) U.14; ) h yields -1-1/2 using a pole (9) conspire t o say x p Again (3.13) 2 k + <= - » axis. of the function at this T^T r u + z ) equation t o r u n from yields V at 2 2 k real i s an i n t e g e r from the behaviour i svalid (3.8) a certainty b y means V which when assumes t h e form concerning i s considered the negative along seems point i s now v a l i d .2*"i(a+5/2) . 2 - 1 / 2 ( ( 1 +*-) ). (3.15) u. f o r a < - 2, 14 3.6 v (v) - 4 2 r(-a-2) £la+i+3) 1 a l ^a 5/2 + r + /, , k V " " a l-a+4-l) U 2 ^ as k - 2 - u 2 which shows rather of that a zero V^(k) v ^(k) h a s no b r a n c h of order when - a -2 a = - 2 point at f o rintegral remains (3.16) 2 k a < , 2 = - u. , b u t - 2. The structure uninvestigated. 2 The only remaining infinity,and equation this Since point. singularity (3.1l) shows i n t h e complex-k the behaviour the hypergeometric functions plane of v ^ ( k i n (3.1l) i s at ) a ^ both 2 approach 1 as V^(k) If either k approaches a k~ " a as of the quantities non-positive integer here, V^(k) a k" ~ a the f i r s t pole, depending on t h e value one k of the form —a—3 the in V^(k) 2 or, i n special properties McMillan shows of a be of a. 1 has a + \i cases, ^(k) a t either k k at —a—4 2 k 2 disappears, and a branch point o f the form or a k at k = Oi 2 = - u. , a n d o n e o f t h e f o r m , at k = 0 (^ - a ) / 2 i s a i n (3.1l) singularity 2 —a 2 ) ~ ~ or 2 = °°. and a t k 2 This = - agrees u. 2 with derived (1963). Considering (3.7) (k w i l l (3,17) |k| - ». a s at infinity case, -* « . k term 4 the point summary, i n the general (a + I + 4)/2 Thus, In infinity; 3 now V^(k) two p o s s i b l e as a f u n c t i o n sources o f complex of singularities I, i n the equation I plane 15 for 2i 2 )k I < p, : 3.7 t h e hypergeometric function; poles f o r non-positive-integral values i.e. I = - function i . e . may b e F(a b;ciiz) P (c y~~~ 9 showing £ = - a seen 3, - a - 4 ... simple parameter, 5 7 ) a n d t h e gamma has simple t o be i l l u s o r y „. ( \ ~ 'l ~ e poles f o r non-positive ( s e eE I , page i n ^ ( k ) b y means 2.) The o f E I 2.1.3 ( 1 6 ) v ( \ l - c F(a+l-c, b+l-cs 2-c; z ) ; l ~ c p"( 2 - c ) a b v that which has of the third 5 / 2 , ...; ( s e eE I , p a g e i n t h e numerator, integers;, former 3/2, - which (3.18) z F(a,b;c;z) / p ( c ) i s finite f o ra l l c, a n d thus "that 2 r u is finite in the complex—t the f o ra l l v a l u e s numerator plane Thus i n equation + -t + I - or n o f I, of equation a where + 3/2) (3.7). 3 = - ( n+ i s a non-negative the only source of ( 3 . 7 ) i s t h e gamma I t gives a simple singularities function i n pole f o r - n a + 3) integer. These same poles are exhibited 2 by a l l the analytic in t h e complex-k provides in .« McMillan continuations plane. second-order (1963). except Each o f these pole i n when k i s a t a first-order ( k ) . These poles poles singulari ofV^(k) a r e mentioned 16, 3. Most of V^(k) have V^(k) as separately t h e gamma f u n c t i o n s essential I -* 00 singularities i s thus i n the next involved rather chapter. at i n the -L = °°„ complicated, equations The and f o r behaviour shall be 8 of treated 17 4.1 Chapter In 4. order scattering series as Asymptotic to derive amplitude, expression employed, on The as \l\ -* a^(k) a double-dispersion f o r the shall permits 00 of total scattering by Omnes (1963). be found such (<lj -» f o r the transformation amplitude In that as relation the Sommerfeld-Watson described the potential Behaviour this t h e Sommerf e l d - W a t s o n on w i l l the be chapter, the behaviour total restrictions of transformation a to ^(k) be employed. The terms series expression of the partial-wave f o r the total amplitudes scattering amplitude i n i s i 00 f(k , c o s («•)) = > (21 + 1) a , fck) P , ( + 1=0 Using the Sommerfeld-Watson „ f(k% • ' , COs(ft)) = k 2 _ ^. a.(k J half-plane 2 ) 2 ° 4 J I % -(2*+l) Cl* ' P 2 \.' J , >- 1/2, and the j i f the residue a^(k) Squires of satisfies (1962)J the j written cos(S)) : ~ '- B (4.2) (k )(-.cos(*)<) 2 2 th 8.(k ) Regge = 2 pole. be (k )) l i m pole 0 ^a.(k^) J a^(k), (4.1) J sin(TTa of (_ SmfnO J site P cn-n^/) 2 . may a,(k) ' -1/2-ti•~. this ( 2 a ( k ) + 1 ) ^.(k ) *a i s the ReU) transformation, ' ¥ j where cos(G-)). ' This t h e f o l l o w i n g two o f a , (k) ^ (l - i n the a,(k ))x 2 .V transformation i s conditions, given by valid 18 (l) 0(<t ) n a^(k) = (2) Conditions that In EI shall conditions behaviour of (Z."_i)~ ~ a now X F ( a+ X » a - and TT/2J (4.3) -t - + i » on V(r) satisfied. r -» 0 must be c + as given I t will of by (3.1) i n order be found severely 1 + Xj a - b + that the restricted. a ^ ( k ) as \l\ — », r . r(a-c+l+X)r(c-b+X). x (1+0 (X- )) the upper exp(T iTT ) ) jx| - as 1 o r lower sign 1 + 2X; a+X r(a-b+l+2X) ((l-z+Jz^i) (z) J i s a constant (16) i s usedi X The EI as be n where to i n v e s t i g a t e the behaviour ft l m as 2 ( 4 . 3 ) may ,a+b where < be p l a c e d V(r) order 2.3.2 (l) (k) = 0 U~ / ) 3 a<t \ l\ — as - TT/2 < a r j 4.2 1 / 2 d/2) " C ( z fz^i) - i (l+z-Zz -!) 2 c-a-b-1/2 », i n exp(+ (4.4) i n ) i s chosen - according as 0. expression i n brackets c a n be t i d i e d up 1.18 ( 4 ) ; r(z+a) _ a-b r u + b f - 2 ( l + 0 ( z ^ ) ) as I somewhat by using 1-9 4.3 and Stirling's T(z) to = e" obtain 2 X 2 Z ( E I 1.18 z /2~7 1 l / , b = a+2 2 "> = •2 into the - no greater considering the k », , (4,4)) = ^rj—• w a + J, Q. ^i° 2 2 '*" n ( ( l - e u a (3.7); ^3/2 , n (4,4); and using E I 1.18 (4) again; y777. sii±2 2 ) z + f z ^ l ) exp(+ ))" a ITT 5 / 2 (l+0(^" )) 1 (4.5) = - i n (4.5) has than 1 z a (4.6) the happy f o r a l l values (4.6) t o be property of z. that This conformal-mapping may i thas be function, magnitude seen the by inverse of which i s 2z which |z |- quantity appears mapping as «. w(z) which 1 c = equation x (^1 (1+0U- )) a ( i nequation . zjgll as (2)), 2 t/2, 2 y-— V k ) Now 2 = X substituting - Z "" / Setting and formula maps entire = - the unit z plane (w + circle 1/w), and i t s i n t e r i o r (Churchill (i960)). i n the Thus, w plane the magnitude onto of 20 4.4 w(z), 1. + the quantity Therefore, » v ^(k) for a l lvalues Since ^ ( k ) particular, used i n D^(k-), that a^(k) f i r s t i t follows follows unit be noticed +/z that - 2 the circle. that the that (k) of a that along line from branch line f o r the integral the Re{l) ^(k) path exceeds approaches k, and, i n integration disappears approaches Re(l). amply of of therein denominator branch; l ) ( z other z = - gets The satisfies -J z 1, and resulting Squires' to usual branch trouble 1) - 2 of must z branch line line has w(z) happens 1, w(z) be = + hypergeometric = i s always excluded \, This function by on or avoiding corresponds i n equation outside the to (4.4) a from 2 *~ ^° line of the 1-z ~ hypergeometric 2 branch i f since branch 1 into 2 1-z a as the for large one This branch branch to (4.5), never exponentially occurring as i n t o i t sother (z the i n uniformly for a l l values implies disappearance will wander the I k. values This power condition. It i t the decreases behaves exponential to decreases of f o r the exponentially. to raised i n a^(k) a l r e a d y been from k" a > i function. 2 = - recognized; avoiding the spurious branch of w(z) In determining the behaviour of V u. and I t corresponds 2 to k thus = the - «, which necessity c r e a t e s no new problems. (k) as I m ( - t ) -» + °° with of 21. 4.5 Re(<t) held raised t o t h e power such The constant, a quantity ^a+3/2^ a < - A g b 3/2. e f o r e > a Thus, since, Im(t) + order - 9/4. case, because of the real (k) 3/2 D^(k) here order that i s thus requirement grows no quantities the magnitude part given by the factor provided implies Squires' 2 faster of o f t h e power. ImU) f o r large violates which a > satisfy - a < 5/2 < (1963) t o be able since found - The p r e s e n t i s thus - D^(k) 5/2. than requirement N^(k) converge, Thus, as i ti s necessary a has been ( 4 . 2 ) t o be v a l i d severely i n the 9/4. to perform work a potential not sufficiently potential (4.7) i tn e c e s s a r y validly he used i n f o r equation , i tmust transformation. however, the integral i n order Gushing and a < the requirement restricted; in 3/2 (4.5) case second - - 2 to consider -• + <». impose used ^ ( k )a V a > only i n this 3 < when i n this In to function Squires' or behaviour i n equation behaviour 2a The I i s a significant i ti s unnecessary t o use an ^-dependent the Sommerfeld-Vatson i s not inconsistent which i s singular potential J0(r / 2 ) 3 with h i s , f o r at the origin. small r , Thus, the 22 4 . 6 number of parameters required denumerable The i n f i n i t y v a l i d i t y of to (4.5) to investigate the of the variables complex three; i n the p o t e n t i a l g>u-, and thus having analytic k properties 2 and cos i s reduced from a. been of f (k secured, , cosG-) i t i s now as a used function 23 Chapter 5; 5.1 2 f ( k , cos ©•) The A n a l y t i c P r o p e r t i e s o f i n the Complex - cos Q- P l a n e . In t h i s c h a p t e r , the a n a l y t i c properties of the scattering a m p l i t u d e as a f u n c t i o n o f t h e c o s i n e o f t h e s c a t t e r i n g studied, and The r e s u l t s the f u n c t i o n (1963) and M i t r a earlier worksi The convergence, (1963) b u t d i f f e r An explicit in the p r e s e n t work. The r e s p e c t s from the effort, form i s g i v e n f o r t h e w e i g h t analytic properties complex-cos i n two or o t h e r w i s e , of the d i s p e r s i o n i n t e g r a l s i s determined i n the p r e s e n t (2) integral. g i v e n h e r e a r e c o n s i s t e n t w i t h t h e e a r l i e r work by Cushing (1) i s e x p r e s s e d as a d i s p e r s i o n angle are 0- p l a n e may of the s c a t t e r i n g be o b t a i n e d from o f t h e argument o f B o t t i n o f u n c t i o n i n the integrals amplitude i n the e q u a t i o n (4.2) by means ( 1 9 6 2 ) , pages 988 to 989. -l/2+i» f(k , cos(ft)) = § 2 dl • • £=-1/2-1f { 2 l + / 0 thence t h a t f ( k , cos ©•) I n g e n e r a l , P^j(z) the r e a l a r e no when a x i s from has two -1 to £ V " s i n (TT t) a ( k ) converges i s analytic branch l i n e s other s i n g u l a r i t i e s . a. ) i s an even i n t e g e r . 1 <»» C 0 8 + pole terms. ( T h e r e i t i s shown t h a t t h e i n t e g r a l 2 and l f o r a l l values of i n the domain o f i n the and a n o t h e r from z p l a n e , one —» The b r a n c h l i n e from 2 Thus, f ( k , cos ©•) to -1 -1, 4 # 2 ) cos ©• P ^ ( - c o s ©•). along There to I d i s a p p e a r s i s analytic for 24 cos ©• n o t on t h e p o s i t i v e real conditions(inequations (4.3)) the t h e same in integral i n (4.2), the present A cos O Cauchy's axis from 1 are sufficient result may to °°. Since Squires' f o r convergence be obtained of similarly case, dispersion relation w i l l 5.2 now be f o r derived, f ( k , c o s ©•) using t h e method as a f u n c t i o n o f of Mitra (1963). Using theorem, n (t+i€)-p.(t-ie) P t=-«° o-rr + l i 1 m / d ,- (5.1) m (Te 1 ( p X ) cp=0 Considering across by now the branch means only line, of EI 3.7(6), P,(z) = «• /du the f i r s t i . e . which (z + / z p ^("t + integral, the 16) - ^ ( t - discontinuity i € ) , may be obtained yields 2 - T cos u)*, (5.2) 0 valid be f o r a l l values varied axis of the parameters. continuously to a value just from above a value along just the path In this below shown expression, the negative belows l e t real z 25 Since this path + \Jz the branch line Jz of 2 - 1, t h e p h a s e o f • l~2 z avoids 5.3 - 1 integrand cos u passes passes through 2n through I 2?T radians, and t h a t of the radians. - co < t < ~ 1 , m P^(t + = - i<E) - P^(t - i€) / d u ( t + ft - 1 c o s u) 2 (1 - 1 exp(27rU)) 0 Substituting = e x - p i i n l ) - e x p ( - Ul) = - 2 i s i n {-nt) P ^ ( - t ) . into _ - - d u ( _ t + y 2_ t 1 C Q S u ) . (5.3) (5.1), r P ( v\ v^-z) J sin(-rU) ^ J (t) i t - z P / I n t e g r a l around infinite circle. K a t + ,~ ° , ,\ 4 ; t=l Mitra for (1963) i t t o be The The is has not studied valid, integral lower limit nonsingular coefficient converge the integrals of along at t = of a t the upper i sshown the branch of the integral P^(t) It the validity As i s 0(t"""^) limit = sr(l) i n appendix line shall no as representations converge. be trouble f o r the i n f i n i t e considered because t a i l ; P^(t) since t -• <=, t h e i n t e g r a l f i r s t . the w i l l provided as I I (5.4) must gives 1. of this that t - «. (5.5) 26> P^t) Thus, = max (0(t ), The integral neglect equation the satisfied (II.l) that for - Re Thu?, be A The neglected of i s valid of the IT review (l) < (II.1) - 0. i n f i n i t e circle contribution i n i s I i f condition (5.5) i s the derivation regardless the i n f i n i t e of circle of equation arg(t), may be and neglected 0. i n equation - 1 < relation (4.2)j a.(k (5.4) i s v a l i d i t can ) < 0. form, hence also These this be f o r a l l values used i n the terms w i l l restriction Regge however need of n o t be be made following. this equation 2 may 1*1 Re of as contribution U) < 1 < j u s t i f i e d . the l a t t e r provided Hence, f(k be arg(t). i n "undispersed" the into now - contribution the dispersion occurring l e f t in that the 1 < pole t e r m s the contribution shows thus t of (P^T)) f o r any as 1 converges f o r ( 5 . 1 ) may 0 Hence, 0(t-^ )) / the 5.4 , representation (4.2) to of ^ ^ ( ~ cos(©0) may be substituted yield cos + Regge pole (5.6) terms + t=l Regge pole terms, 2? 5.5 where -l/2+i« c(k,t) = - ^ d£ (21+1) a ^ ( k ) P ^ ( t ) , (5.7) -l/2-i» which i s t h e d e s i r e d s p e c t r a l r e p r e s e n t a t i o n i n The cos($). convergence o f t h e i n t e g r a l r e p r e s e n t i n g now be i n v e s t i g a t e d . o(k,t) will E I 3.6.1 (3) says P ^ ( t ) = F U + 1, -I j 1 ; and t h e behaviour of t h i s f u n c t i o n f o r l a r g e I may be i n v e s t i g a t e d by means of E I 3.2.2 ( 1 7 ) : F ( a + X, b - X; c; ^jr) T(l-b+X)r(c) r(i/2)Plc-b+x) = x (( TJT~l) x 2 a + b - 1 ( . ^)-c+l/2 u-z+yz i ; l z + ( r^ c-a-b-l/2 u+z+yz i ; l } h + z + exp(+ i-rr ( c - l / 2 ) ) ( z - / z ^ l ) * ) ( l + 0 ( X " ) ) X as IXl 1 S e t t i n g a = 1, b = 0, c = 1, and z = t ; -1/4 F U + 1 , -4, 1; i f i ) = P ^ ( t ) = jl i ± ^ i i ((t /t -l)^ 00 . 2 2 + ± Since constant, i ( x t 1 = 0 (imU)) as (1 + 0 U" )). Im(<t) -»'+ • 1 with ReU) held 28= + 1) = 0 a^(k) = 0 (£ ) from equation P^(t) = 0 U" / ) f o r ReU) (21 and 0 integrand i s condition a + 1 + 2 a + 3 2 ^' ' ) / condition a < - 7/4 the restrictions - becomes i s necessary as 2 I m ( ^ ) -» + (4.5), held ». A constant; the sufficient i s thus 7/2 < a < (This - 1, 9/4. necessary and imposed on f o r k =0 sufficient.) a earlier and f o r This by t=°°. further Otherwise, strengthens excluding the case 9/4. This dispersion scattering decreases is 2 f o r convergence or = - U), (lm(-0 2a a 5.6 amplitude as dominated instance, representation reflects the property of the i n the local the integral c o s ( O - ) -• » , by case, and t h a t the behaviour t h e r i g h t m o s t Regge the paper that by Mandelstam pole term a of the jj(k) i n Theoretical term amplitude ( s e e ,f o r Physics (1963), <x (k) R page as 413). cos In both cases, the amplitude behaves like ( c o s ©•) 29* Chapter 6. The Analytic 6.1 Properties of f ( k , cos &) i n the k plane. In f(k this , cos chapter, a double-dispersion (©•)). I t takes the relation i s derived for form (6.1) + As i n chapter Cushing. examined 5, Here and Regge the pole form however derived the explicit, terms. i s identical convergence closed forms of the f o r the to that of Mitra integrals weight and involved functions i s are derived. 2 In pole equation terms i s (5.6), a^(k), weight the which only function appears of i n the k other definition than of i n the o(k,t), the function. ^(k) h a s two s o u r c e s o f b r a n c h l i n e s i n t h e 2 2 complex - k plane: the f u n c t i o n ( k ) , which i s cut from 2 2 2 k = -u. to k -.co nd also along the entire p o s i t i v e real a = axis; the and the positive For (3.9), the which a integral real i n D^(k), which has a branch line along axis. discontinuity across the i s valid Re(k for 2 ) < - negative real 2 u. / 2 , may be k 2 used axis, to equation give 30 6.2 , rx— V. (/k a^(/k^ + i€) - a ^ U k ^ - i€) = g ^ 1 2 / ( — j — ) / „^-a * -a—t-1 2 ' v , 2 The ~ ' 5 a k ' 1 2 k +u. 2 2 2 p 2 \~| 1 - 2 - i€) ( k ) (—2—) J contribution of this + i€) - V ( / k * 2 ^ (6.2) ' exp(-4n-j(a+2)) . 5J(kl singularity t o a^(k) i s given by a , ( / k ^ i € ) - a,(/k' -i€) , . • _ v.2 2 0 a(k- ) 2 2 2rTi k' Since 2 Z (6.3) 2 = -u. |a^(Jk , 2 +i€) - a^(Jk' -i€) ( 2 < | a ^ ( Jk' + i € ) / 2 + |a^(/k^i6)| , 2 the integral reveals 0(a^(k)k i n (6.3) i s ) k as -• . 00 Equation (3.17) that 2 V (k) = 0(k" ~ ) 2a 6 as k - »; (3.17) </ and the behaviour of a ^(k) i s similarly bounded because D^(k) 2 approaches 2a 0(k~ 1 as k approaches - . 0 0 Thus, the integral 8 ~ ) f o r large k , and the i n f i n i t e - 1, t a i l of integrand (6.3) converges f o r -2a or This i s imposes - 8 < a > no new - 7/2. r e s t r i c t i o n on a. 2 The singularity of the integrand at k = - 2 u. i s 316.3 0( ( k + [ A ) ~ 2 2 2 a ~ ). Consequently, 4 this portion of the integral converges for - 2a or Again, a i s not Equation the phase from one side phase positive of real The a < - ) 1, 3/2. restricted. which V^(k) the - i s valid changes positive changes axis the 2 radians to same 2 ^C, 2 R e ( k ) >- 2nl by real by for the as k other. amount as k reveals ! passes Thus, 2 crosses the i t s axis. integral in D^(k) can / f f § \ 2^1 showing of 2 ( 2 V. k of 4 > further (3.16), that - that there be written in the form 4 V<*>. i s a branch line ( 6 along i t s path of - 4 ) integration, 2 the positive real k axis, with discontinuity D ^ / k ^ i e ) - D ^ ( / k - i € ) = 2 g i k V^ (k) = 2 i k N^(k). 1 2 2 2 Therefore, along ^(/]?+i€) Njk) ~ positive a ^ ( k ) ( 1 k axis, 2 1 ^ , ( ) exp(2fri k D^(k) D^(k) = real a (/k Ti€) N D-^k) = the ~ D^k) + 2 i k exp(2Tfi + 2 i k exp(2^"i £) l) Njk) *,) N^(k) } (6.5) 32 6.4 The contribution the convergence of of this branch which must line now to be a ^(k) i s thus given by established. 2 In i t order to i s necessary latter i s not e s t a b l i s h the to determine entirely behaviour that obvious 6f of a D^(k). since, i n ^(k) The the as k -• + °°, of the behaviour limit under consideration, 2 k goes D^(k). i t to infinity However, follows Hence, the for large decreasing as V^ (k) the integral was contour (3.17) of and integration used the work Lanz k + of < a > - new ) i n the as the integrand k. Hence, i n defining (1964), - (6.8) - i n the (6.7) also integrand has i s 2 of Since the (k) this same ' is behaviour 0((k )~ ~^); 2 a for 1 or 3; restrictions shown representation, 1 a large -4 - of ^(k) i s i d e n t i c a l to that 2 a 3 i s , i t approaches 0((k ) ). converges -a no (k function, for 2 As equation =1+0 behaviour 2 k 5 that a again from the that D^(k) and along the on a derivation condition for are of the necessary. cos convergence of (©•) the - spectral integral i n 33 6.5 (6.7) i s s u f f i c i e n t f o r disappearance of the c o n t r i b u t i o n of the infinite c i r c l e to the Cauchy i n t e g r a l f o r The a^(k). r e s u l t i s t h a t the cut s t r u c t u r e of a^(k) may be v a l i d l y r e p r e s e n t e d by a^(k) where y = girl f ^ by expression i s given Substituting 2 ( k .2) / + (6.2) J m£i and }f cos < • » = t and k (k ,2," by (6.6). + i n t o equation (5.6) and i n t e r c h a n g i n g of i n t e g r a t i o n over f(k , ^ll the o r d e r y e i l d e the r e s u l t 2 f U ,t) 2 t=l q = -(x ^ 00 t=l t-cos(fr) Q ^ q =0 2 k -q + ( 2 q , ) + Regge t p o l e terms (6.9) where -l/2+i» Q_(k ,t) 2 = - -L- <\i (2-t+l) P ( t ) ( a ( y k ' + i € ) - a ^ ( y k ^ - i € ' ) ) j 2 2 -1/2-1- 4 t t -l/2+i« J = " -h k fcT(a+W) ^ + l ^2a+3 a 2 x p ( (2, l) « P (t) l - e x p ( y ( a 2 ) ) + T 4 I-a i= a f + * -1/2-ico 4 7 T i ' ' ' P ( +2) p ^a+£+3xr^a+-t-+4x. " j z-su-l-1 azidL. _ _ , a 1 ; kk^+u_ u_ 2 2 2 } s"] 2 kfx" '" .2' 8 a K 2 (6.10) 34 6.6 and 1/2+i e (k ,t) = - 03 i 2 + d£ 47T (2-t + 1) P^(t)(a^(k+i€)-a^(k-i€)) l/2-i« 1 2 l/2-i» (6.11) By i t may are t h e same m e t h o d be shown used the integrals i n discussing i n equations formula (6.10) (6.3) and (6.11) both 0 The ( (2^+1) P ^ ( t ) a ^ ( k ) ) integral for a < v a l i d i t y - need 9/4 latter question terms special may pole arises cancel cases. considerations quantity This with increasing from the integral i s happens shown to equation from converge (5.7). the dispersion The integral, or not the left-hand of the dispersion may be integral obviated a t t h e end o f c h a p t e r Re(cos(&))« ». the eliminated. possibility decreases, i proven. of whether that mentioned the amplitude terms -» + has been following (6.9) i s thus by- o m i t t i n g I as i n the discussion f o r any subtractions The pole of this of equation Thus, to that a s was and that Thus, to cancel of the i n other b y means than of the The contribution terms increases, i fthe contribution to the cut that f o r one o f most from 5. cuts pole the poles 35 6 7 0 particular value former a for of cos sufficiently the large latter value w i l l of yet cos dominate the 36 7.1 Chapter In 7„ The this Origin chapter, of the "Extra" Branch the investigation analytic properties complex k fashion, suitable It out t h a t i n the n o n l o c a l case turns has f o r fixed angle f o r both a cut not only along of the scattering along the positive shall local be Bottino amplitude as (1962) axis; the scattering k of the of general and n o n l o c a l p o t e n t i a l imaginary Plane. a function f o l l o w e d i n a more the negative imaginary by Cut i n the k scattering. amplitude axis, that i s , that i t has but a also c u t on 2 both is sheets shown of the complex i n detail The radial - k plane. The reason f o r this difference below. Schroedinger equation says CO i|j"U) ( i ^ ± i i + + k ) 2 *(z) + dz« z, large V(z,z') the centrifugal disappears approaches V^(z,z') 2 i|((z ) = ! the faster term than disappears, \|t(z) and i f the f o r large z, then potential the form *"(z) + k 2 f(z) = 0. « Thus, f o r large A For 0. 0 2 For z« z, exp a potential the general ( ik z) + B solution exp (— i k z ) . satisfying "VXzj.z') = « ( approaches e x p ( - |j, z ) ) the form equation 37/ for large for z, the solution w i l l |lm ( k ) | Pour (for or, solutions k, ? thus always approach the above form < conveniences, q> ( X 7.2 of the radial t h e symbol z ) , cp (-X, f o r t h e sake k, X = z), f Schroedinger I + 1/2 ( X , k, equation will may be used be i n the defined following) z ) , f ( X , ke""'"', z ) ; of brevity, cp+, <p_, f , f_ ; + where <p(X, k, z) - z X + 1 / as 2 z - 0, (7.1) f The are (X kj, z) ~ p k z) as z -* <*>. f ' s a r e known a s t h e " J o s t solutions". In (1962), accordance defined g U l ; * , ^ ~ by Bottino shall in terms i t i s shown t h a t , fuctions f( X,-k)exp(ikz) ± 2 i to equation useful f(-A,-k) of the four f(X,k) + i k prove f(X,k) the Jost f(+Xj,+k) relation: f(±^,k)exp(-ikz) i s equivalent It with the 2 This exp(~i - f(-X,k) solutions f o r the local f(-X,-k) - an s z ^ . . (7.3) k (2.9) of Bottino to obtain a (1962). expression f o r the quantity f(X,-k) defined i n (7.l). In Bottino (1962) case, f(-X,k) f(X,-k) = 4 i X k. (7.3) 38. However, i tw i l l • identity does be seen;that, 7.3 i n thenonlocal case, this important not hold. Now l e t (7.4) Since f+ ~ become J o s t exist. since e x p (+ i k functions independent potential solutions of the local this among A,B,C, a n d D 9 the solutions and any three equation; pairwise but with a linearly nonlocal may n o t b e t h e c a s e . o f two f u n c t i o n s , say g ( z ) and h ( z ) , i s by 1 h ( g l e t g ( z ) and z ) )= g ( z ) Mai - h ( z ) M*l h ( z ) be two s o l u t i o n s o f t h e . equation; CO 0 00 h"(z) + 0 Multiplying and the f i r s t sub t r a c t i n g equation by h ( z ) and t h e second by g ( z ) yields CO _d dz f A,B,C, a n d D a r e c o n s t a n t s , , the coefficients relationship exists v( (z), Now z -» » , t h e q u a n t i t i e s z -* °°, p r o v i d e d case, The ¥ro n s k ia n defined as as I n the local a linear z) dz' 0 R(z,z«) [g(z) h(z')- h(z) g(z')]. 39 In general, t h e r e f o r e , t h eW r o n s k i a n R(z,z') as i nthe local appears of the local i s a function then t h e Wronskian difference equations i sa constant. between (7.4), - W(f + f Af+ + Bf-) a BW(f ,f_), + DV(f+,fJ, ? W(f^<p+) ta - A W ( f + ,£«.), W(f„ cp_) = - C W ( f , f J . 5 8 8 ~ Thus equations. W(f+ cp„) B *P+ + V(£+ f * (7.5) <p Taking (7.1) - - vif ,f_) r + the limit z -» °» and (7.2) reveals fU,±k) + + w(f ,f_) r + i n equations - • _/ (7.5) and using equations that = l 2 i k i m W(f±,q>+) + z — ~ W(f ,f_) + and (7.6) f(-*,±k) = 2 i k The denominators equation (7,l)s p t h e properties of the solutions and nonlocal Schroedinger + > I f , however l case, W(f cp+) o f z. = R(z) 6(z-z ), an important Prom 7.4 of these l i m + W(f+,y.) z — quantities W(f+,f_) may b e c o m p u t e d with t h e a i do f 40, 7.5 lim ¥(f+,f_) -»eo = ¥(exp(-ikz), exp(ikz)) = 2ik. (7.7) z Thus. f(X,+k) = + l i m Y(f+,cp ) + 2-»eo ~~ y and f(-X,+k) This One shows sees with that also + p that functions i nt h e l o c a l the definition Substituting 7 = + l i m ¥(f cp_) z-*°° the Jost exist case, f provided these (2.1) of Bottino equations (7.8) and expressions cp exist. are identical (1962). (7.5) into the expression ¥(<p+,cp_) yields V(<p+,<p_) = W(f ,f.) thus, V(f_,<p_) V(f ,f_) ¥(f ,f_) + + and ¥(f+,<p+) using equations f(X,k) f(-\,-k) _ V(f_ <p+) V(f+,<p-) ¥(f+pf~) V(f+,f„) f + (7.6), - f(-A,k) f(X.-k) (2 ik ) 2 = using also In lim Z-.CO W(<p+.q>-) ¥(f+,f-J = l i m z-» 0 -kr- l i m ¥(<p ,q>_), + (7.9) (7,7). the local case, ¥(cp „cp_) + i s constant, ¥(<p ;cp„) = l i m V(q> ,tp_) = ¥ ( z + + Z"*0 A + 1 / 2 , Z ~ A + and then 1 / 2 ) = - 2\ (7.10) 41 so that f(X,k) which the f(-X,-k) i s equation f(-X,k) - (2,5) f(X,-k) = 4 of Bottino (1962). f(-X,k) f(X,-k) i X k, (7.11) Thus, i n the local case, quantity f(X,k) is 7.6 analytic functions over cp Using f(-X,-k) - + the entire and f+ X case S(X,k) f o r the > S i n exactly = ||x^) e x assuming the Sommerfeld-Vatson validity of (1962), this i n a particular f(E,cos(e)) = ~ x r when the Yukawa case , / 9 which as may be i n the local derived case,, 1/2)), transformation X P. dX manner UTT (X - P nonlocal J matrix, t h e same step i n the local , p l a n e s , evenk are not. the usual relation i n the n o n l o c a l k and may case be performed- i s shown i n earlier by chapters) (The Bottino yields (cos(G)) ' exp(-i/r(X+l/2)) ( S ( X , k ) - l ) cosirX) —i» + + Now only that c o m p o n e n t of the i cos pole (^X)] integrand which terms + is pole an even terms. (7.12) function of 42 7 , 7 contributes to X According X; to EI 3.3.1 paragon odd, and only P X - l / 2 cos In taking is and 2 t (TT c o f(-X,-k) - ^ s s a n e v e n and function of X i s the ^ s X ) of the odd p a r t , -f(X,k) f(-A.-k) o f o r i t sevenness, H x ^ T k ) - s i n (TTX) contributes. f(X.k) c Therefore, t h e odd p a r t (- unaltered, ( i s famous of oddity. X is cos(G-)). ( l ) , cos(rrX) further, very f ( E , / f(X,-k) f ( - X , k ) + i c o s (TTX) ) i cos (^X) becomes, u s i n g f(X,-k) out, sin(7rX) equation (7.9), drops lim z-»°° V ( ( p q > . ) = _ + t f(X,-k) ( e 7 > 1 3 ) 4 i k f ( - X , - k ) f ( X , - k ) * Thus, f ( E , cos(s)) X j 1 = - ^ P dx x x / (cos(^)) ;twx) . (sin (7rX) pole terms. -i=° lim + z ~°° ¥ ( P+ T-) ) ( ? + (7.14) 4ik f(-A,-k)f(X,-k) (The pole Bottino In lim Z -»CD terms are inexplicably missing from equation (6.16) of (1962)). the local V(cp ,cp_) + = case, - 2 X , equation and thus (7.10) that may the only be used source t o show of that singularities 43 k plane in the in the denominator. In the nonlocal in the numerator contribute scattering equations of f ( + X, -k) cp i n the X f and portion be equations used of the Each (7.8), k i n t h e same shall case, however, additional functions 9 the singularities i t may be seen that functions to the the domain i s the i n t e r s e c t i o n of the domains and Integral and may the i n t e g r a l i n (7.7) are the J o s t amplitude. Prom + from 7.8 shall manner and f_ analytici and planes. t o show X of of now be s e t up as i n Appendix analyticity k I f o r the s o l u t i o n s of the Bottino of the solutions i n a cp effort, restric ted planes. o f t h e i n t e g r a l equations shall be w r i t t e n i n the form CO 0 This one to a i s may identical apply to the present the effect that, given region, case equation, an bound to also f o r the function i t i s sufficient original upper t o e q u a t i o n ( l I . l) i n B o t t i n o i n form i.e« g , Q Thus, their subsequent argument g ( X , k, z) t o be analytic i n f o r the free t o be (1962), analytic eigeofunctions i n the region, of the and f o r exist f o r the i n t e g r a l CO (7.16) 0 where \g Q ( A , k, k, z) f o r a l l X, k, z i n the region. 44 The for integral the nonlocal equation f o r ^ ( r ) derived = j ^ k z ) - by McMillan( 1963) case i s 00 t^(z) 7.9 09 i k ^ d z » z» ^dz' 2 0 z' j ^ ( k z " 2 < ) h ^ ( l ) (kz ^) 1 0 V ^ ( z % z « ) ,^(z») where Thus, z"^ = min (z,z")» z (z,z "). =s m a x f o rsmall z, 00. f^(z) y ^(kzMl-ik » CO dz" z " ^ 2 0 z' h^^tkz") V ^ z ^ z ' ) 2 0 ilr^U') ) r r / 2ft£+3/2) 1 and M since j^(z) ~ I z ) ~ z " IZ+T ^ 2 rtftW (7.17) 1 z ^ a l / 2 S ~* ° * Z °° ^ i °° / « i z " k z " 2 0 ^ ~dz- z ' h / ^ k z " ) 2 0 V (z",z') ( | (z") ) as ( z- 0. I The coefficient ^ ( z ) a of equation normalization For the i s not a function (2.20) constant, case, of may b e i d e n t i f i e d which formula with i s immaterial of deriving z. Thus, <p(X, k , z) f o rp r e s e n t the analytic properties (7.16) solution up t o purposes. of cp i n becomes 00 ^ d z ' ^ d z " 0 z t h e purpose nonlocal CO /k/ of 0 j z " 2 z« 2 ^ ( k z 1 ^ ) h^^Uz'^) x gMlUf,' W V^(z",z«) l< z) k, (7.18) 45 where M(-t, k, j,(kz) and be has s i n g u l a r i t i e s space component contribution may that the First, V separately, <p for Re(l) For page > - and equally the convergence and w i l l conditions z V nl = ^ ( z "~ 1/2, or the infinite k w i l l yield ^ 0, 2 ) ) , this i n need Any a s ,a V(z")6(z"-z )/z" k and f o r (1962), Re(X) > of the inner becomes of the integral of the inner integral, from converges Morse (1953), 622, h ^ z ) Therefore, ~ \ exp(i (z - J (i+l))) z -» °>. as (7.19) the integrand i s 0(z" 2 z"" V(z",z') = 0 0 1 exp(i (exp(- k z") V ( z [i z " ) ) ( z "exp(z"(i k , , as - ,z )) as 1 z " -» ji))). ro , z" - 0. integral; 0. z" - , 1 0. Re(X) > tail behaved the result of Bottino portion 4), page manifest itself the integrand as (EII of the integrations. f o r convergence (z",z') ^ well 0 0 of the integral s o l u t i o n i s a n a l y t i c f o ra l l 2 kz = p o t e n t i a l of t h e form and stipulating (z",z') t o be of the potential, which 0(z" For kz = 0 f o r f o r t h e end p o i n t s to find (7.17) using hence, to the nonlocal be t r e a t e d i.e. only only be a s s u m e d variables; investigated local For jj^(kz)[. z) > the potential w i l l both 7.10 «. the integrandis 7.11 Therefore, Thus, the integral the inner converges integral i s found f o r Re(i k - u.) < t o converge 0, or Ira(k) > -u.. f o r Re(X)> 0, Im ( k ) > - u.. For treatment MU, Thus, provided of the outer i n t e g r a l ,l e t k , z ' ) = /d^Ckz')). the inner integral (7.20) converges, t h e integrand o fthe outer i s 0 ( z Precisely V(x",z') The of the |j ^ ( k z ' ) | = jer(z'~^/ ), 2 infinite Vjz",z>)) as z'- the integral t a i l 0. integral; converges of the outer f o r f o r Re(A) > integral provides a 0. strengthening requirements. |j^(kz)J ~ ~ cos(z (exp(i(kz - J = - ^ U+l))/ U+l))) + as z - ». exp(-i(kz - | U+l))))| t h e integrand i s 0(z' For 2 as i n t h e d i s c u s s i o n o f t h e i n n e r MU,k,z) = and | 2 V(z",z') 0(z' Therefore, 2 max ( |exp(i k z ' ) | , /exp(- = 0 (exp(- \iz')) max( )exp(z'(- i tconverges as 1 z ' -• , t h i s 0 0 [x + i k ) ) / , f o r - fi, < i kz )()• V(z",z*)). quantity i s f e x p ( z ( - u. - i k ) ) | ) ) . Im(k)< j i . ! , 47 Thus, - [i, < cp the function Im(k) < 7,12 i sanalytic provided for t h epotential Re(X) > i s no more 0, k ^ 0, a n d singular than -5/2 z ' as e i t h e r like exp(- with was space \iz) variable as either variable theresult of Bottino found t o be a n a l y t i c r e s t r i c t i o n onX). &{z~ ) as The z -• reason This for the loss involves only example, equation however, there An for °°. thelocal the entire k This case, plane to a local of analyticity for i t decays contrasts i nw h i c h (with cp t h e same potential which i s (1,4) of Bottino i s an extra t o ensure i n t e g r a l equation 0 0 cp(z) to (1962)). integral, to i n the local may b e z, found (See, f o r I n thenonlocal thepotential integral ». convergence for i sthat, the solution an i n t e g r a t i o n from a l l t h eway from be imposed and provided 0, which must 0, approaches result applies an i n t e g r a l equation runs (1962) over case, which approaches Hence, extra of the i n f i n i t e theJost case, i t s e l f , restrictions t a i l . solution i s CO f(X,k,z) = exp(-i kz) - d ^ (exp(ik?)exp(-ikz) z - exp(-ik?) exp(ikz)) ca d?'( x X 2 6(?-?')+? V(?,g') , 2 5 "0 CO = f(X, k, ? • ) ) exp(-ikz) - J Z k f CO d?' f(X,k,5') d? ( e x p ( i k ? ) exp(-ikz) 0 - x (*%±^ % exp(-ik?) 6(5-?') + V 2 exp (ikz)) V(5,?')). 48 7.13 This may be appendix I identical derived i n exactly of Bottino i n form (1962) t h e same m a n n e r to derive to equation (7.15) as that h i s equation used ( l . 6). i n I t i s with CO L = - 2~fk d ?( z e x p ( - i k z ) - expH-ktj) P (ik£) e x exp(ikz)) 2 I f ( A , k , z) For to be a n a l y t i c , i t i s s u f f i c i e n t f o r t h e r e to exist an upper bound f o r the i n t e g r a l OS J~ 0 CO d ? ' ^ d? / ( e x p ( i k S ) e x p ( - i k z ) - e x p ( - i k O e x p ( i k z ) ) ? z k where ; l] M(X, k , ? ' ) The since * ] c e n t r i f u g a l c o n t r i b u t i o n must be i t i s identical bounded potential, term, may f o r a l l which also v(5 '))| (7.21) f ? considered to the corresponding A i n appendix and w i l l be 2 ( = |exp(-i k ?•)/. case, Bottino's discussion is + ? ' «(M') / , 2 be lm(K) < 0. manifested as included i n this contribution I I shows Any a separately that local similar result provided i n the but, local i t s contribution component of the delta-function i t i s no more _2 singular The except at z than remaining % = », at the origin. i n t e g r a l of equation ?' = °°, a n d ?' = 0, (7.2l) assuming i s not singular the potential i s well 49 behaved a t a l l other Considering 0 (max points f i r s t (|exp(ik?)/, 7.14 i n the the inner integration. integral, |exp(-ik?)() the integrand i s V(?,?')) as ? - <». If V(?,?«) the = 0 (exp(- ?)) as $ - », quantity i s 0(max This (/exp((ik quantity decays - [i) ? )| , |exp((-ik exponentially, - and hence |i) I )| )). the integral converges, for - If the inner 0(?' If then the bottom decays Thus, (K) < = as o(?'" 5 / 2 ) end o f t h e outer supposition, exp(-ikS') nicely f o r considering \i. converges, V(?,?')) t h e same 0( Im integral V(?,?') Under which 2 [i, < ?' - as then integrand i s 0. ?' - 0, integral always the outer V(?,?')) the outer as converges. integrahdis ?' - 0, I m ( k ) < |x. only the nonlocal contribution, f ( A , k, z) 50; i s a n a l y t i c f o r a l l A, and f o r 7.15 - \i < Im(k) < u. t h a t the c e n t r i f u g a l c o n t r i b u t i o n and any other no more s i n g u l a r than i t may on k to The fuither. - |x < Im(k) < However, i n order local contribution converge, onemust strengthen p o t e n t i a l may restriction 0. r e g i o n s of a n a l y t i c i t y of the s o l u t i o n s may Let the z = p exp(io), p a and r e a l , and be a n a l y t i c a l l y continued be extended assume t h a t the i n t o the complex z p l a n e . Then the r a d i a l Schroedinger equation may be w r i t t e n CO p 0 V(p e x p ( i o ) , z ' ) #(z') = 0. (7.22) This i s j u s t the o r i g i n a l e q u a t i o n w i t h a complex p o t e n t i a l and complex k. in I f the p o t e n t i a l order V(z,?) t h a t the general tji' = a(o) is 0 ( e x p ( - u. z ) ) s o l u t i o n may exp(-ikp) As b e f o r e , + P(o) exp(ikp), be d e f i n e d . has the behaviour The new \a\ <7T/2 i n order that d i s a p p e a r f o r l a r g e z. four s o l u t i o n s may cp'U, z; approach the form i t i s n e c e s s a r y to impose the r e s t r i c t i o n the l a s t term may for large k', p) Jost s o l u t i o n i s defined A+l/2 by The new cp solution 51- exp(-i k' p) f'(X,k'p) where 7.16 k' = k e x p ( i o ) . The o r i g i n a l J o s t s o l u t i o n f ( X , k, z ) , defined f o r z r e a l , may- be a n a l y t i c a l l y continued i n t o the complex-z plane provided the p o t e n t i a l i t s e l f may be so continued. I t s behaviour w i l l then become f(X, Since k, z) exp(-ikz) + y^ (o) a ( z f ( X , k, z) exp(-ikz) P(o) exp(ikz). i s a n a l y t i c i n the complex-z plane, oc(c) exp(-2 i k z ) + P(o) must be a n a l y t i c f o r large p. for large /z| and P(o) p, i t must therefore be a constant. B(o) = 0 , for p = 0 , a ( o ) exp(-ikz). f(X, k, z) Since Im(k) < 0 , the f i r s t term disappears i t s e l f must be a n a l y t i c f o r large p. Since i t i s not a f u n c t i o n of Since For exp(ikz) i s i t s e l f analytic, and therefore constant. a(o) must also be a n a l y t i c , .'. f ( X , k, z) — ~ ~ ^ e x p ( - i k z ) z for a l l z} .'. f ( X , k, z) = f ' ( X , k», p) . The s i t u a t i o n f o r the cp s o l u t i o n s i s much simpler. conditions f o r cp and cp 1 (7.23) The boundary are i d e n t i c a l up to a phase f a c t o r , and hence so are the functions themselves: tp'(X, k, p) = exp(-i a(X+l/2)) <p(X, k, z ) . (7.24) 52 7.17 Substituting (7.8), and using from the b i l i n e a r i t y f'U,k«) (where the f'(X,k') old Jost The t i c i t y order to the the exp ( z ( - [i + as p-* . i s the Jost convergence as real a such o* of the be may fi < take on when - there and Im(k) remain any Re(k) [i < + decay (cos(o) f are Im(k) < i n general tails of analy- i s that, the i n integrals exponentially i s not + of i p sin(o))(- \x, + i ( k ) i s (j, + i k ) ) I m ( k ) 4 - s i n (o) analytic only Re(k)). i f there exists and tan(cr) < f i n i t e , 0. (- Thus, analysis of complex exp(p(cos(c) Re(k) ^ infinite 7.12). one i s . z coefficient \a\ <'*T/2, that satisfied Thus, cp making s i n (a)) u. c o s ( o ) + solutions - tan(a) - must equation (7.25) i f t h e new the z-»°°, b u t part into fU,k), i n the previous of quantity which = the Wronskian, s o l u t i o n s by Re((cos(o) + i Thus, of the (7.22) function f o r equation introduced various i k ) ) The 00 and exp(-c(X+l/2)) = change obtain involved, (7.2l) function i s analytic only of equations real I f u. (7.26) value. Re(k) = Thus, 0, (7<$26) c a n i t reduces always to \i. branch lines f o r each solution along 53 both imaginary axes Precisely as i n the case and centrifugal for the Jost may start Thus the extra from i \i along closer i t i s seen branch + i °°. give rise (1962), the to an a d d i t i o n a l the positive to the that to discussed Bottino contributions solution even + 7.18 imaginary k local branch axis line which origin. two circumstances cut of the scattering conspire to amplitude engender i n the complex-k planes I The inconstancy of a Vronsk i a n o f two solutions of the nonlocal equation. If (7.10) is the Vronsk could analytic the case II be used even numerator could The when the cp an were constant, expression f o r solutions which i s necessary which are not. expression i n equation the negative imaginary k integral equation ¥(cp+,cp_) themselves a n y s i n g u l a r i t i e s . ( The d e n o m i n a t o r cut along extra to obtain solutions of the parenthetic n o t have has a i a n o f two even Thus, (7.14) i n the local axis.) t o accommodate the nonlocal potential. In (7.21) - ^< for order may that converge, I m ( k ) <[j, large z faster the analytic this restriction axes i n both neither than i n general cp based engenders and t h e f i n formulae (7.18) to introduce the exp(ikz) e x p ( - \iz) continuation the integral i ti s necessary so t h a t when k the potential nor damps on branch restriction exp(-ikz) may the integrand. equation (7.12) cuts solutions. explode Even i s employed, along both Thus, and both imaginary the 54 7.19 numerator and (7.14) have the branch denominator lines along of the both parenthetic imaginary effort axes. i n equation 55 8.1 Chapter In this properties by thesis, advantage class here y i e l d a relatively McMillan for by analytic (1963) the case proving amplitudes f i r s t also been the case the the involve f o r t h e more using by Regge performed class general case relation involving (1963) integrals here are essentially local potential case by, f o r example, total scattering point the origin also amplitude given shown goes explicitly beyond h i s scattering angular This by momentum extension (1963) f o ra has has more investigated obtained. amplitude found f o r t h e same a s t h o s e Bottino i n the present of which, They amplitude. but neither of the scattering amplitude i nthe decay. and Cushing been interactions work (1962)). the potentials been complex considered branch have that of these f o rthe total potentials, of the spectral properties f o rstrong but the present of separable analytic class an e x p o n e n t i a l (seeBottino by Mitra The as has f o rthe partial-wave dispersion a technique form of the partial-wave here, f o r scattering potentials. i n closed reasonable form of the analytic p o t e n t i a l s lies i n t h e f a c t and the p a r t i c u l a r properties a double convergence The simple considered exploited restricted the the potentials i s made amplitude nonlocal can be w r i t t e n i s physically that and t o t a l separable (1963), sense investigation of separable, amplitude b y McMillan The a detailed of studying partial-wave studied Summary. of the partial-wave a particular shown 8, a s i s shown case (1962), found f o r except contains i n detail an i n the that extra last 56. chapter, lies Schroedinger the fact potential it that i n the fact equation converges sheets has a Wronsk i a n f o r ' t h e n o n l o c a l i s not i n general the extra integral i s a general amplitude that 8.2 only property a branch which a constant, accommodates for a restricted of nonlocal line of the complex-energy along class plane. together with the nonlocal of solutions. potentials that the negative radial real the axis Thus, scattering of both 57 Appendix I . Convergence lol of the Integrals defining D^(k) and Vk) In this appendix, J i t i s shown t h a t the integrals CO V^k) = dr r j ^ k r ) V ^ r ) 2 ( I . l ) 0 and r D j k ) = / Q converge - ^ U ) o dq q^ -\ j k -q (1.2) if V j r ) = * ( r ." 5 /) 2 as and relevant properties according to Morse (1953) M The The In worst this Thus, ~ 2T(W/2) ~ — case a (-t+1)) r), j S Z as V ^ ( r ) = «f(r°"^) 0 functions " ° (kr) ( I z -• <=°. end i s when approaches a finite a t the lower as r -» 0 , ' 5 ) (1.6) i n ( I . l ) i s f i r s t of the lower the requirement f o r convergence 2 (l 4) 1573 a r e f > " f o r convergence (1.3) of the spherical Bessel of the integral ( f o rs m a l l r or ( cos (z - convergence case page 0 r -» » , as The h r - considered. t = 0; constant. limit i s 58 V.(p) = *(r~ ) as 1 convergence at the upper l i m i t r V ( r ) =s «(r~^) (1.7) r -* 0, as 3 j, . ( k r ) =_ 0 r \(- r ~ )i Since 1.2 r -» », the requirement f o r of ( I . l ) i s as r •* <», as r V or Vjr) = #(r~ ) 2 Convergence «. (1.8) of the i n t e g r a l i n (1.2) w i l l now be s t u d i e d . I t w i l l be found that somewhat stronger r e s t r i c t i o n s on the p o t e n t i a l w i l l be necessary to ensure Prom (1,5) and (1.6), J,U) for I > 0 , where convergence. C ^ ^ 0 < z < oo < i s an a r b i t r a r y , f i n i t e constant and 0 < b < 1. Thus, CO lv k ) l 2 dr r If 2 C (kr) 0 V.(r) 'l b (1.9) k and D (k)| < 1 t + 0 f^j£ (i) 2b 0 ™ 2b Thus, the i n t e g r a n d i n (1.2) i s " bounded by i n t e g r a l i n (1,9) converges; V.(r) , p r o v i d e d the that i s , p r o v i d e d = *(r ~ ) 3 as r - 0 and as r -» °°. b q (i.io) 59 1.3 For where convergence t h e i n t e g r a n d may a t the lower limit be t h e most singular, of the integral i n (l„2), unfavourable value of 2 k i s zero, and t h i s value i s assumed i n this paragraph. The 2 b integrand i s then b < Thus, 1/2. that 0 < b < V 0(q~ With and of ( r )= f"«(r° ) ) - / \ (1.9), 1/2 < as q (1.8), i i as 1 t a i l (I.10) / 2 leads b thus ) Thus, to r - studied. bounded i n order (1.9) converge leads as by that q f o r some as r-*». V ^ ( r ) must 2 imposed i n (1.7), satisfy as r - 0 as r -» » , (1.13) and out by M c M i l l a n (1963). case, by a similar argument, t o be ™2 VAT) b 0 a l l the restrictions and ( I . l l ) , the local t a i l (1.12) ) 5 In , to 1 given «=• 2 b the 2 V, ( r ) = < r ( r ~ / ) as such r - co. t h e i n t e g r a n d becomes that / (1.10) o f ( 1 . 2 ) i s now i ti s s u f f i c i e n t 5 b, f o r some r -» 0 1/2. b <L 1 . converge f o r (I.11) ) f o r to satisfy (I.10), on b > _ = jer(r~" order 9 « V, ( r ) = ^ ( r " In and t h e r e f o r e converges as / / converges (1.2) converge, that 3 0, bounds of the i n f i n i t e the integral sue h these 0 ( r " q i n (1.9) must 1/2. Convergence again as the integral * Using ) = «(r~ ) and as r -» 0 as r -* <=. the requirement turns 60 I I c l Appendix I I . E I 3 o 9 , 2 and (19) E I ( t ) = max P when The A s y m p t o t i c Behaviour o f P ^ ( t ) as 3 . 9 . 2 (0(t ) ^iiT " )) 1 t 0 yield (20) l J t | -» o= It! as 1 - «. (lid) 1 / 2 . I n t h i s appendix, i t i s shown t h a t t h i s l a t t e r B e l t ) ?i c o n d i t i o n may be r e l a x e d , EI 3 . 3 . 1 (8) tar^ul = and El 3 « 9 < > 2 for all and 1 - 0(t^" ) as 1 Itl - « (II.3) ( 9 . 8 ) for Re{l)~ - l / ; Im(-t) / 0 „ For P where t Combining the two c o n f i r m s e q u a t i o n -to (II.2) ( Q ( t ) - Q_^_ (t)) » yields (21) Q^t) says I = - 1 / 2 , E I 3 . 1 4 ( 5 ) may be useds the case ~ l / 2 K ^ c o s ^ = n (sin(©-/2)) K i s t h e complete e l l i p t i c Substituting into P - l / 2 EII13,8 (l 2z) + (1) =|y (II.4) i n t e g r a l of the f i r s t k i n d . yields J i + z ' s i n 2 , ' 0 specializing to the case z r e a l and p o s i t i v e . ( I I » 5 ) 6 II.2 1 asin(z P , / o = , l / 2 ) ( l + 2 z ) = !• + . _ /—i / 2 dcp(~sin(cp) + — vising) - 0(z Although that the binomial last - 3 / 2 )). integrand expansion used ^"(^ (II.6) i s singular at doesn't converge cp = a s i n ( z ~ 1 there, singularity the ) "1/2 is removable point. because Thus, b o t h i n t e g r a l s asin equation the integrand (ll l) 0 (x) i s proven x approaches a r e bounded and, as x -* 0 , for this case also. 2 using ' at this because 62. I I I . l Bibliography Bateman M a n u s c r i p t P r o j e c t ( E r d e l y i e t a l ) , H i g h e r Functions, McGraw-Hill, 1953. Bateman Blatt, Manuscript Project, Tables McGraw-Hill, 1954. J.M., Bottino, A., Churchill, Cushing, Dicke, Lanz, Lomon, A.M. J.T., Nuovo T. Regge, Cimento, Nuovo and a n d M. McMillan, M., McMillan, M. , Prosperi, McMillan, Ph.D. Thesis, Nuovo Annals McGill Review, 23, 439, 1963. 130. 2 ^ 7 . 1963. Omnes, R., a n d M. F r o i s s a r t , M a n d e l s t a m Benjamin, 1963. Salam, A., Director, Agency, a n d H. F e s h b a c h , M e t h o d s McGraw-Hill, 1953. F. , A n n a l s 3_3, 3 4 8 1 , 1 9 6 4 . 29., 4 1 5 3 , 1 9 6 3 . P.M., Tabakin, Mechanics, U n i v e r s i t y , 1961. Morse, of Theoretical theory Theoretical Physics, Vienna, 1963. Cimento, 23, 954, 1962. t o Quantum of Physics, A.N. , P h y s i c a l E.J. , Nuovo Physics, Applications, Cimento, Mitra, Squires, Nuclear 27, 2364, 1963. Nuovo CimentoJ Transforms, Cimento, and I.R. W i t t k e , Introduction Addison-Wesley, 1961. a n d G.M, E. Longoni, Theoretical R.V., Complex V a r i a b l e s McGraw-Hill, 1960. R.H., L, and V . J . Veisskopf, W i l e y , 1952.* of Integral Transcendental 2_5, 2 4 2 , . of Physics, Physics, and Regge International 1962 , 30_, 5 1 , 1 9 6 4 . Poles, Atomic Energy
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Analytic properties of the scattering amplitude for interaction via nonlocal potentials Davis, Ronald Stuart 1965
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Title | Analytic properties of the scattering amplitude for interaction via nonlocal potentials |
Creator |
Davis, Ronald Stuart |
Publisher | University of British Columbia |
Date Issued | 1965 |
Description | The derivation of a partial-wave amplitude for scattering by a separable, nonlocal potential given by MCMillan in Nuovo Cimento 29, 4153 (1963) is reviewed. Using his results, an exact expression for the amplitude is derived for a potential of the form -g V(r)V(r¹), where V(r) = ra e-μr , and its analytic properties are studied. The asymptotic behaviour of the amplitude as |ℓ| → ∞ (where ℓ is the usual angular-momentum parameter) is derived, and is shown to permit a Sommerfeld-Watson transformation to be performed on the series expression for the total scattering amplitude in terms of the partial-wave amplitudes. By means of this transformation, a double-dispersion relation is derived for the total amplitude in both the complex-energy and complex-cos θ planes. Explicit forms are derived for the weight functions, and the convergence of the integrals involved is studied. In addition to the usual branch cuts along the positive, real energy and cos θ axes, an extra cut along the negative , real energy axis is found which is not present for the local case. Its origin is traced to the fact that the Wronskian of two solutions of the nonlocal radical Schroedinger equation is not necessarily a constant, as it is in the purely local case; and to the conditions necessary to ensure convergence of the extra integral in the nonlocal Schroedinger equation. |
Subject |
Potential theory (Mathematics) Scattering (Physics) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085302 |
URI | http://hdl.handle.net/2429/38011 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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