UBC Theses and Dissertations

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UBC Theses and Dissertations

Analytic properties of the scattering amplitude for interaction via nonlocal potentials Davis, Ronald Stuart 1965

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THE ANALYTIC PROPERTIES OF THE SCATTERING AMPLITUDE FOR INTERACTION V I A NONLOCAL POTENTIALS by RONALD STUART DAVIS B . S c , U n i v e r s i t y o f A l b e r t a , 1963 A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e d e p a r t m e n t o f PHYSICS a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF B R I T I S H COLUMBIA A p r i l 196^ I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y , I f u r t h e r a g r e e t h a t p e r -m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t . c o p y i n g o r p u b l i -c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n * D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r 8, C a n a d a ABSTRACT The derivation of a partial-wave amplitude for scattering by a separable, nonlocal potential given by M°Millan i n Nuovo Cimento 29, 4153 (1963) i s reviewed. Using his re s u l t s , an exact expression for the amplitude i s derived for a potential of the form -g V ( r ) V ( r f ) , where V(r) = r a e""'Ar, and i t s analytic properties are studied. The asymptotic behaviour of the amplitude as \t\ -* » (where t i s the usual angular-momentum parameter) i s derived, and i s shown to permit a Sommerfeld-Watson transformation to be performed on the series expression for the t o t a l scattering amplitude i n terms of the partial-wave amplitudes. By means of this transformation, a double-dispersion r e l a t i o n i s derived for the t o t a l amplitude i n both the complex-energy and complex-cos ©• planes. E x p l i c i t forms are derived for the weight functions, and the convergence of the integrals involved i s studied. In addition to the usual branch cuts along the positive, real energy and cos ©• axes, an extra cut along the negative , real energy axis i s found which i s not present for the l o c a l case. Its o r i g i n i s traced to the fact that the Wronskiah' of two solutions of the nonlocal radi c a l Schroedinger equation i s not necessarily a constant, as i t i s i n the purely l o c a l case; and to the conditions necessary to ensure convergence of the extra in t e g r a l i n the nonlocal Schroedinger equation. i v ACKNOWLEDGEMENTS I am indebted to Dr. J , M. M°Millan f o r suggesting the problem and f o r generous a s s i s t a n c e with i t . T h i s work was supported by the N a t i o n a l Research C o u n c i l of Canada. i i i TABLE OF CONTENTS I I n t r o d u c t i o n 1.1 II A closed form f o r the pa r t i a l - w a v e s c a t t e r i n g amplitude 2.1 I I I The a n a l y t i c p r o p e r t i e s of v ^ ( k ) and v ^ ( k ) f o r a 3.1 p a r t i c u l a r p o t e n t i a l IV The asymptotic behaviour of a^(k) as \l\ -* 0 0 4.1 V The a n a l y t i c p r o p e r t i e s of f ( k , cos ©•) i n the 5.1 complex-cos ©• plane VI The a n a l y t i c p r o p e r t i e s of f ( k ,cos ©•) i n the 6,1 2 complex-k plane VII The o r i g i n of the " e x t r a " cut i n the k plane 7,1 V I I I Summary 8.1 Appendices s I Convergence of the i n t e g r a l s d e f i n i n g D^(k) and 1.1 I I The asymptotic behaviour of P^(t) as |t| -» » I I . 1 B i b l i o g r a p h y . 111,1 1 1.1 C h a p t e r 1, I n t r o d u c t i o n I n t h i s t h e s i s , a d e t a i l e d i n v e s t i g a t i o n i s made o f t h e a n a l y t i c p r o p e r t i e s o f t h e p a r t i a l - w a v e and t o t a l a m p l i t u d e f o r s c a t t e r i n g by a p a r t i c u l a r c l a s s o f s e p a r a b l e , n o n l o c a l p o t e n t i a l s . The a d v a n t a g e o f s u c h p o t e n t i a l s , a s shown by M c M i l l a n ( 1 9 6 3 ) , i s t h a t i t i s p o s s i b l e t o o b t a i n an e x p l i c i t , c l o s e d f o r m f o r t h e p a r t i a l -wave s c a t t e r i n g a m p l i t u d e . F o r t h e p a r t i c u l a r t y p e o f p o t e n t i a l u s e d h e r e , t h i s a m p l i t u d e h a s a r e l a t i v e l y s i m p l e f o r m , so t h a t i t i s p o s s i b l e t o d e r i v e an e x p l i c i t f o r m f o r t h e t o t a l s c a t t e r i n g a m p l i t u d e , and t o e x p r e s s i t i 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 . S e p a r a b l e , n o n l o c a l p o t e n t i a l s a r e n o t e n t i r e l y d e v o i d o f p h y s i c a l s i g n i f i c a n c e . Lomon and M c M i l l a n ( 1 9 6 3 ) h a v e shown t h a t t h e B o u n d a r y C o n d i t i o n M o d e l f o r t h e n u c l e a r - n u c l e a r i n t e r a c t i o n c a n be r e f o r m u l a t e d i n t e r m s o f s e p a r a b l e p o t e n t i a l , and T a b a k i n ( 1 9 6 4 ) , a s s u m i n g t h e n u c l e a r - n u c l e a r p o t e n t i a l t o be s e p a r a b l e , h a s o b t a i n e d a d e t a i l e d f i t t o t h e a v a i l a b l e n u c l e a r - n i l c l e a r s c a t t e r i n g d a t a up t o 310 mev. C u s h i n g ( 1 9 6 3 ) and M i t r a ( 1 9 6 3 ) , u s i n g p o t e n t i a l s s i m i l a r t o t h o s e u s e d h e r e , h a v e a l s o p r o v e n d o u b l e - d i s p e r s i o n r e l a t i o n s . T h i s w o r k goes b e y o n d t h e i r s , h o w e v e r , by p r e s e n t i n g e x p l i c i t , c l o s e d f o r m s f o r t h e s p e c t r a l f u n c t i o n s i n v o l v e d , and by i n v e s t i g a t i n g t h e c o n v e r g e n c e o f t h e s p e c t r a l i n t e g r a l s . I t a l s o e x t e n d s t h e w o r k o f o t h e r s by a p p l y i n g t h e method o f B o t t i n o ( 1 9 6 2 ) t o an i n v e s t i g a t i o n o f t h e o r i g i n o f c e r t a i n a n a l y t i c p r o p e r t i e s o f t h e s c a t t e r i n g a m p l i t u d e f o r g e n e r a l n o n l o c a l p o t e n t i a l s . C h a p t e r 2 - A c l o s e d F o r m f o r t h e P a r t i a l - W a v e S c a t t e r i n g A m p l i t u d e I n t h i s c h a p t e r , t h e i n t e g r a l e q u a t i o n s g i v e n b y M c M i l l a n ( 1 9 6 3 ) f o r t h e s c a t t e r i n g w a v e f u n c t i o n a n d t h e p a r t i a l - w a v e a m p l i t u d e f o r i s c a t t e r i n g v i a a s e p a r a b l e , n o n l o c a l p o t e n t i a l a r e d e r i v e d . . T h e S c h r o e d i n g e r e q u a t i o n w i t h a n o n l o c a l p o t e n t i a l i s V 2 + k 2 * ( r ) - f dr' V ( r \ r ' ) = 0 ( 2 . 1 ) a l l r e a l r a n d t h e c o r r e s p o n d i n g i n t e g r a l e q u a t i o n f o r t h e s c a t t e r i n g - w a v e f u n c t i o n i s * ( r ) = e x p ( i k « r ) + j d r ' J*dr" G ( r , r ' ) V ( r ' , r H ) f ( r " ) ( 2 . 2 ) J a l l r e a l r a n d r w h e r e t h e G r e e n ' s f u n c t i o n G i s 0(?,?') = / dk1 e x p ( i k - ( r - r ) ) , ( 2 < 3 ) / k - k ^ + i e a l l r e a l k T h e f i r s t t e r m i n e q u a t i o n ( 2 . 2 ) r e p r e s e n t s a n i n c o m i n g p l a n e w a v e a n d t h e s e c o n d t e r m g i v e s t h e e f f e c t d u e t o t h e p o t e n t i a l . T h e G r e e n ' s f u n c t i o n g i v e n b y e q u a t i o n ( 2 . 3 ) c o n t a i n s + i € t o y i e l d o u t g o i n g w a v e s a t i n f i n i t y . E q u a t i o n ( 2 . 2 ) i s a g e n e r a l i z a t i o n o f t h e i n t e g r a l e q u a t i o n f o r s c a t t e r i n g g i v e n i n M o r s e ( 1 9 5 3 ) , p a g e 1 0 7 7 , a n d may b e d e r i v e d s i m i l a r l y . A m u c h m o r e e l e g a n t d e r i v a t i o n i s g i v e n b y M c M i l l a n ( 1 9 6 1 ) , c h a p t e r 3.1 a n d a p p e n d i x I . E q u a t i o n ( 2 . 2 ) may b e e x p r e s s e d a s a r a d i a l e q u a t i o n b y e x p a n d i n g e a c h o f t h e q u a n t i t i e s c o n c e r n e d i n t e r m s o f s p h e r i c a l h a r m o n i c s ; t h e a p p r o p r i a t e e x p a n s i o n s a r e ^ . 2.2 1=0 m=-l <t=0 m=-<I 00 -t e x p ( i £ .?) = 4 f f V V i * j ( k r ) T * ( k ) T ( r ) , - B l a t t ( 1 9 5 2 ) 1=0 m ^ t A p p e n d i x A w h i c h d e f i n e t h e new f u n c t i o n s t ^ ( r ) a n d V ^ ( r , r ) • ( j ^ ( k r ) i s t h e u s u a l s p h e r i c a l B e s s e l f u n c t i o n ) . When t h e i n t e g r a l s o f e q u a t i m s ( 2 . 2 ) a n d ( 2 . 3 ) a r e e x p r e s s e d i n s p h e r i c a l p o l a r f o r m b y m e a n s o f t h e e x p r e s s i o n s ( 2 . 4 ) , t h e a n g u l a r p o r t i o n s o f t h e i n t e g r a t i o n s may a l l b e s e p a r a t e d a n d e v a l u a t e d b y m e a n s o f t h e o r t h o n o r m a l i t y r e l a t i o n f o r s p h e r i c a l h a r m o n i c s ( D i c k e ( 1 9 6 1 ) , p a g e 1 8 9 ) : s p h e r e E q u a t i o n ( 2 , 3 ) t h u s l e a d s t o a f a m i l y o f e q u a t i o n s o f t h e f o r m ^ ( r ) = . j ^ ( k r ) + J d r r d G j ^ r ) jT d r r ' V ^ ( r , r ) ^ ( r ) ( 2 . 6 ) w h e r e G j r . r ) = / d k k J ( 2 . 7 ) 0 k - k + i € a n d I = 0 , 1 , 2, ... » ( t h e " p h y s i c a l " v a l u e s o f I.). T h e m e t h o d o f M c M i l l a n ( 1 9 6 1 ) , a p p e n d i x I , may b e u s e d t o show 1 •* a d e n o t e s t h e s p h e r i c a l p o l a r c o o r d i n a t e a n g l e s o f a i n some — » a r b i t r a r y c o o r d i n a t e s y s t e m ; a d e n o t e s t h e m a g n i t u d e o f a . 2.3 t h a t G ^ ( r , r ' ) = - i k j ^ k r j h ^ 1 * ( k r ^ ( 2 . 8 ) w h e r e = m a x ( r , r ' ) . ^ mm x ' ' A n e x p r e s s i o n f o r t h e p a r t i a l - w a v e s c a t t e r i n g a m p l i t u d e a ^ ( k ) may b e d e r i v e d f r o m t h e a s y m p t o t i c f o r m o f J r ^ ( r ) . S u b -s t i t u t i n g e x p r e s s i o n ( 2 . 8 ) f o r t h e G r e e n ' s f u n c t i o n i n e q u a t i o n (2.6) g i v e s isAr) = j ( k r ) - i k h ( 1 ) ( k r ) / d r ' r ' 2 / d r " r " 2 j , ( k r ' ) / 1 "v / \ x V ^ ( r , r ) * 4 ( r ) i k ^ ( k r ) f d r ' r ' 2 y d r " r " 2 h ^ 1 J ( k r ' ) i io / " »2 ( l ) - 1 * ' r  ^ h * 1 ' x V ^ ( r ' , r " ) ^ ( r " ) ... ( 2 . 9 ) . A s r -* °°, t h e i n t e g r a l i n t h e l a s t t e r m o f ( 2 . 9 ) v a n i s h e s a n d , u s i n g h * ( k r ) -» j ~ e x p ( i k r ) a s r » °°, ( M o r s e , ( 1 9 5 3 ) ) t h e s e c o n d t e r m b e c o m e s 1 " * ^ E i L i - i / d r ' r ' 2 / d r " r " 2 J ^ k r ' ) V ^ r ' . r " ) ^ ( r " ) ( 2 . 1 0 ) a s r -• », w h i c h i s p a r t o f t h e o u t g o i n g , s p h e r i c a l , s c a t t e r e d w a v e w i t h a n g u l a r momentum e q u a l t o £. T h e c o e f f i c i e n t o f i " e x p ( i k r ) / r i s 5> 2.4 d e f i n e d t o b e t h e p a r t i a l - w a v e s c a t t e r i n g a m p l i t u d e , i . e . , a ^ ( k ) =JT d r r d r r ^ j ^ ( k r ) V ^ ( r , r ) ^ ( r ) . ( 2 . 1 l ) T h e e q u a t i o n s u p t o t h i s p o i n t may b e c o n v e r t e d t o t h e c o r r e s -p o n d i n g e x p r e s s i o n s f o r t h e m o r e f a m i l i a r l o c a l p o t e n t i a l b y w r i t i n g V ( r , r * ' ) = V ( r ) 6 ( r - r ' ) o r , u s i n g e q u a t i o n ( 2 . 4 ) , V ( r , r * ) = ^ « ( * - * ' ) r s i n c e r r 1=0 m=l T h e c o n d i t i o n i s now i m p o s e d t h a t t h e p o t e n t i a l b e s e p a r a b l e ; t h a t i s , t h a t i t may b e w r i t t e n i n t h e f o r m V ^ ( r , r ' ) = - g V j r ) ( r ' ) ( 2 . 1 2 ) w h e r e g i s a c o n s t a n t . T h i s f o r m i s c h o s e n p r i m a r i l y f o r m a t h e m a t i c a l c o n v e n i e n c e , s i n c e u n d e r t h i s c o n d i t i o n , e q u a t i o n ( 2 . 6 ) c a n b e s o l v e d e x a c t l y , a s i s now s h o w n . S u b s t i t u t i n g e x p r e s s i o n ( 2 . 1 2 ) f o r V ^ ( r , r ) i n e q u a t i o n ( 2 . 6 ) y i e l d s 2.5 (r) = 3^(kr) - gjf dr' r ' 2 G ^ ( r , r ' ) V ^ r ' ) ^ d r " r " 2 V*"* x ^ ( r ) ... ( 2 . 1 3 ) T h e d o u b l e i n t e g r a l t h u s b e c o m e s a p r o d u c t o f two s i n g l e i n t e g r a l s a n d t h e k e r n e l o f t h e e q u a t i o n b e c o m e s d e g e n e r a t e . L e t t h e s e c o n d i n t e g r a l now b e d e n o t e d d r r ^ V j r ) ^ ( r ). ( 2 . 1 4 ) S u b s t i t u t i n g t h e r i g h t - h a n d s i d e o f e q u a t i o n ( 2 . 1 3 ) f o r ^ ( r ) i n ( 2 . 1 4 ) y i e l d s CO 00 00 I I o I A = / dr r 2 V t ( r ) j^tkr) - g A ^ dr r 2 V t ( r ) m ^ dr r 2 G ^ r . r ) w h i c h may b e s o l v e d a l g e b r a i c a l l y t o y i e l d A - ^ ^ ^ ( k r > . <2.„> CO O ^ t O I t 1 + g / d r r 2 / d r r G , ( r , r ) V ( r ) V ( r ) 0 0 . * • S u b s t i t u t i n g t h i s e x p r e s s i o n f o r t h e s e c o n d i n t e g r a l o f e q u a t i o n ( 2 . 1 3 ) y i e l d s d r r G , ( r , r ) v ^ ( r )J d r r ^ ( r Jj^Ckr ) t ^ ( r ) = ^ ( k r ) 1 + g / d r t r ' 2 V ( t ( r , ) < ( / B d r V 2 G ^ ( r ' ,r" )V^(r") ( 2 . 1 6 ) 7 2.6 Now new f u n c t i o n s a r e d e f i n e d : GO V j k ) =J^ d r r V ^ r ) j ^ ( k r ) , ( 2 . 1 7 ) a n d D , ( k ) = 1 + g / d r V 2 V ^ ( r ' ) ^ d r V 2 G^(r',r") V ^ r " ) ( 2 . 1 8 ) = ! + 2 £ ^ ° dq'q V-^ (q) ( 2 # 1 9 ) "T^O .2 2 k - q E q u a t i o n ( 2 . 1 6 ) may t h e n b e w r i t t e n r 4 ( r ) = ^ ( k r ) - D ^ ( k ) ^ t • * ( 2 . 2 0 ) T h e n a . ( k ) = / a r r 1 d r r h{ d r r ' d r r j . ( k r ) V ( r , r ) * . ( r ) ( 2 . 1 1 ) V' ~J0 -o = -gj^ d r V 2 j ^ ( k r ' ) V ^ ( r ' ) ^ d r V 2 V ^ r " ) ^ ( r " ) ( 2 . 2 1 ) V ( k ) w h e r e ( 2 . 1 2 ) , ( 2 . 1 4 ) , ( 2 . 1 7 ) , a n d ( 2 . 1 9 ) h a v e b e e n u s e d . D e f i n i n g now N j k ) = g V ^ 2 ( k ) , ( 2 . 2 3 ) e q u a t i o n s ( 2 . 1 9 ) a n d ( 2 . 2 2 ) may b e r e w r i t t e n D, ( k ) = 1 - 2 r d c L V*> ( 2 . 2 4 ) a n d K*J0 2 , 2 q - k' N . ( k ) ( k ) = -fe-pjv . ( 2 . 2 5 ) a « t V i W " D , ( k ) * S 2 . 7 In appendix I i t i s shown t h a t the 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 of I i f V - t ( r ) = M r - 5 / 2 ) as r - 0 (1.3) and a s r - * ° > . (1.4) 2 C h a p t e r 3: T h e A n a l y t i c P r o p e r t i e s o f v ^ ( k ) a n d o f ( k ) f o r a P a r t i c u l a r P o t e n t i a l . I n t h i s c h a p t e r , s e v e r a l c l o s e d f o r m s f o r v ^ ( k ) a r e d e r i v e d i n t e r m s o f h y p e r g e o m e t r i c a n d L e g e n d r e f u n c t i o n s f o r t h e s p e c i a l c a s e V ^ ( r ) = r a e x p (- u . r ) , ( 3 . l ) . w h e r e a a n d u. a r e r e a l a n d u. > 0. B y m e a n s o f t h e s e f o r m u l a e , t h e a n a l y t i c p r o p e r t i e s o f v ^ ( k ) a n d ( k ) a r e d e r i v e d . I t w i l l b e s e e n t h a t t h e g e n e r a l p r o p e r t i e s o f a ^ ( k ) d e r i v e d i n M c M i l l a n ( 1 9 6 3 ) a p p e a r e x p l i c i t l y i n t h e p r e s e n t c a s e , t h i s b e i n g o n e o f t h e m a i n m o t i v a t i o n s o f t h e p r e s e n t w o r k , A p a r t i c u l a r c a s e o f ( 3 . 1 ) h a s b e e n s t u d i e d b y C u s h i n g ( 1 9 6 3 ) , a n d c o m p a r i s o n s w i t h h i s w o r k w i l l b e p o i n t e d o u t i n d u e c o u r s e . A c c o r d i n g t o B a t e m a n ( 1 9 5 4 ) , V o l u m e I I , p a g e 2 9 : f d x x ^ - 3 / 2 e x p (-.ax) J v ( x y ) ( x y ) l / 2 QMV). p ^ , J4±M±L, V + 1; - ^ ) ( 3 . 2 ) " 2 V ^ r ( v + l ) 2 ' 2 a 2 S / F r ( ^ v ) p - v ( a — . ( 3 < 3 ) ( a 2 + y 2 ) ^ / 2 ]j a2 + y2 w h e r e P i s a " h y p e r g e o m e t r i c f u n c t i o n " , d e f i n e d b y P ( a , b ; c j z ) = 2-x 771—TP Z ( 3 . 4 ) k = 0 V c ; k K * w h e r e (a) f c = ^ f f i = ( a + n ) ( 3 « 5 ) 10 3.2 ( S e e , f o r i n s t a n c e , E I 2 . 1 . 1 ( l ) f f . 4 ) , a n d w h e r e P ~ V i i s a n a s s o c i a t e d L e g e n d r e f u n c t i o n ( E I c h a p t e r 3 ) . Th e m o s t u s e f u l p r o p e r t i e s o f t h e h y p e r g e o m e t r i c f u n c t i o n i n t h e p r e s e n t w o r k a r e t h a t P ( a , b j c ; 0 ) = 1, ( 3 . 6 ) a n d t h a t t h e s e r i e s d e f i n i n g i t c o n v e r g e s a n d i s a n a n a l y t i c f u n c t i o n o f z i f | z | < 1. A l s o , t h e r e a r e s e v e r a l a n a l y t i c c o n t i n u a t i o n s a v a i l a b l e f o r i t , some o f w h i c h a r e u s e d i n t h e f o l l o w i n g e U s i n g ( 3 . 1 ) , ( 3 . 2 ) , a n d ( 3 . 3 ) , v ( k ) = F ( ^ ± 2 , S±§±± ; ^ 3 / 2 ; - 4) ( 3 . 7 ) 1 2 - ^ V T U + 3 / 2 ) 2 2 | i . _ / l f " r ( a + £ + 3 ) p--t-l/2 ^ u ) ( 3 . 8 ) V 2 k a+5/2 a+3/2 / 2 2 ( , 2 + k 2 ) 2 V " Th e a b b r e v i a t i o n s E I a n d E I I a r e u s e d f o r v o l u m e s 1 a n d 2 r e s p e c t i v e l y o f " H i g h e r T r a n s c e n d e n t a l F u n c t i o n s " b y t h e m a k e r s o f r e f e r e n c e s ( B a t e m a n ( 1 9 5 4 ) ) . T h e s u b s e q u e n t n u m e r a l s a r e e q u a t i o n n u m b e r s i n t h o s e w o r k s . 11 3.3 E q u a t i o n ( 3 . 7 ) i s v a l i d o n l y i n t h e d o m a i n l k 2 l < u 2 b e c a u s e t h i s i s t h e r e g i o n i n w h i c h t h e s e r i e s d e f i n i n g t h e h y p e r g e o m e t r i c f u n c t i o n a l w a y s c o n v e r g e s . H o w e v e r , t h e r e a r e s e v e r a l a n a l y t i c c o n t i n u a t i o n s a v a i l a b l e f o r i t w h i c h we now u s e . U s i n g t h e a n a l y t i c c o n t i n u a t i o n g i v e n b y E I 2.10 ( 4 ) o n t h e h y p e r g e o m e t r i c f u n c t i o n i n e q u a t i o n ( 3 . 7 ) y i e l d s V ( k ) = ^ T(a+-M-3) ( r ( - a - 2 ) i a + < t + 3 g(a+U3,aU+4 . 1 " 2l+1 r ( ^ ) r ( ^ i)k a + 3 2 : 2 k , _T(a_+2) (l4*4) p(i=£f=a=i=l, - a - 1 ; r { a i | ± 3 ) r ( ^ ± t l ) ( A a + 3 ,x 2 )). ( 3 . 9 ) k ^ T h i s c o n t i n u a t i o n i s v a l i d i n t h e r e g i o n i n w h i c h 2 k 2 + » 2 k 2 < 1? t h a t i s , . R e ( k ) < - H o w e v e r , a c c o r d i n g t o t h e r e s t r i c t i o n s s p e c i f i e d w i t h E I 2.1Q: ( 4 ) , t h i s r e g i o n i s c u t i n g e n e r a l a l o n g 2 2 t h e n e g a t i v e r e a l k a x i s f r o m -u. t o -®. ( I t w i l l b e s e e n 2 2 l a t e r t h a t t h e s i n g u l a r i t y a t k = —\i may s o m e t i m e s b e a p o l e , ) S i m i l a r l y , s u b s t i t u t i o n o f E I 2.10 ( 6 ) i n t o e q u a t i o n ( 3 . 7 ) r e v e a l s 2~ V ^ ( k ) " 2 ^ V + " ' + 3 r U + 3 / 2 ) U +7 J ( 2 ' 2 ' ^' 2 ) . ( 3 . 1 0 ) i 2 ^ 2 k +\i 2 2 T h i s c o n t i n u a t i o n i s v a l i d i n t h e r e g i o n R e ( k ) > - [i. / 2 . T h e 2 2 a+-t+3 c u t d u e t o t h e f a c t o r ( l + k /\i )~ 2 i s o u t s i d e t h i s r e g i o n . 12 3.4 A n o t h e r c o n t i n u a t i o n , b a s e d o n E I 2,10 ( 2 ) , r e v e a l s t h a t 3 2 v (u) - rU+l+3) / _ _ _ J s Z a _ p/a+£+3 a ^ t + 2 . 1„ ji . V K ' ~ 2 t + l p ( S ± k t i ) p ( i z ^ ) 2 ' 2 ' 2' ' 2 u k ~ a " 4 w/a±£+£ a-4+3- 3. x Y ( a + | ± 3 ) r(i=|=i) 2 ' 2 ' 2 ' " k 2 ( 3 . 1 1 ) T h i s i s v a l i d a n d a n a l y t i c f o r a l l j k 2 | > [ j , 2 e x c e p t a l o n g t h e 2 n e g a t i v e r e a l k a x i s . T h u s , a n a l y t i c c o n t i n u a t i o n s v a l i d b o t h i n s i d e a n d o u t s i d e t h e 2 2 c i r c l e Ik i = u. , a n d b o t h t o t h e l e f t a n d t o t h e r i g h t o f t h e 2 2-s t r a i g h t l i n e R e ( k ) = - \ij 2, h a v e b e e n f o u n d . W i t h t h e i r a s s i s t a n c e , t h e p r o p e r t i e s o f v ^ ( k ) a s a f u n c t i o n o f c o m p l e x k s h a l l now b e s t u d i e d . A s k -• 0 i n e q u a t i o n ( 3 . 3 ) , t h e h y p e r g e o m e t r i c f u n c t i o n a p p r o a c h e s 1; a n d h e n c e , V j k ) a Kl a s k - 0. ( 3 . 1 2 ) T h u s , V^(]a) h a s a b r a n c h p o i n t a t k = 0 f o r n o n - i n t e g r a l - t , a n d a p o l e t h e r e f o r I a n e g a t i v e i n t e g e r ; ( k ) h a s a b r a n c h 2 p o i n t a t k = 0 w h e n 21 i s n o t a n i n t e g e r , a n d a p o l e t h e r e w h a i 21 i s a n e g a t i v e i n t e g e r . T h e b r a n c h l i n e c o r r e s p o n d i n g t o t h i s s i n g u l a r i t y i s g e n e r a l l y c o n s i d e r e d t o go f r o m t h e o r i g i n a l o n g t h e p o s i t i v e r e a l a x i s t o i n f i n i t y . 13 3.5 2 2 E q u a t i o n ( 3 . 5 ) s h o w s a b r a n c h p o i n t a t k = - \i o f t h e f o r m /, 2 2 \ - a -2 ( k + u. ) 2 2 The b r a n c h l i n e f o r t h i s p o i n t i s c o n s i d e r e d t o r u n f r o m k = ~u. 2 - 2 t o k = . - <=° a l o n g t h e n e g a t i v e r e a l k a x i s . T h e b r a n c h p o i n t s e e m s t o d i s a p p e a r w h e n a i s a n i n t e g e r b u t i t i s i m p o s s i b l e t o i n f e r t h i s w i t h c e r t a i n t y f r o m e q u a t i o n ( 3 . 5 ) b e c a u s e t h e r i g h t -h a n d s i d e a s s u m e s t h e f o r m <= - » f o r t h i s c a s e . F u r t h e r i n f o r m a t i o n c o n c e r n i n g t h e b e h a v i o u r o f t h e f u n c t i o n a t t h i s p o i n t may b e o b t a i n e d b y m e a n s o f E I 3.9.2 ( 1 9 ) , p u . / x _ 2 V r ( y + 1/2) v a s | z | - » , V z ) T^ T ru + v - u.) z w h i c h i s v a l i d f o r Re(v) > - 1/2. S u b s t i t u t i n g t h i s i n t o e q u a t i o n ( 3 . 8 ) y i e l d s V ( k ) - / f a + 1 ^ a - 5 / 2 r ( a + 2 ) ( l + 4 ) " a " 2 » ( 3 . 1 3 ) v a l i d f o r a > - 2. T h u s , a s e x p e c t e d , v ^ ( k ) h a s n o b r a n c h p o i n t 2 2 a t k = - | i f o r i n t e g r a l a > - 2, b u t r a t h e r a p o l e o f o r d e r a + 2. F u r t h e r , E I 3.2 ( 2 4 ) a n d E I 3.2 ( 9 ) c o n s p i r e t o s a y vV-(r) _ , 2 ( X z t l + v • ' pz-u^v 1 - u - v . , i _ I ) ( 3 14) Vs5' - r ( l - | A ) ( z2„ 1)tx/2 n 2 ' 2 ' 1 ^ z 2 h U . 1 4 ; S u b s t i t u t i n g i n t o e q u a t i o n ( 3 . 7 ) y i e l d s , r-i P f 2 ? U + l / 2 ) . 2 * " i ( a + 5 / 2 ) -1-1/2 . 2 - 1 / 2 x p ( ( 1 + *-) ) . ( 3 . 1 5 ) -a- -5/2 u. A g a i n u s i n g E I 3.9.2 (19), w h i c h i s now v a l i d f o r a < - 2, 14 3.6 v (v) - 1 £ l a + i + 3 ) r(-a-2) /, , kV a"" 2  4 2 a + l ^ a + 5 / 2 r l-a+4 - l ) U ^ a s k 2 - - u 2 , ( 3 . 1 6 ) 2 2 w h i c h s h o w s t h a t v^(k) h a s n o b r a n c h p o i n t a t k = - u. , b u t r a t h e r a z e r o o f o r d e r - a -2 f o r i n t e g r a l a < - 2. T h e s t r u c t u r e o f V^(k) w h e n a = - 2 r e m a i n s u n i n v e s t i g a t e d . 2 T h e o n l y r e m a i n i n g s i n g u l a r i t y i n t h e c o m p l e x - k p l a n e i s a t i n f i n i t y , a n d e q u a t i o n ( 3 . 1 l ) s h o w s t h e b e h a v i o u r o f v ^ ( k ) a ^ t h i s p o i n t . S i n c e t h e h y p e r g e o m e t r i c f u n c t i o n s i n ( 3 . 1 l ) b o t h 2 a p p r o a c h 1 a s k a p p r o a c h e s i n f i n i t y ; i n t h e g e n e r a l c a s e , V^(k) a k~a"3 a s k -* «. ( 3 , 1 7 ) I f e i t h e r o f t h e q u a n t i t i e s ( a + I + 4 ) / 2 o r (^ - a ) / 2 i s a n o n - p o s i t i v e i n t e g e r h e r e , t h e f i r s t t e r m i n ( 3 . 1 l ) d i s a p p e a r s , a n d V^(k) a k"a~4 a s |k| - ». T h u s , t h e p o i n t a t i n f i n i t y w i l l b e e i t h e r a b r a n c h p o i n t o r a p o l e , d e p e n d i n g o n t h e v a l u e 1 o f a . I n s u m m a r y , V^(k) h a s a s i n g u l a r i t y o f t h e f o r m k a t k = Oi 2 2 — a 2 2 2 o n e o f t h e f o r m (k + \i ) ~ ~ a t k = - u. , a n d o n e o f t h e f o r m — a — 3 — a — 4 2 k o r , i n s p e c i a l c a s e s , k , a t k = °°. T h i s a g r e e s w i t h 2 2 2 t h e p r o p e r t i e s o f a^(k) a t k = 0 a n d a t k = - u. d e r i v e d i n M c M i l l a n ( 1 9 6 3 ) . C o n s i d e r i n g now V^(k) a s a f u n c t i o n o f c o m p l e x I, e q u a t i o n ( 3 . 7 ) s h o w s t w o p o s s i b l e s o u r c e s o f s i n g u l a r i t i e s i n t h e I p l a n e 15 3.7 2 i 2 f o r ) k I < p, : t h e h y p e r g e o m e t r i c f u n c t i o n ; w h i c h h a s s i m p l e p o l e s f o r n o n - p o s i t i v e - i n t e g r a l v a l u e s o f t h e t h i r d p a r a m e t e r , i . e . I = - 3 / 2 , - 5 / 2 , ...; ( s e e E I , p a g e 5 7 ) a n d t h e gamma f u n c t i o n i n t h e n u m e r a t o r , w h i c h h a s s i m p l e p o l e s f o r n o n - p o s i t i v e i n t e g e r s ; , i . e . £ = - a - 3, - a - 4 ... ( s e e E I , p a g e 2.) T h e f o r m e r may b e s e e n t o b e i l l u s o r y i n v ^ ( k ) b y m e a n s o f E I 2 . 1 . 3 ( 1 6 ) F(a9b;ciiz) „. ( a \ ( b \ z l - c F ( a + l - c , b + l - c s 2 - c ; z ) ; ( 3 . 1 8 ) P (c y~~~ ~ v ' l ~ e l ~ c p"( 2 - c ) s h o w i n g t h a t F ( a , b ; c ; z ) / p ( c ) i s f i n i t e f o r a l l c , a n d t h u s " t h a t 2 r u + 3/2) i s f i n i t e f o r a l l v a l u e s o f I, T h u s t h e o n l y s o u r c e o f s i n g u l a r i t i e s i n t h e c o m p l e x — t p l a n e i n e q u a t i o n ( 3 . 7 ) i s t h e gamma f u n c t i o n i n t h e n u m e r a t o r o f e q u a t i o n ( 3 . 7 ) . I t g i v e s a s i m p l e p o l e f o r a + -t + 3 = - n o r I - - ( n + a + 3) w h e r e n i s a n o n - n e g a t i v e i n t e g e r . T h e s e same p o l e s a r e e x h i b i t e d 2 b y a l l t h e a n a l y t i c c o n t i n u a t i o n s e x c e p t w h e n k i s a t a s i n g u l a r i i n t h e c o m p l e x - k p l a n e . E a c h o f t h e s e f i r s t - o r d e r p o l e s o f V ^ ( k ) p r o v i d e s .« s e c o n d - o r d e r p o l e i n ( k ) . T h e s e p o l e s a r e m e n t i o n e d i n M c M i l l a n ( 1 9 6 3 ) . 16, 3. 8 M o s t o f t h e gamma f u n c t i o n s i n v o l v e d i n t h e e q u a t i o n s f o r V ^ ( k ) h a v e e s s e n t i a l s i n g u l a r i t i e s a t -L = °°„ T h e b e h a v i o u r o f V ^ ( k ) a s I -* 0 0 i s t h u s r a t h e r c o m p l i c a t e d , a n d s h a l l b e t r e a t e d s e p a r a t e l y i n t h e n e x t c h a p t e r . 17 4.1 C h a p t e r 4. T h e A s y m p t o t i c B e h a v i o u r o f a^(k) a s (<lj -» I n o r d e r t o d e r i v e 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 f o r t h e t o t a l s c a t t e r i n g a m p l i t u d e , t h e S o m m e r f e l d - W a t s o n t r a n s f o r m a t i o n o n t h e s e r i e s e x p r e s s i o n f o r t h e t o t a l s c a t t e r i n g a m p l i t u d e w i l l b e e m p l o y e d , a s d e s c r i b e d b y Omnes ( 1 9 6 3 ) . I n t h i s c h a p t e r , r e s t r i c t i o n s o n t h e p o t e n t i a l s h a l l b e f o u n d s u c h t h a t t h e b e h a v i o u r o f a^(k) a s \l\ -* 0 0 p e r m i t s t h e S o m m e r f e l d - W a t s o n t r a n s f o r m a t i o n t o b e e m p l o y e d . T h e s e r i e s e x p r e s s i o n f o r t h e t o t a l s c a t t e r i n g a m p l i t u d e i n t e r m s o f t h e p a r t i a l - w a v e a m p l i t u d e s i s i 00 f ( k , c o s («•)) = > (21 + 1 ) a , fck) P , ( + c o s ( G - ) ) . ( 4 . 1 ) 1=0 ' U s i n g t h e S o m m e r f e l d - W a t s o n t r a n s f o r m a t i o n , t h i s may be w r i t t e n „ • ' , ' ¥ 2 4 ° - ( 2 * + l ) a,(k) P (_ c o s ( S ) ) f ( k % COs(ft)) = k J Cl* ' ' cn-n^/) : ~ 2 I % - 1 / 2 - t i - S m f n O •~. P B '- ( 4 . 2 ) _ ( 2 a ( k 2 ) + 1 ) ^.(k 2) *a (k 2)(-.cos(*)<) ^. \ . ' J . J J j , s i n ( T T a (k 2)) 2 t h w h e r e a . ( k ) i s t h e s i t e o f t h e j R e g g e p o l e o f a , (k) i n t h e J ^ h a l f - p l a n e ReU) >- 1 / 2 , a n d 8 . ( k 2 ) = l i m 0 (l - a , ( k 2 ) ) x J ^ a . ( k ^ ) .V a ^ ( k ) , t h e r e s i d u e o f t h e j p o l e . T h i s t r a n s f o r m a t i o n i s v a l i d i f a ^ ( k ) s a t i s f i e s t h e f o l l o w i n g t w o c o n d i t i o n s , g i v e n by S q u i r e s ( 1 9 6 2 ) J 18 4.2 ( l ) a ^ ( k ) = 0(<tn) a s \ l\ — w h e r e n i s a c o n s t a n t a n d - TT/2 < a r j (l) < T T / 2 J ( 2 ) a < t(k) = 0 U~ 3/ 2) a s -t - + i » ( 4 . 3 ) C o n d i t i o n s s h a l l now b e p l a c e d o n V ( r ) a s g i v e n b y ( 3 . 1 ) i n o r d e r t h a t c o n d i t i o n s ( 4 . 3 ) may b e s a t i s f i e d . I t w i l l b e f o u n d t h a t t h e b e h a v i o u r o f V ( r ) a s r -» 0 m u s t b e s e v e r e l y r e s t r i c t e d . I n o r d e r t o i n v e s t i g a t e t h e b e h a v i o u r o f a ^ ( k ) a s \l\ — », E I 2 .3.2 ( 1 6 ) i s u s e d i ( Z . " _ i ) ~ a ~ X F ( a + X » a - c + 1 + X j a - b + 1 + 2X; ,a+b ft r ( a - b + l + 2 X ) . r(a-c+l+X ) r(c-b+X). r d / 2 ) ( z - fz^i) a+X X ( ( l - z + J z ^ i ) exp(T iTT ) ) 1 / 2 " C (l+z -Zz 2 - ! ) i c - a - b - 1 / 2 x ( 1 + 0 ( X - 1 ) ) a s jx| - » , ( 4 . 4 ) w h e r e t h e u p p e r o r l o w e r s i g n i n e x p ( + i n - ) i s c h o s e n a c c o r d i n g a s l m (z) J 0. T h e e x p r e s s i o n i n b r a c k e t s c a n b e t i d i e d u p s o m e w h a t b y u s i n g E I 1.18 ( 4 ) ; r ( z+a) _ a-b r u + b f - 2 ( l + 0 ( z ^ ) ) a s I 1-9 4.3 a n d S t i r l i n g ' s f o r m u l a ( E I 1.18 ( 2 ) ) , T ( z ) = e " Z z Z - l / 2 /2~7 (1+0U- 1)) a s | z | - », t o o b t a i n 2 2 X " " 1 / 2 S e t t i n g X = t/2, a ( i n e q u a t i o n ( 4 , 4 ) ) = ^ r j — • , a+2 , w b = 2 "> c = a + J , 2 k 2 a n d y - — = • 2 2 '*"n e Q . u a ^ i ° n ( 4 , 4 ) ; s u b s t i t u t i n g i n t o e q u a t i o n ( 3 . 7 ) ; a n d u s i n g E I 1.18 ( 4 ) a g a i n ; V k ) . zjgll ^ 3 / 2 , y 7 7 7 . 2 ) s i i± 2 x ( ( l - z + f z ^ l ) e x p ( + ITT ) ) " a - 5 / 2 ( l + 0 ( ^ " 1 ) ) a s (^ 1 - « . ( 4 . 5 ) Now t h e q u a n t i t y w ( z ) = - z ( 4 . 6 ) w h i c h a p p e a r s i n ( 4 . 5 ) h a s t h e h a p p y p r o p e r t y t h a t i t h a s m a g n i t u d e n o g r e a t e r t h a n 1 f o r a l l v a l u e s o f z . T h i s may b e s e e n b y c o n s i d e r i n g ( 4 . 6 ) t o b e a c o n f o r m a l - m a p p i n g f u n c t i o n , t h e i n v e r s e m a p p i n g o f w h i c h i s 2 z = - (w + 1 / w ) , w h i c h maps t h e u n i t c i r c l e a n d i t s i n t e r i o r i n t h e w p l a n e o n t o t h e e n t i r e z p l a n e ( C h u r c h i l l ( i 9 6 0 ) ) . T h u s , t h e m a g n i t u d e o f 20 4.4 w ( z ) , t h e q u a n t i t y r a i s e d t o t h e p o w e r I i n ( 4 . 5 ) , n e v e r e x c e e d s 1. T h e r e f o r e , v ^ ( k ) d e c r e a s e s e x p o n e n t i a l l y a s Re{l) a p p r o a c h e s + » f o r a l l v a l u e s o f k. S i n c e ^ ( k ) d e c r e a s e s u n i f o r m l y f o r a l l v a l u e s o f k, a n d , i n p a r t i c u l a r , f o r t h e v a l u e s o c c u r r i n g a l o n g t h e p a t h o f i n t e g r a t i o n u s e d i n D ^ ( k - ) , i t f o l l o w s t h a t t h e i n t e g r a l t h e r e i n d i s a p p e a r s e x p o n e n t i a l l y . T h i s i m p l i e s t h a t t h e d e n o m i n a t o r a p p r o a c h e s 1, a n d t h a t a ^ ( k ) b e h a v e s a s ( k ) f o r l a r g e Re(l). T h e r e s u l t i n g e x p o n e n t i a l d i s a p p e a r a n c e o f a ^ ( k ) a m p l y s a t i s f i e s S q u i r e s ' f i r s t c o n d i t i o n . I t w i l l b e n o t i c e d t h a t o n e g e t s i n t o t r o u b l e i f w ( z ) h a p p e n s t o w a n d e r i n t o i t s o t h e r b r a n c h ; s i n c e ( z + / z 2 - l ) ( z - J z 2 - 1) = 1, i t f o l l o w s t h a t t h e o t h e r b r a n c h o f w ( z ) i s a l w a y s o n o r o u t s i d e t h e u n i t c i r c l e . T h i s b r a n c h m u s t b e e x c l u d e d b y a v o i d i n g t h e b r a n c h l i n e f r o m z = - 1 t o z = + \, T h i s c o r r e s p o n d s t o a b r a n c h l i n e f o r t h e h y p e r g e o m e t r i c f u n c t i o n i n e q u a t i o n ( 4 . 4 ) f r o m 2 2 1-z *~ ^° 1-z ~ a > i t h e u s u a l b r a n c h l i n e o f t h e h y p e r g e o m e t r i c f u n c t i o n . I t c o r r e s p o n d s 2 2 2 t o a b r a n c h l i n e i n a ^ ( k ) f r o m k " = - u. t o k = - «, w h i c h b r a n c h l i n e h a s a l r e a d y b e e n r e c o g n i z e d ; a n d t h u s t h e n e c e s s i t y o f a v o i d i n g t h e s p u r i o u s b r a n c h o f w ( z ) c r e a t e s no new p r o b l e m s . I n d e t e r m i n i n g t h e b e h a v i o u r o f V ( k ) a s Im(-t) -» + °° w i t h 21. 4.5 Re(<t) h e l d c o n s t a n t , i t i s u n n e c e s s a r y t o c o n s i d e r q u a n t i t i e s r a i s e d t o t h e p o w e r I i n e q u a t i o n ( 4 . 5 ) b e c a u s e t h e m a g n i t u d e o f s u c h a q u a n t i t y i s a f u n c t i o n o n l y o f t h e r e a l p a r t o f t h e p o w e r . T h e s i g n i f i c a n t b e h a v i o u r i n t h i s c a s e i s t h u s g i v e n b y t h e f a c t o r ^ a + 3 / 2 ^ A g b e f o r e > a ^ ( k ) a V 2 ( k ) f o r l a r g e ImU) p r o v i d e d a < - 3/2. T h u s , S q u i r e s ' s e c o n d r e q u i r e m e n t i m p l i e s 2 a + 3 < - 3/2 o r a < - 9/4. T h e b e h a v i o u r w h e n a > - 3/2 v i o l a t e s S q u i r e s ' 2 r e q u i r e m e n t s i n c e , i n t h i s c a s e , D ^ ( k ) g r o w s n o f a s t e r t h a n N ^ ( k ) a s Im(t) -• + <». I n o r d e r t h a t t h e i n t e g r a l i n D ^ ( k ) c o n v e r g e , i t i s n e c e s s a r y t o i m p o s e t h e r e q u i r e m e n t a > - 5 / 2 . T h u s , a h a s b e e n s e v e r e l y r e s t r i c t e d ; i n o r d e r f o r e q u a t i o n ( 4 . 2 ) t o b e v a l i d i n t h e p o t e n t i a l u s e d h e r e , i t m u s t s a t i s f y - 5/2 < a < - 9/4. ( 4 . 7 ) G u s h i n g ( 1 9 6 3 ) f o u n d i t n e c e s s a r y t o u s e a n ^ - d e p e n d e n t p o t e n t i a l i n o r d e r t o b e a b l e v a l i d l y t o p e r f o r m t h e S o m m e r f e l d - V a t s o n t r a n s f o r m a t i o n . T h e p r e s e n t w o r k i s n o t i n c o n s i s t e n t w i t h h i s , h o w e v e r , s i n c e h e u s e d a p o t e n t i a l w h i c h i s J 0 ( r 3 / 2 ) f o r s m a l l r , a n d w h i c h i s t h u s n o t s u f f i c i e n t l y s i n g u l a r a t t h e o r i g i n . T h u s , t h e 22 4 . 6 n u m b e r o f p a r a m e t e r s r e q u i r e d i n t h e p o t e n t i a l i s r e d u c e d f r o m d e n u m e r a b l e i n f i n i t y t o t h r e e ; g>u-, a n d a . T h e v a l i d i t y o f ( 4 . 5 ) t h u s h a v i n g b e e n s e c u r e d , i t i s now u s e d t o i n v e s t i g a t e t h e a n a l y t i c p r o p e r t i e s o f f ( k , cosG-) a s a f u n c t i o n 2 o f t h e c o m p l e x v a r i a b l e s k a n d c o s 23 5.1 2 Chapter 5; The A n a l y t i c P r o p e r t i e s of f ( k , cos ©•) i n the Complex - cos Q- Plane. In t h i s chapter, the a n a l y t i c p r o p e r t i e s of the s c a t t e r i n g amplitude as a f u n c t i o n of the cosine of the s c a t t e r i n g angle are studied, and the f u n c t i o n i s expressed as a d i s p e r s i o n i n t e g r a l . The r e s u l t s given here are c o n s i s t e n t with the e a r l i e r work by Cushing (1963) and M i t r a (1963) but d i f f e r i n two respects from the e a r l i e r worksi (1) The convergence, or otherwise, 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 present e f f o r t , (2) An e x p l i c i t form i s given f o r the weight f u n c t i o n i n the i n t e g r a l s i n the present work. The a n a l y t i c p r o p e r t i e s of the s c a t t e r i n g amplitude i n the complex-cos 0- plane may be obtained from equation (4.2) by means of the argument of B o t t i n o (1962), pages 988 to 989. -l/2+i» f ( k 2 , cos(ft)) = § f dl { 2 l + l ) a £ ( k ) V " C 0 8 < » » + pole terms. • / 0 • s i n (TT t) £=-1/2 -1- ( 4 # 2 ) There i t i s shown that the i n t e g r a l converges f o r a l l values of cos ©• 2 and thence t h a t f ( k , cos ©•) i s a n a l y t i c i n the domain of P^(-cos ©•). In general, P^j(z) has two branch l i n e s i n the z plane, one along the r e a l a x i s from -1 to 1 and another from —» to -1, There are no other s i n g u l a r i t i e s . The branch l i n e from -1 to I disappears 2 when a. i s an even i n t e g e r . Thus, f ( k , cos ©•) i s a n a l y t i c f o r 24 5.2 c o s ©• n o t o n t h e p o s i t i v e r e a l a x i s f r o m 1 t o °°. S i n c e S q u i r e s ' c o n d i t i o n s ( i n e q u a t i o n s ( 4 . 3 ) ) a r e s u f f i c i e n t f o r c o n v e r g e n c e o f t h e i n t e g r a l i n ( 4 . 2 ) , t h e same r e s u l t may b e o b t a i n e d s i m i l a r l y i n t h e p r e s e n t c a s e , A d i s p e r s i o n r e l a t i o n f o r f ( k , c o s ©•) a s a f u n c t i o n o f c o s O w i l l now b e d e r i v e d , u s i n g t h e m e t h o d o f M i t r a ( 1 9 6 3 ) . U s i n g C a u c h y ' s t h e o r e m , n P ( t + i € ) - p . ( t - i e ) t=-«° o-rr ,-m ( 5 . 1 ) + l i m 1 / d ( T e 1 ( p ) X cp=0 C o n s i d e r i n g now o n l y t h e f i r s t i n t e g r a l , t h e d i s c o n t i n u i t y a c r o s s t h e b r a n c h l i n e , i . e . p ^ ( " t + 1 6 ) - ^ ( t - i € ) , may b e o b t a i n e d by m e a n s o f E I 3 . 7 ( 6 ) , w h i c h y i e l d s P,(z) = «• / d u (z + / z 2 - T c o s u ) * , ( 5 . 2 ) 0 v a l i d f o r a l l v a l u e s o f t h e p a r a m e t e r s . I n t h i s e x p r e s s i o n , l e t z b e v a r i e d c o n t i n u o u s l y f r o m a v a l u e j u s t b e l o w t h e n e g a t i v e r e a l a x i s t o a v a l u e j u s t a b o v e a l o n g t h e p a t h s h o w n b e l o w s 25 5.3 2 S i n c e t h i s p a t h a v o i d s t h e b r a n c h l i n e o f Jz - 1, t h e p h a s e o f l~2 • z + \Jz - 1 c o s u p a s s e s t h r o u g h 2?T r a d i a n s , a n d t h a t o f t h e i n t e g r a n d p a s s e s t h r o u g h 2n I r a d i a n s . m - co < t < ~ 1, P ^ ( t + i<E) - P ^ ( t - i € ) = - / d u ( t + ft2 - 1 c o s u)1 ( 1 - exp(27rU)) 0 = e x - p i i n l ) - e x p ( - Ul) J d u ( _ t + y t 2 _ 1 C Q S u ) . = - 2 i s i n {-nt) P ^ ( - t ) . ( 5 . 3 ) S u b s t i t u t i n g i n t o ( 5 . 1 ) , r P ( t ) P ( v\ _ s i n ( - r U ) / iK I n t e g r a l a r o u n d , ~ ,\ v^-z) - - ^ J a t t - z + i n f i n i t e c i r c l e . ° , 4 ; t = l M i t r a ( 1 9 6 3 ) h a s n o t s t u d i e d t h e v a l i d i t y o f t h i s r e p r e s e n t a t i o n s f o r i t t o b e v a l i d , t h e i n t e g r a l s o f ( 5 . 4 ) m u s t c o n v e r g e . T h e i n t e g r a l a l o n g t h e b r a n c h l i n e s h a l l b e c o n s i d e r e d f i r s t . T h e l o w e r l i m i t o f t h e i n t e g r a l g i v e s n o t r o u b l e b e c a u s e P ^ ( t ) i s n o n s i n g u l a r a t t = 1. A s f o r t h e i n f i n i t e t a i l ; s i n c e t h e c o e f f i c i e n t o f i s 0(t"""^) a s t -• <=, t h e i n t e g r a l w i l l c o n v e r g e a t t h e u p p e r l i m i t p r o v i d e d P ^ ( t ) = sr(l) a s t - «. ( 5 . 5 ) I t i s shown i n a p p e n d i x I I t h a t 26> 5.4 P ^ t ) = max ( 0 ( t / ) , 0 ( t - ^ 1 ) ) a s 1*1 - ( I I . 1 ) T h u s , t h e i n t e g r a l c o n v e r g e s f o r - 1 < Re (l) < 0. T h e n e g l e c t o f t h e c o n t r i b u t i o n o f t h e i n f i n i t e c i r c l e i n e q u a t i o n ( 5 . 1 ) may now b e j u s t i f i e d . T h e c o n t r i b u t i o n i s s a t i s f i e d f o r a n y a r g ( t ) . A r e v i e w o f t h e d e r i v a t i o n o f e q u a t i o n ( I I . l ) s h o w s t h a t t h e l a t t e r i s v a l i d r e g a r d l e s s o f a r g ( t ) , a n d t h u s t h a t t h e c o n t r i b u t i o n o f t h e i n f i n i t e c i r c l e may b e n e g l e c t e d f o r - 1 < R e U) < 0. T h u ? , t h e d i s p e r s i o n r e l a t i o n ( 5 . 4 ) i s v a l i d f o r a l l v a l u e s o f t o c c u r r i n g i n e q u a t i o n ( 4 . 2 ) j i t c a n a l s o b e u s e d i n t h e R e g g e pole t e r m s p r o v i d e d - 1 < a . ( k ) < 0. T h e s e t e r m s w i l l h o w e v e r b e l e f t i n " u n d i s p e r s e d " f o r m , h e n c e t h i s r e s t r i c t i o n n e e d n o t b e made i n t h e f o l l o w i n g . H e n c e , t h i s r e p r e s e n t a t i o n o f ^ ^ ( ~ cos(©0) may b e s u b s t i t u t e d i n t o e q u a t i o n ( 4 . 2 ) t o y i e l d 0 ( P ^ T ) ) a s IT I H e n c e , t h e c o n t r i b u t i o n may b e n e g l e c t e d i f c o n d i t i o n ( 5 . 5 ) i s 2 f ( k , c o s + R e g g e p o l e t e r m s ( 5 . 6 ) + R e g g e p o l e t e r m s , t = l 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 the desired s p e c t r a l representation i n cos($). The convergence of the i n t e g r a l representing o(k,t) w i l l now be i n v e s t i g a t e d . EI 3.6.1 (3) says P^(t) = F U + 1, -I j 1 ; and the behaviour of t h i s f u n c t i o n f o r large I may be i n v e s t i g a t e d by means of EI 3.2.2 (17): F(a + X, b - X; c; ^ jr) = T ( l - b + X ) r ( c ) 2 a + b - 1 ( l . z + ^ ) - c + l / 2 ( l r ^ } c - a - b - l / 2 r ( i / 2 ) P l c - b + x ) u-z+yz i ; u+z+yz i ; x ( ( z +TJT~l)x-h + exp(+ i-rr ( c - l / 2 ) ) ( z - / z ^ l ) * X ) ( l + 0 ( X " 1 ) ) as IXl 00. S e t t i n g a = 1, b = 0, c = 1, and z = t ; 2 -1/4 FU+1, -4, 1; i f i ) = P ^ ( t ) = jl i ± ^ i i ( ( t + / t 2 - l ) ^ ± i ( t x (1 + 0 U " 1 ) ) . Since 1 = 0 ( i m U ) ) as Im(<t) -»'+ • w i t h ReU) held constant, 28= 5.6 (21 + 1 ) = 0 U ) , a ^ ( k ) = 0 ( £ 2 a + 3 ) f r o m e q u a t i o n ( 4 . 5 ) , a n d P ^ ( t ) = 0 U " 1 / 2 ) f o r R e U ) h e l d c o n s t a n t ; t h e i n t e g r a n d i s 0 ( l m ( - 0 2 a + ^ ' / ' 2 ) a s I m ( ^ ) -» + ». A s u f f i c i e n t c o n d i t i o n f o r c o n v e r g e n c e i s t h u s 2 a + 7/2 < - 1, o r a < - 9/4. ( T h i s c o n d i t i o n b e c o m e s n e c e s s a r y f o r k =0 a n d f o r t=°°. O t h e r w i s e , a < - 7/4 i s n e c e s s a r y a n d s u f f i c i e n t . ) T h i s f u r t h e r s t r e n g t h e n s t h e r e s t r i c t i o n s i m p o s e d o n a e a r l i e r b y e x c l u d i n g t h e c a s e a = - 9/4. T h i s d i s p e r s i o n r e p r e s e n t a t i o n r e f l e c t s t h e p r o p e r t y o f t h e s c a t t e r i n g a m p l i t u d e i n t h e l o c a l c a s e , t h a t t h e i n t e g r a l t e r m d e c r e a s e s a s cos(O-) -• », a n d t h a t t h e b e h a v i o u r o f t h e a m p l i t u d e i s d o m i n a t e d b y t h e r i g h t m o s t R e g g e p o l e t e r m a j j ( k ) ( s e e , f o r i n s t a n c e , t h e p a p e r b y M a n d e l s t a m i n T h e o r e t i c a l P h y s i c s ( 1 9 6 3 ) , < x R ( k ) p a g e 4 1 3 ) . I n b o t h c a s e s , t h e a m p l i t u d e b e h a v e s l i k e ( c o s ©•) a s c o s 29* 6.1 C h a p t e r 6. T h e A n a l y t i c P r o p e r t i e s o f f ( k , c o s &) i n t h e k I n t h i s c h a p t e r , 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 f ( k , c o s (©•)). I t t a k e s t h e f o r m A s i n c h a p t e r 5, t h e f o r m d e r i v e d i s i d e n t i c a l t o t h a t o f M i t r a a n d C u s h i n g . H e r e h o w e v e r t h e 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 i n v o l v e d i s e x a m i n e d a n d e x p l i c i t , c l o s e d f o r m s f o r t h e w e i g h t f u n c t i o n s a r e d e r i v e d . 2 I n e q u a t i o n ( 5 . 6 ) , t h e o n l y f u n c t i o n o f k o t h e r t h a n i n t h e p o l e t e r m s i s a ^ ( k ) , w h i c h a p p e a r s i n t h e d e f i n i t i o n o f o ( k , t ) , t h e w e i g h t f u n c t i o n . a ^ ( k ) h a s t w o 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 c o m p l e x - k p l a n e : t h e f u n c t i o n ( k ) , w h i c h i s c u t f r o m 2 2 2 -k = - u . t o k = -.co a n d a l s o a l o n g t h e e n t i r e p o s i t i v e r e a l a x i s ; a n d t h e i n t e g r a l i n D ^ ( k ) , w h i c h h a s a b r a n c h l i n e a l o n g t h e p o s i t i v e r e a l a x i s . 2 F o r t h e d i s c o n t i n u i t y a c r o s s t h e n e g a t i v e r e a l k a x i s , e q u a t i o n 2 2 ( 3 . 9 ) , w h i c h i s v a l i d f o r R e ( k ) < - u. / 2 , may b e u s e d t o g i v e p l a n e . ( 6 . 1 ) + R e g g e p o l e t e r m s . 30 6.2 , rx— 1 V . 2 ( / k 2 + i€) - V 2 ( / k 2 - i€) a^(/k^ + i€) - a^Uk^ - i€) = g ^  p ( k ) * / ( — j — ) / (—2—) ^ ( 6 . 2 ) 2 2 2 ' „ ^ - a - a — t - 1 , k +u. \~| 1 - e x p ( - 4 n - j ( a + 2 ) ) . * v 2 ' 2 5 ~ a ' 1 ' k 2 J 5J(kl T h e c o n t r i b u t i o n o f t h i s s i n g u l a r i t y t o a^(k) i s g i v e n b y 2rTi 0 a , ( / k ^ i € ) - a,(/k' 2-i€) a(k- 2) , . ( 6 . 3 ) 2 2 k ' Z = -u. •2 _ v.2 S i n c e | a ^ ( J k , 2 + i € ) - a ^ ( J k ' 2 - i € ) ( < |a^( Jk' 2 + i € ) / + | a ^ ( / k ^ i 6 ) | , 2 t h e i n t e g r a l i n ( 6 . 3 ) i s 0(a^(k)k ) a s k -• 0 0. E q u a t i o n ( 3 . 1 7 ) r e v e a l s t h a t V 2(k) = 0(k" 2 a~ 6) a s k - »; ( 3 . 1 7 ) </ a n d t h e b e h a v i o u r o f a^(k) i s s i m i l a r l y b o u n d e d b e c a u s e D^(k) 2 a p p r o a c h e s 1 a s k a p p r o a c h e s - 0 0. T h u s , t h e i n t e g r a l i s 2 a 8 0(k~ ~ ) f o r l a r g e k , a n d t h e i n f i n i t e t a i l o f i n t e g r a n d ( 6 . 3 ) c o n v e r g e s f o r - 2 a - 8 < - 1, o r a > - 7/2. T h i s i m p o s e s n o new r e s t r i c t i o n o n a . 2 2 The s i n g u l a r i t y o f t h e i n t e g r a n d a t k = - u. i s 31-6.3 0( ( k 2 + [ A 2 ) ~ 2 a ~ 4 ) . C o n s e q u e n t l y , t h i s p o r t i o n o f t h e i n t e g r a l c o n v e r g e s f o r - 2 a - 4 > - 1 , o r a < - 3 / 2 . A g a i n , a i s n o t f u r t h e r r e s t r i c t e d . E q u a t i o n ( 3 . 1 6 ) , w h i c h i s v a l i d f o r R e ( k 2 ) > - ^C, r e v e a l s 2 2 ! t h a t t h e p h a s e o f V ^ ( k ) c h a n g e s b y 2nl r a d i a n s a s k p a s s e s f r o m o n e s i d e o f t h e p o s i t i v e r e a l a x i s t o t h e o t h e r . T h u s , t h e 2 ( 2 2 p h a s e o f V. k ) c h a n g e s b y t h e same a m o u n t a s k c r o s s e s i t s p o s i t i v e r e a l a x i s . The i n t e g r a l i n D^(k) c a n b e w r i t t e n i n t h e f o r m 2 ^ 1 / f f § \ 4 V<*>. ( 6 - 4 ) s h o w i n g t h a t t h e r e i s a b r a n c h l i n e a l o n g i t s p a t h o f i n t e g r a t i o n , 2 t h e p o s i t i v e r e a l k a x i s , w i t h d i s c o n t i n u i t y D ^ / k ^ i e 1 ) - D ^ ( / k 2 - i € ) = 2 g i k V^ 2(k) = 2 i k N^(k). (6 . 5 ) 2 T h e r e f o r e , a l o n g t h e p o s i t i v e r e a l k a x i s , ^(/]?+i€) - a 1 ( / k 2 T i € ) N j k ) N ^ , ( k ) e x p ( 2 f r i l) = D-^k) ~ D^(k) + 2 i k N j k ) D^(k) exp(2Tfi *,) = a ^ ( k ) ( 1 ~ D^k) + 2 i k N ^ ( k ) } exp(2^"i £) 32 6.4 T h e c o n t r i b u t i o n o f t h i s b r a n c h l i n e t o a ^ ( k ) i s t h u s g i v e n b y t h e c o n v e r g e n c e o f w h i c h m u s t now b e e s t a b l i s h e d . 2 I n o r d e r t o e s t a b l i s h t h e b e h a v i o u r o f a ^ ( k ) a s k -• + °°, i t i s n e c e s s a r y t o d e t e r m i n e t h a t 6 f D ^ ( k ) . T h e b e h a v i o u r o f t h e l a t t e r i s n o t e n t i r e l y o b v i o u s s i n c e , i n t h e l i m i t u n d e r c o n s i d e r a t i o n , 2 k g o e s t o i n f i n i t y a l o n g t h e c o n t o u r o f i n t e g r a t i o n u s e d i n d e f i n i n g D ^ ( k ) . H o w e v e r , f r o m e q u a t i o n ( 3 . 1 7 ) a n d t h e w o r k o f L a n z ( 1 9 6 4 ) , i t f o l l o w s t h a t D ^ ( k ) = 1 + 0 ( k - 1 ) a s k - + ( 6 . 8 ) 2 H e n c e , t h e b e h a v i o u r o f a ^ ( k ) i s i d e n t i c a l t o t h a t o f ( k ) 2 2 a 3 ' f o r l a r g e k 5 t h a t i s , i t a p p r o a c h e s 0 ( ( k ) ) . S i n c e t h i s i s a d e c r e a s i n g f u n c t i o n , t h e i n t e g r a n d i n ( 6 . 7 ) a l s o h a s t h e same b e h a v i o u r a s V ^ 2 ( k ) f o r l a r g e k. H e n c e , t h e i n t e g r a n d i s 0 ( ( k 2 ) ~ a ~ ^ ) ; t h e i n t e g r a l c o n v e r g e s f o r - a -4 < - 1 o r a > - 3; a n d a g a i n no new r e s t r i c t i o n s o n a a r e n e c e s s a r y . A s w a s s h o w n i n t h e d e r i v a t i o n o f t h e c o s (©•) - s p e c t r a l r e p r e s e n t a t i o n , t h e c o n d i t i o n f o r c o n v e r g e n c e o f t h e i n t e g r a l i n 33 6.5 (6.7) i s s u f f i c i e n t for disappearance of the contribution of the i n f i n i t e c i r c l e to the Cauchy integral for a^(k). The re s u l t i s that the cut structure of a^(k) may be v a l i d l y represented by a^(k) = girl f ^ll y ( k.2) + / m£i J ( k,2," where ^ i s given by expression (6.2) and }f+ by (6.6). S u b s t i t u t i n g i n t o equation (5.6) and interchanging the order of i n t e g r a t i o n over t and k 2 yeilde the re s u l t f ( k 2 , cos <•» = f U 2 , t ) t=l q = -(x ^ 00 t-cos(fr) 2 ^ Q + ( q 2, t) + Regge t=l q =0 k -q pole terms (6.9) where - l / 2 + i » Q _ ( k 2 , t ) = - -L- j <\i (2-t+l) P ( t ) ( a ( y k ' 2 + i€)-a^ ( y k ^ 2 - i € ' ) ) 4 t t -1/2-1-- l / 2 + i « = " -h J « ( 2 , + l ) P 4 ( t ) l - e x p ( T y ( a + 2 ) ) 4 7 T -1/2-ico * k a f c T ( a + W ) i ' ' ' P ( a+2) k f x " 8 ' " 2  2 ^ + l ^2a+3 p ^ a+£+3xjr^ a+-t-+4x. " K .2' I-a -su-l-1 , k2+u_2 s"]2 (6.10) 2 x p ( i = a f zazidL. _ a _ 1 ; k ^ u _ } 34 6.6 a n d 1 / 2 + i 03 e + (k 2 , t ) = - i 4 7 T l/2-i« d£ (2-t + 1 ) P ^ ( t ) ( a ^ ( k + i € ) - a ^ ( k - i € ) ) 1 2 l/2-i» ( 6 . 1 1 ) B y t h e same m e t h o d a s was u s e d i n d i s c u s s i n g f o r m u l a ( 6 . 3 ) i t may b e s h o w n t h a t t h e i n t e g r a l s i n e q u a t i o n s ( 6 . 1 0 ) a n d ( 6 . 1 1 ) T h e i n t e g r a l o f t h i s l a t t e r q u a n t i t y h a s b e e n s h o w n t o c o n v e r g e f o r a < - 9/4 i n t h e d i s c u s s i o n f o l l o w i n g e q u a t i o n ( 5 . 7 ) . T h e v a l i d i t y o f e q u a t i o n ( 6 . 9 ) i s t h u s p r o v e n . T h u s , by- o m i t t i n g p o l e t e r m s f r o m t h e d i s p e r s i o n i n t e g r a l , t h e n e e d f o r a n y s u b t r a c t i o n s i s e l i m i n a t e d . T h e q u e s t i o n a r i s e s o f w h e t h e r o r n o t t h e l e f t - h a n d c u t s o f t h e p o l e t e r m s may c a n c e l t h a t o f t h e d i s p e r s i o n i n t e g r a l i n o t h e r t h a n s p e c i a l c a s e s . T h i s p o s s i b i l i t y may b e o b v i a t e d b y m e a n s o f t h e c o n s i d e r a t i o n s m e n t i o n e d a t t h e e n d o f c h a p t e r 5. T h e c o n t r i b u t i o n t o t h e a m p l i t u d e d e c r e a s e s , a n d t h a t o f m o s t p o l e t e r m s i n c r e a s e s , w i t h i n c r e a s i n g Re(cos(&))« T h u s , i f t h e c o n t r i b u t i o n t o t h e c u t f r o m t h e i n t e g r a l h a p p e n s t o c a n c e l t h a t f r o m t h e p o l e s f o r o n e a r e b o t h 0 ( ( 2 ^ + 1 ) P ^ ( t ) a ^ ( k ) ) a s I -» + i ». 35 6 0 7 p a r t i c u l a r v a l u e o f c o s t h e l a t t e r w i l l y e t d o m i n a t e t h e f o r m e r f o r a s u f f i c i e n t l y l a r g e v a l u e o f c o s 36 7.1 C h a p t e r 7„ T h e O r i g i n o f t h e " E x t r a " B r a n c h C u t i n t h e k P l a n e . I n t h i s c h a p t e r , t h e i n v e s t i g a t i o n b y B o t t i n o ( 1 9 6 2 ) o f t h e a n a l y t i c p r o p e r t i e s o f t h e s c a t t e r i n g a m p l i t u d e a s a f u n c t i o n o f c o m p l e x k f o r f i x e d a n g l e s h a l l b e f o l l o w e d i n a m o r e g e n e r a l f a s h i o n , s u i t a b l e f o r b o t h l o c a l a n d n o n l o c a l p o t e n t i a l s c a t t e r i n g . I t t u r n s o u t t h a t i n t h e n o n l o c a l c a s e t h e s c a t t e r i n g a m p l i t u d e h a s a c u t n o t o n l y a l o n g t h e n e g a t i v e i m a g i n a r y k a x i s , b u t a l s o a l o n g t h e p o s i t i v e i m a g i n a r y a x i s ; t h a t i s , t h a t i t h a s a c u t o n 2 b o t h s h e e t s o f t h e c o m p l e x - k p l a n e . T h e r e a s o n f o r t h i s d i f f e r e n c e i s s h o w n i n d e t a i l b e l o w . T h e r a d i a l S c h r o e d i n g e r e q u a t i o n s a y s CO i|j "U) + ( i ^ ± i i + k 2 ) * ( z ) + dz« z « 2 V ^ ( z , z ' ) i | ( ( z ! ) = 0. 2 0 F o r l a r g e z, t h e c e n t r i f u g a l t e r m d i s a p p e a r s , a n d i f t h e p o t e n t i a l V ( z,z') d i s a p p e a r s f a s t e r t h a n \|t(z) f o r l a r g e z , t h e n t h e e q u a t i o n a p p r o a c h e s t h e f o r m * " ( z ) + k 2 f ( z ) = 0. « T h u s , f o r l a r g e z , t h e g e n e r a l s o l u t i o n a p p r o a c h e s t h e f o r m A e x p ( i k z ) + B e x p (— i k z ) . F o r a p o t e n t i a l s a t i s f y i n g "VXzj.z') = «( e x p ( - |j, z ) ) 37/ 7.2 f o r l a r g e z, t h e s o l u t i o n w i l l t h u s a l w a y s a p p r o a c h t h e a b o v e f o r m f o r |lm ( k ) | < P o u r s o l u t i o n s o f t h e r a d i a l S c h r o e d i n g e r e q u a t i o n may b e d e f i n e d ( f o r c o n v e n i e n c e s , t h e s y m b o l X = I + 1/2 w i l l b e u s e d i n t h e f o l l o w i n g ) q> ( X ? k, z ) , cp (-X, k, z ) , f ( X , k, z ) , f ( X , ke""'"', z ) ; o r , f o r t h e s a k e o f b r e v i t y , cp+, <p_, f + , f _ ; w h e r e <p(X, k , z ) - z X + 1 / 2 a s z - 0, f ( X p k j , z ) ~ e x p ( ~ i k z ) a s z -* <*>. (7 .1) T h e f ' s a r e k n o w n a s t h e " J o s t s o l u t i o n s " . I n a c c o r d a n c e w i t h B o t t i n o ( 1 9 6 2 ) , t h e J o s t f u c t i o n s f ( + X j , + k ) a r e d e f i n e d b y t h e r e l a t i o n : g U l ; * , ^ ~ f ( ± ^ , k ) e x p ( - i k z ) + f ( ± X , - k ) e x p ( i k z ) a s z ^ . . ( 7 . 3 ) 2 i k 2 i k T h i s i s e q u i v a l e n t t o e q u a t i o n ( 2 . 9 ) o f B o t t i n o ( 1 9 6 2 ) . I t s h a l l p r o v e u s e f u l t o o b t a i n a n e x p r e s s i o n f o r t h e q u a n t i t y f ( X , k ) f ( - A , - k ) - f ( - X , k ) f ( X , - k ) i n t e r m s o f t h e f o u r s o l u t i o n s d e f i n e d i n ( 7 . l ) . I n B o t t i n o ( 1 9 6 2 ) i t i s shown t h a t , f o r t h e l o c a l c a s e , f ( X , k ) f ( - X , - k ) - f ( - X , k ) f ( X , - k ) = 4 i X k. ( 7 . 3 ) 38. 7.3 H o w e v e r , i t w i l l b e s e e n ; t h a t , i n t h e n o n l o c a l c a s e , t h i s i m p o r t a n t • i d e n t i t y d o e s n o t h o l d . Now l e t S i n c e f + ~ e x p (+ i k z ) a s z -» », t h e q u a n t i t i e s A,B,C, a n d D b e c o m e J o s t f u n c t i o n s a s z -* °°, p r o v i d e d t h e s o l u t i o n s 9 a n d f e x i s t . I n t h e l o c a l c a s e , t h e c o e f f i c i e n t s A,B,C, a n d D a r e c o n s t a n t s , , s i n c e a l i n e a r r e l a t i o n s h i p e x i s t s a m ong a n y t h r e e p a i r w i s e l i n e a r l y i n d e p e n d e n t s o l u t i o n s o f t h e l o c a l e q u a t i o n ; b u t w i t h a n o n l o c a l p o t e n t i a l t h i s may n o t b e t h e c a s e . T h e ¥r o n s k i a n o f t w o f u n c t i o n s , s a y g ( z ) a n d h ( z ) , i s d e f i n e d b y 1 (7.4) v ( g ( z ) , h ( z ) ) = g ( z ) Mai - h ( z ) M*l . Now l e t g ( z ) a n d h ( z ) b e t w o s o l u t i o n s o f t h e e q u a t i o n ; CO 0 00 h " ( z ) + 0 M u l t i p l y i n g t h e f i r s t e q u a t i o n b y h ( z ) a n d t h e s e c o n d b y g ( z ) a n d s u b t r a c t i n g y i e l d s CO _ d d z d z ' R ( z , z « ) [ g ( z ) h ( z ' ) - h ( z ) g ( z ' ) ] . 0 39 7.4 I n g e n e r a l , t h e r e f o r e , t h e W r o n s k i a n i s a f u n c t i o n o f z . I f , h o w e v e r p R ( z , z ' ) = R ( z ) 6 ( z - z l ) , a s i n t h e l o c a l c a s e , t h e n t h e W r o n s k i a n i s a c o n s t a n t . T h u s a p p e a r s a n i m p o r t a n t d i f f e r e n c e b e t w e e n t h e p r o p e r t i e s o f t h e s o l u t i o n s o f t h e l o c a l a n d n o n l o c a l S c h r o e d i n g e r e q u a t i o n s . P r o m e q u a t i o n s ( 7 . 4 ) , W ( f + > c p + ) - W ( f + f A f + + B f - ) a B W ( f + , f _ ) , W(f+ ?cp„) B W(f^<p+) ta W(f„5cp_) = D V ( f + , f J , - A W(f + ,£«.), - C W ( f + , f J . *P+ 8 8 ~ V ( £ + f * < p- - v i f + , f _ ) r+ + w ( f + , f _ ) r - • _/ ( 7 . 5 ) T a k i n g t h e l i m i t z -» °» i n e q u a t i o n s ( 7 . 5 ) a n d u s i n g e q u a t i o n s ( 7 . 1 ) a n d ( 7 . 2 ) r e v e a l s t h a t fU,±k) = l i m + W(f±,q>+) z — ~ W ( f + , f _ ) 2 i k and ( 7 . 6 ) f ( - * , ± k ) = l i m + W ( f + , y . ) 2 i k z — W ( f + , f _ ) T h e d e n o m i n a t o r s o f t h e s e q u a n t i t i e s may b e c o m p u t e d w i t h t h e a i d o f e q u a t i o n ( 7 , l ) s 40, 7.5 l i m z-»eo T h u s . ¥ ( f + , f _ ) = ¥ ( e x p ( - i k z ) , e x p ( i k z ) ) = 2 i k . ( 7 . 7 ) f ( X , + k ) = + l i m Y ( f + , c p + ) 2-»eo ~~ a n d y ( 7 . 8 ) f ( - X , + k ) = + l i m ¥ ( f + p c p _ ) z-*°° 7 T h i s s h o w s t h a t t h e J o s t f u n c t i o n s e x i s t p r o v i d e d f a n d cp e x i s t . One s e e s a l s o t h a t i n t h e l o c a l c a s e , t h e s e e x p r e s s i o n s a r e i d e n t i c a l w i t h t h e d e f i n i t i o n ( 2 . 1 ) o f B o t t i n o ( 1 9 6 2 ) . S u b s t i t u t i n g e q u a t i o n s ( 7 . 5 ) i n t o t h e e x p r e s s i o n ¥(<p+,cp_) y i e l d s V(<p+,<p_) = ¥ ( f+,<p+) V ( f_,<p_) _ V ( f_ f<p+) V ( f+,<p-) W ( f + , f . ) V ( f + , f _ ) ¥ ( f + , f _ ) ¥ ( f + p f ~ ) V ( f + , f „ ) a n d t h u s , u s i n g e q u a t i o n s ( 7 . 6 ) , f ( X , k ) f ( - \ , - k ) - f ( - A , k ) f ( X . - k ) = l i m W(<p+.q>-)  0 z-» ¥(f+,f-J ( 2 i k ) 2 = -kr- l i m ¥(<p +,q>_), ( 7 . 9 ) u s i n g a l s o (7,7). I n t h e l o c a l c a s e , ¥(cp+„cp_) i s c o n s t a n t , a n d t h e n l i m ¥(<p+;cp„) = l i m V(q>+,tp_) = ¥ ( z A + 1 / 2 , Z ~ A + 1 / 2 ) = - 2\ ( 7 . 1 0 ) Z-.CO Z"*0 4 1 7.6 s o t h a t f ( X , k ) f ( - X , - k ) - f ( - X , k ) f ( X , - k ) = 4 i X k , ( 7 . 1 1 ) w h i c h i s e q u a t i o n (2,5) o f B o t t i n o (1962). T h u s , i n t h e l o c a l c a s e , t h e q u a n t i t y f ( X , k ) f ( - X , - k ) - f ( - X , k ) f ( X , - k ) i s a n a l y t i c over t h e e n t i r e X a n d k p l a n e s , e v e n k w h e n t h e functions cp+ a n d f + a r e n o t . U s i n g t h e u s u a l r e l a t i o n f o r t h e > S m a t r i x , w h i c h may b e d e r i v e d i n the nonlocal c a s e i n e x a c t l y t h e same m a n n e r a s i n t h e l o c a l c ase,, S ( X , k ) = | | x ^ ) e x P UTT (X - 1/2)) , a s s u m i n g t h e S o m m e r f e l d - V a t s o n t r a n s f o r m a t i o n may b e p e r f o r m e d - ( T h e v a l i d i t y of t h i s s t e p i n t h e l o c a l Y u k a w a c a s e i s s h o w n b y B o t t i n o ( 1 9 6 2 ) , i n a p a r t i c u l a r n o n l o c a l c a s e i n e a r l i e r c h a p t e r s ) y i e l d s , x r X P. , / 9 ( c o s ( G ) ) f ( E , c o s ( e ) ) = ~ J dX c o s i r X ) ' e x p ( - i / r ( X + l / 2 ) ) ( S ( X , k ) - l ) —i» + p o l e t e r m s + i c o s ( ^ X ) ] + p o l e t e r m s . ( 7 . 1 2 ) Now o n l y t h a t c o m p o n e n t of t h e i n t e g r a n d w h i c h i s a n e v e n f u n c t i o n o f 42 7 , 7 X c o n t r i b u t e s t o f ( E , cos(G-)). A c c o r d i n g t o E I 3.3.1 ( l ) , ( c o s ^ s a n e v e n f u n c t i o n o f X; f u r t h e r , c o s ( r r X ) i s f a m o u s f o r i t s e v e n n e s s , a n d X i s t h e v e r y p a r a g o n o f o d d i t y . T h e r e f o r e , X P X - l / 2 t c o s ^ c o s (TT X ) i s o d d , a n d o n l y t h e o d d p a r t o f (- H x ^ T k ) - s i n (TTX ) + i c o s (TTX) ) c o n t r i b u t e s . I n t a k i n g t h e o d d p a r t , i c o s (^X) d r o p s o u t , s i n ( 7 r X ) i s u n a l t e r e d , a n d - f ( X , k ) / f ( X , - k ) b e c o m e s , u s i n g e q u a t i o n ( 7 . 9 ) , l i m f ( X . k ) f ( - A . - k ) - f ( - X , k ) f ( X , - k ) = z-»°° V ( ( p + t q > . ) _e ( 7 > 1 3 ) 2 f ( - X , - k ) f ( X , - k ) 4 i k f ( - X , - k ) f ( X , - k ) * T h u s , 1 X P x / ( c o s ( ^ ) ) . f ( E , c o s ( s ) ) = - ^ j dx x ; t w x ) ( s i n ( 7 r X ) -i=° l i m + z~°° ¥ ( ( P + ? T - ) ) + p o l e t e r m s . ( 7 . 1 4 ) 4 i k f ( - A , - k ) f ( X , - k ) ( T h e p o l e t e r m s a r e i n e x p l i c a b l y m i s s i n g f r o m e q u a t i o n ( 6 . 1 6 ) o f B o t t i n o ( 1 9 6 2 ) ) . I n t h e l o c a l c a s e , e q u a t i o n ( 7 . 1 0 ) may b e u s e d t o show t h a t l i m V ( c p + , c p _ ) = - 2 X , a n d t h u s t h a t t h e o n l y s o u r c e o f s i n g u l a r i t i e s Z - » C D 43 7.8 i n t h e k p l a n e f r o m t h e i n t e g r a l i n ( 7 . 7 ) a r e t h e J o s t f u n c t i o n s i n t h e d e n o m i n a t o r . I n t h e n o n l o c a l c a s e , h o w e v e r , t h e 9 f u n c t i o n s i n t h e n u m e r a t o r may c o n t r i b u t e a d d i t i o n a l s i n g u l a r i t i e s t o t h e s c a t t e r i n g a m p l i t u d e . P r o m e q u a t i o n s ( 7 . 8 ) , i t may b e s e e n t h a t t h e d o m a i n o f a n a l y t i c i o f f ( + X, - k ) i s t h e i n t e r s e c t i o n o f t h e d o m a i n s o f f _ a n d cp + i n t h e X a n d k p l a n e s . I n t e g r a l e q u a t i o n s s h a l l now b e s e t u p f o r t h e s o l u t i o n s cp a n d f i n t h e same m a n n e r a s i n A p p e n d i x I o f t h e B o t t i n o e f f o r t , a n d s h a l l b e u s e d t o show a n a l y t i c i t y o f t h e s o l u t i o n s i n a r e s t r i c t e d p o r t i o n o f t h e X a n d k p l a n e s . E a c h o f t h e i n t e g r a l e q u a t i o n s s h a l l b e w r i t t e n i n t h e f o r m T h i s i s i d e n t i c a l i n f o r m t o e q u a t i o n ( l I . l) i n B o t t i n o ( 1 9 6 2 ) , T h u s , o n e may a p p l y t o t h e p r e s e n t c a s e a l s o t h e i r s u b s e q u e n t a r g u m e n t t o t h e e f f e c t t h a t , f o r t h e f u n c t i o n g ( X , k, z ) t o b e a n a l y t i c i n a g i v e n r e g i o n , i t i s s u f f i c i e n t f o r t h e f r e e e i g e o f u n c t i o n s o f t h e o r i g i n a l e q u a t i o n , i.e« g Q , t o b e a n a l y t i c i n t h e r e g i o n , a n d f o r a n u p p e r b o u n d t o e x i s t f o r t h e i n t e g r a l CO 0 CO ( 7 . 1 6 ) 0 w h e r e \gQ ( A , k, k, z ) f o r a l l X, k, z i n t h e r e g i o n . 44 7.9 The i n t e g r a l e q u a t i o n f o r ^ ( r ) d e r i v e d b y M c M i l l a n ( 1 9 6 3 ) f o r t h e n o n l o c a l c a s e i s 00 09 t ^ ( z ) = j ^ k z ) - i k ^ d z » z » 2 ^ d z ' z ' 2 j ^ ( k z " < ) h ^ ( l ) ( k z 1 ^ ) 0 0 V^(z%z«) ,^(z») w h e r e z " ^ = m i n (z,z")» z =s max ( z , z " ) . T h u s , f o r s m a l l z , 00. CO f ^ ( z ) » ^ ( k z M l - i k y dz" z " 2 ^ z ' 2 h ^ ^ t k z " ) V ^ z ^ z ' ) 0 0 i l r ^ U ' ) ) ( 7 . 1 7 ) r r 1 / 2 z 1 a n d s i n c e j ^ ( z ) ~ 2 f t £ + 3 / 2 ) ^ a S Z ~* °* I l / 2 °° °° M z ) ~ z " I Z + T rtftW ^ i k / « i z " z " 2 ^ ~ d z - z ' 2 h / ^ k z " ) ^ 0 0 V ( ( z " , z ' ) | ( ( z " ) ) a s z - 0. I T h e c o e f f i c i e n t o f z i s n o t a f u n c t i o n o f z . T h u s , s o l u t i o n ^ ( z ) o f e q u a t i o n ( 2 . 2 0 ) may b e i d e n t i f i e d w i t h <p(X, k, z ) u p t o a n o r m a l i z a t i o n c o n s t a n t , w h i c h i s i m m a t e r i a l f o r p r e s e n t p u r p o s e s . F o r t h e p u r p o s e o f d e r i v i n g t h e a n a l y t i c p r o p e r t i e s o f cp i n t h e n o n l o c a l c a s e , f o r m u l a ( 7 . 1 6 ) b e c o m e s CO 00 /k/ ^ d z ' ^ d z " j z " 2 z « 2 ^ ( k z 1 ^ ) h ^ ^ U z ' ^ ) V ^ ( z " , z « ) 0 0 W M U , k, z) xglf' l< ( 7 . 1 8 ) 45 7 . 1 0 w h e r e M(-t, k, z ) > j j ^ ( k z ) [ . j , ( k z ) h a s s i n g u l a r i t i e s o n l y f o r k z = 0 a n d k z = 0 0 (EII p a g e 4 ) , a n d t h e p o t e n t i a l w i l l b e a s s u m e d t o be e q u a l l y w e l l b e h a v e d i n b o t h s p a c e v a r i a b l e s ; h e n c e , t h e c o n v e r g e n c e o f t h e i n t e g r a l n e e d b e i n v e s t i g a t e d o n l y f o r t h e e n d p o i n t s o f t h e i n t e g r a t i o n s . A n y l o c a l c o m p o n e n t o f t h e p o t e n t i a l , w h i c h w i l l m a n i f e s t i t s e l f a s ,a c o n t r i b u t i o n t o t h e n o n l o c a l p o t e n t i a l o f t h e f o r m V ( z " ) 6 ( z " - z 1 ) / z " , may b e t r e a t e d s e p a r a t e l y , a n d w i l l y i e l d t h e r e s u l t o f B o t t i n o ( 1 9 6 2 ) , i . e . t h a t t h e <p s o l u t i o n i s a n a l y t i c f o r a l l k a n d f o r R e ( X ) > 0 . F i r s t , t o f i n d c o n d i t i o n s f o r c o n v e r g e n c e o f t h e i n n e r i n t e g r a l ; u s i n g ( 7 . 1 7 ) a n d s t i p u l a t i n g k ^ 0 , t h e i n t e g r a n d b e c o m e s 0 ( z " 2 znl V ( z " , z ' ) ) a s z " - 0 . F o r V ( z " , z ' ) = ^ ( z " ~ ^ 2 ) , t h i s p o r t i o n o f t h e i n t e g r a l c o n v e r g e s f o r Re(l) > - 1 / 2 , o r R e ( X ) > 0 . F o r t h e i n f i n i t e t a i l o f t h e i n n e r i n t e g r a l , f r o m M o r s e ( 1 9 5 3 ) , p a g e 6 2 2 , h ^ z ) ~ \ e x p ( i ( z - J (i+l))) a s z -» °>. ( 7 . 1 9 ) T h e r e f o r e , t h e i n t e g r a n d i s 0 ( z " 2 z " " 1 e x p ( i k z " ) V ( z , , , z 1 ) ) a s z " - «. F o r V ( z " , z ' ) = 0 ( e x p ( - [i z " ) ) a s z " -» ro, t h e i n t e g r a n d i s 0 ( z " e x p ( z " ( i k - j i ) ) ) . 7 . 1 1 T h e r e f o r e , t h e i n t e g r a l c o n v e r g e s f o r R e ( i k - u.) < 0, o r Ira(k) > -u.. T h u s , t h e i n n e r i n t e g r a l i s f o u n d t o c o n v e r g e f o r R e ( X ) > 0, Im ( k ) > - u.. F o r t r e a t m e n t o f t h e o u t e r i n t e g r a l , l e t M U , k, z') = /d^Ckz')). (7.20) T h u s , p r o v i d e d t h e i n n e r i n t e g r a l c o n v e r g e s , t h e i n t e g r a n d o f t h e o u t e r i s 0 ( z | 2 | j ^ ( k z ' ) | V j z " , z > ) ) a s z '- 0. P r e c i s e l y a s 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 i n t e g r a l ; f o r V ( x " , z ' ) = j e r ( z ' ~ ^ / 2 ) , t h e i n t e g r a l c o n v e r g e s f o r R e ( A ) > 0. The i n f i n i t e t a i l o f t h e o u t e r i n t e g r a l p r o v i d e s a s t r e n g t h e n i n g o f t h e r e q u i r e m e n t s . M U , k , z ) = | j ^ ( k z ) J ~ ~ c o s ( z - ^ U + l ) ) / a s z - ». = ( e x p ( i ( k z - J U + l ) ) ) + e x p ( - i ( k z - | U + l ) ) ) ) | a n d t h e i n t e g r a n d i s 0 ( z ' 2 max ( | e x p ( i k z ' ) | , / e x p ( - i k z 1 ) ( ) • V ( z " , z * ) ) . , F o r V ( z " , z ' ) = 0 ( e x p ( - \iz')) a s z ' -• 0 0, t h i s q u a n t i t y i s 0 ( z ' 2 max( ) e x p ( z ' ( - [x + i k ) ) / , f e x p ( z ! ( - u. - i k ) ) | ) ) . T h e r e f o r e , i t c o n v e r g e s f o r - fi, < I m ( k ) < j i . 47 7,12 T h u s , t h e f u n c t i o n cp i s a n a l y t i c f o r R e ( X ) > 0, k ^ 0, a n d - [i, < I m ( k ) < p r o v i d e d t h e p o t e n t i a l i s n o m o r e s i n g u l a r t h a n - 5 / 2 z ' a s e i t h e r s p a c e v a r i a b l e a p p r o a c h e s 0, a n d p r o v i d e d i t d e c a y s l i k e e x p ( - \iz) a s e i t h e r v a r i a b l e a p p r o a c h e s °°. T h i s c o n t r a s t s w i t h t h e r e s u l t o f B o t t i n o ( 1 9 6 2 ) f o r t h e l o c a l c a s e , i n w h i c h cp w a s f o u n d t o b e a n a l y t i c o v e r t h e e n t i r e k p l a n e ( w i t h t h e same r e s t r i c t i o n o n X ) . T h i s r e s u l t a p p l i e s t o a l o c a l p o t e n t i a l w h i c h i s &{z~ ) a s z -• 0, T h e r e a s o n f o r t h e l o s s o f a n a l y t i c i t y i s t h a t , i n t h e l o c a l c a s e , a n i n t e g r a l e q u a t i o n f o r t h e s o l u t i o n cp(z) may b e f o u n d w h i c h i n v o l v e s o n l y a n i n t e g r a t i o n f r o m 0 t o z , ( S e e , f o r e x a m p l e , e q u a t i o n ( 1 , 4 ) o f B o t t i n o ( 1 9 6 2 ) ) . I n t h e n o n l o c a l c a s e , h o w e v e r , t h e r e i s a n e x t r a i n t e g r a l , t h e p o t e n t i a l i n t e g r a l i t s e l f , w h i c h r u n s a l l t h e w a y f r o m 0 t o ». H e n c e , e x t r a r e s t r i c t i o n s m u s t b e i m p o s e d t o e n s u r e c o n v e r g e n c e o f t h e i n f i n i t e t a i l . A n i n t e g r a l e q u a t i o n f o r t h e J o s t s o l u t i o n i s C O f ( X , k , z ) = e x p ( - i k z ) - d ^ ( e x p ( i k ? ) e x p ( - i k z ) z - e x p ( - i k ? ) e x p ( i k z ) ) ca 2 d ? ' ( X 6 ( ? - ? ' ) + ? , 2 V ( ? , g ' ) f ( X , k, ? • ) ) x " 0 5 C O C O = e x p ( - i k z ) - J Z k f d ? ' f ( X , k , 5 ' ) d ? ( e x p ( i k ? ) e x p ( - i k z ) 0 - e x p ( - i k ? ) e x p ( i k z ) ) x ( * % ± ^ 6(5-?') + V2 V(5,? ' )) . % 48 7 . 1 3 T h i s may b e d e r i v e d i n e x a c t l y t h e same m a n n e r a s t h a t u s e d i n a p p e n d i x I o f B o t t i n o ( 1 9 6 2 ) t o d e r i v e h i s e q u a t i o n ( l . 6). I t i s i d e n t i c a l i n f o r m t o e q u a t i o n ( 7 . 1 5 ) w i t h CO L = - 2~fk d ? ( e x P ( ik£) e x p ( - i k z ) - expH-ktj) e x p ( i k z ) ) z 2 I F o r f ( A , k, z) 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 there to e x i s t an upper bound f o r the i n t e g r a l OS CO J~ d ? ' ^ d? / (exp( ikS) e x p ( - i k z ) - e x p ( - i k O e x p ( i k z ) ) ? , 2 ( 0 z k ; l]] * / « ( M ' ) + ? ' 2 v ( 5 f ? ' ) ) | ( 7 . 2 1 ) where M(X, k , ? ' ) = | e x p ( - i k ? • ) / . T h e c e n t r i f u g a l c o n t r i b u t i o n m u s t b e c o n s i d e r e d s e p a r a t e l y b u t , s i n c e i t i s i d e n t i c a l t o t h e c o r r e s p o n d i n g c o n t r i b u t i o n i n t h e l o c a l c a s e , B o t t i n o ' s d i s c u s s i o n i n a p p e n d i x I I s h o w s t h a t i t s c o n t r i b u t i o n i s bounded f o r a l l A a n d l m ( K ) < 0. A n y l o c a l c o m p o n e n t o f t h e p o t e n t i a l , w h i c h w i l l b e m a n i f e s t e d a s a s i m i l a r d e l t a - f u n c t i o n t e r m , may a l s o b e i n c l u d e d i n t h i s r e s u l t p r o v i d e d i t i s n o m o r e _2 s i n g u l a r t h a n z a t t h e o r i g i n . T h e r e m a i n i n g i n t e g r a l o f e q u a t i o n ( 7 . 2 l ) i s n o t s i n g u l a r e x c e p t a t % = », ?' = °°, a n d ? ' = 0, a s s u m i n g t h e p o t e n t i a l i s w e l l 49 7.14 b e h a v e d a t a l l o t h e r p o i n t s i n t h e i n t e g r a t i o n . C o n s i d e r i n g f i r s t t h e i n n e r i n t e g r a l , t h e i n t e g r a n d i s 0 ( m a x ( | e x p ( i k ? ) / , | e x p ( - i k ? ) ( ) V ( ? , ? ' ) ) a s ? - <». I f V ( ? , ? « ) = 0 ( e x p ( - ? ) ) a s $ - », t h e q u a n t i t y i s 0(max ( / e x p ( ( i k - [i) ? )| , | e x p ( ( - i k - | i ) I )| )). T h i s q u a n t i t y d e c a y s e x p o n e n t i a l l y , a n d h e n c e t h e i n t e g r a l c o n v e r g e s , f o r - [i, < Im ( K ) < \i. I f t h e i n n e r i n t e g r a l c o n v e r g e s , t h e n t h e o u t e r i n t e g r a n d i s 0 ( ? ' 2 V ( ? , ? ' ) ) a s ? ' - 0. I f V ( ? , ? ' ) = o ( ? ' " 5 / 2 ) a s ?' - 0, t h e n t h e b o t t o m e n d o f t h e o u t e r i n t e g r a l a l w a y s c o n v e r g e s . U n d e r t h e same s u p p o s i t i o n , t h e o u t e r i n t e g r a h d i s 0( e x p ( - i k S ' ) V ( ? , ? ' ) ) a s ?' - 0, w h i c h d e c a y s n i c e l y f o r I m ( k ) < |x. T h u s , c o n s i d e r i n g o n l y t h e n o n l o c a l c o n t r i b u t i o n , f ( A , k, z ) 50; 7.15 i s a n a l y t i c f o r a l l A, and f o r - \i < Im(k) < u. However, i n order that 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 l o c a l c o n t r i b u t i o n no more s i n g u l a r than i t may converge, onemust strengthen the r e s t r i c t i o n on k to - |x < Im(k) < 0. The regions of a n a l y t i c i t y of the s o l u t i o n s may be extended f u i t h e r . Let z = p e x p ( i o ) , p and a r e a l , and assume that the p o t e n t i 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. Then the r a d i a l Schroedinger equation may be w r i t t e n This i s j u s t the o r i g i n a l equation with a complex p o t e n t i a l and complex k. I f the p o t e n t i a l V(z,?) i s 0(exp(- u. z)) f o r large z; i n order that the general s o l u t i o n may approach the form i t i s necessary to impose the r e s t r i c t i o n \a\ <7T/2 i n order that the l a s t term may disappear f o r large z. As before, four s o l u t i o n s may be defined. The new cp s o l u t i o n has the behaviour CO p 0 V(p exp(io),z') #(z') = 0. ( 7 . 2 2 ) tji' = a(o) exp(-ikp) + P(o) exp(ikp), cp'U, k', p) A+l/2 The new Jost s o l u t i o n i s defined by 51-7.16 f'(X,k ' p ) exp(-i k' p) where k' = k exp(io). The o r i g i n a l Jost solution f(X, k, z ) , defined for z r e a l , may-be a n a l y t i c a l l y continued into the complex-z plane provided the potential i t s e l f may be so continued. Its behaviour w i l l then become f(X, k, z) ( z y ^ a ( o ) exp(-ikz) + P ( o ) exp(ikz). Since f(X, k, z) exp(-ikz) i s analytic i n the complex-z plane, oc(c) exp(-2 ikz) + P(o) must be analytic for large p. For Im(k) < 0 , the f i r s t term disappears for large /z| and P(o) i t s e l f must be analytic for large p. Since i t i s not a function of p, i t must therefore be a constant. Since B(o) = 0 , for p = 0 , f(X, k, z) a ( o ) exp(-ikz). Since exp(ikz) i s i t s e l f analytic, a(o) must also be analytic, and therefore constant. .'. f(X, k, z) —~~^exp(-ikz) z for a l l z} .'. f(X, k, z) = f'(X, k», p) . (7.23) The situation for the cp solutions i s much simpler. The boundary conditions for cp and cp1 are i d e n t i c a l up to a phase factor, 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 S u b s t i t u t i n g f r o m e q u a t i o n s ( 7 . 2 l ) a n d ( 7 . 2 2 ) i n t o e q u a t i o n ( 7 . 8 ) , a n d u s i n g t h e b i l i n e a r i t y o f t h e W r o n s k i a n , f ' U , k « ) = e x p ( - c ( X + l / 2 ) ) f U , k ) , ( 7 . 2 5 ) ( w h e r e f ' ( X , k ' ) i s t h e J o s t f u n c t i o n f o r e q u a t i o n 7 . 1 2 ) . T h u s , t h e o l d J o s t f u n c t i o n i s a n a l y t i c i f t h e new o n e i s . T h e o n l y c h a n g e i n t r o d u c e d i n t h e p r e v i o u s a n a l y s i s o f a n a l y -t i c i t y o f t h e v a r i o u s s o l u t i o n s b y m a k i n g z c o m p l e x i s t h a t , i n o r d e r t o o b t a i n c o n v e r g e n c e o f t h e i n f i n i t e t a i l s o f t h e i n t e g r a l s i n v o l v e d , t h e q u a n t i t y w h i c h m u s t d e c a y e x p o n e n t i a l l y i s n o t e x p ( z ( - [i + i k ) ) a s z-»°°, b u t e x p ( p ( c o s ( c ) + i s i n ( o ) ) ( - \x, + i ( k ) ) a s p-* 0 0. T h e r e a l p a r t o f t h e c o e f f i c i e n t o f p i s R e ( ( c o s ( o ) + i s i n (a)) (- (j, + i k ) ) = - u. cos(o) + ( c o s ( o ) I m ( k ) 4 - s i n (o) R e ( k ) ) . T h u s , t h e s o l u t i o n s cp a n d f a r e a n a l y t i c o n l y i f t h e r e e x i s t s a o* s u c h t h a t \a\ <'*T/2, a n d - fi < I m ( k ) + R e ( k ) tan(cr) < u . ( 7 . 2 6 ) t a n ( a ) may t a k e o n a n y f i n i t e , r e a l v a l u e . T h u s , (7<$26) c a n a l w a y s b e s a t i s f i e d w h e n R e ( k ) ^ 0. I f R e ( k ) = 0, i t r e d u c e s t o - [i < I m ( k ) < \i. T h u s , t h e r e r e m a i n i n g e n e r a l b r a n c h l i n e s f o r e a c h s o l u t i o n a l o n g 53 7.18 b o t h i m a g i n a r y a x e s f r o m + i \i t o + i °°. P r e c i s e l y a s i n t h e c a s e d i s c u s s e d B o t t i n o ( 1 9 6 2 ) , t h e l o c a l a n d c e n t r i f u g a l c o n t r i b u t i o n s g i v e r i s e t o a n a d d i t i o n a l b r a n c h l i n e f o r t h e J o s t s o l u t i o n a l o n g t h e p o s i t i v e i m a g i n a r y k a x i s w h i c h may s t a r t e v e n c l o s e r t o t h e o r i g i n . T h u s i t i s s e e n t h a t t w o c i r c u m s t a n c e s c o n s p i r e t o e n g e n d e r t h e e x t r a b r a n c h c u t o f t h e s c a t t e r i n g a m p l i t u d e i n t h e c o m p l e x - k p l a n e s I T h e i n c o n s t a n c y o f a V r o n s k i a n o f t w o s o l u t i o n s o f t h e n o n l o c a l e q u a t i o n . I f t h e V r o n s k i a n o f t w o s o l u t i o n s w e r e c o n s t a n t , e q u a t i o n ( 7 . 1 0 ) c o u l d b e u s e d t o o b t a i n a n e x p r e s s i o n f o r ¥(cp+,cp_) w h i c h i s a n a l y t i c e v e n w h e n t h e cp s o l u t i o n s t h e m s e l v e s a r e n o t . T h u s , t h e n u m e r a t o r o f t h e p a r e n t h e t i c e x p r e s s i o n i n e q u a t i o n ( 7 . 1 4 ) c o u l d n o t h a v e a n y s i n g u l a r i t i e s . ( T h e d e n o m i n a t o r e v e n i n t h e l o c a l c a s e h a s a c u t a l o n g t h e n e g a t i v e i m a g i n a r y k a x i s . ) I I T h e e x t r a i n t e g r a l w h i c h i s n e c e s s a r y t o a c c o m m o d a t e t h e n o n l o c a l p o t e n t i a l . I n o r d e r t h a t t h e p o t e n t i a l i n t e g r a l i n f o r m u l a e ( 7 . 1 8 ) a n d ( 7 . 2 1 ) may c o n v e r g e , i t i s n e c e s s a r y t o i n t r o d u c e t h e r e s t r i c t i o n - ^< I m ( k ) <[j, s o t h a t n e i t h e r e x p ( i k z ) n o r e x p ( - i k z ) may e x p l o d e f o r l a r g e z f a s t e r t h a n e x p ( - \iz) damps t h e i n t e g r a n d . E v e n w h e n t h e a n a l y t i c c o n t i n u a t i o n b a s e d o n e q u a t i o n ( 7 . 1 2 ) i s e m p l o y e d , t h i s r e s t r i c t i o n i n g e n e r a l e n g e n d e r s b r a n c h c u t s a l o n g b o t h i m a g i n a r y k a x e s i n b o t h t h e cp a n d t h e f s o l u t i o n s . T h u s , b o t h t h e 54 7.19 n u m e r a t o r a n d t h e d e n o m i n a t o r o f t h e p a r e n t h e t i c e f f o r t i n e q u a t i o n ( 7 . 1 4 ) h a v e b r a n c h l i n e s a l o n g b o t h i m a g i n a r y a x e s . 55 8.1 C h a p t e r 8, Summary. I n t h i s t h e s i s , a d e t a i l e d i n v e s t i g a t i o n i s made o f t h e a n a l y t i c p r o p e r t i e s o f t h e p a r t i a l - w a v e a n d t o t a l a m p l i t u d e f o r s c a t t e r i n g b y a p a r t i c u l a r c l a s s o f s e p a r a b l e , n o n l o c a l p o t e n t i a l s . T h e a d v a n t a g e o f s t u d y i n g s e p a r a b l e p o t e n t i a l s l i e s i n t h e f a c t t h a t t h e p a r t i a l - w a v e a m p l i t u d e c a n b e w r i t t e n i n c l o s e d f o r m a s h a s b e e n s h o w n b y M c M i l l a n ( 1 9 6 3 ) , a n d t h e p a r t i c u l a r c l a s s o f t h e s e p o t e n t i a l s s t u d i e d h e r e i s p h y s i c a l l y r e a s o n a b l e f o r s t r o n g i n t e r a c t i o n s i n t h e s e n s e t h a t t h e p o t e n t i a l s i n v o l v e a n e x p o n e n t i a l d e c a y . T h e y a l s o y i e l d a r e l a t i v e l y s i m p l e f o r m f o r t h e p a r t i a l - w a v e a m p l i t u d e . The a n a l y t i c p r o p e r t i e s o f t h e p a r t i a l - w a v e a m p l i t u d e g i v e n b y M c M i l l a n ( 1 9 6 3 ) f o r t h e m o r e g e n e r a l c a s e h a v e b e e n s h o w n e x p l i c i t l y f o r t h e c a s e c o n s i d e r e d h e r e , b u t t h e p r e s e n t w o r k g o e s b e y o n d h i s b y p r o v i n g 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 f o r t h e t o t a l s c a t t e r i n g a m p l i t u d e s u s i n g a t e c h n i q u e i n v o l v i n g c o m p l e x a n g u l a r momentum f i r s t e x p l o i t e d b y R e g g e ( s e e B o t t i n o ( 1 9 6 2 ) ) . T h i s e x t e n s i o n h a s a l s o b e e n p e r f o r m e d b y M i t r a ( 1 9 6 3 ) a n d C u s h i n g ( 1 9 6 3 ) f o r a m o r e r e s t r i c t e d c l a s s o f s e p a r a b l e p o t e n t i a l s , b u t n e i t h e r h a s i n v e s t i g a t e d t h e c o n v e r g e n c e o f t h e s p e c t r a l i n t e g r a l s o b t a i n e d . T h e a n a l y t i c p r o p e r t i e s o f t h e s c a t t e r i n g a m p l i t u d e f o u n d f o r t h e c a s e c o n s i d e r e d h e r e a r e e s s e n t i a l l y t h e same a s t h o s e f o u n d f o r t h e l o c a l p o t e n t i a l c a s e b y , f o r e x a m p l e , B o t t i n o ( 1 9 6 2 ) , e x c e p t t h a t t h e t o t a l s c a t t e r i n g a m p l i t u d e i n t h e p r e s e n t c a s e c o n t a i n s a n e x t r a b r a n c h p o i n t t h e o r i g i n o f w h i c h , a s i s s h o w n i n d e t a i l i n t h e l a s t 56. 8.2 c h a p t e r , l i e s i n t h e f a c t t h a t a Wronsk i a n f o r ' t h e n o n l o c a l r a d i a l S c h r o e d i n g e r e q u a t i o n i s n o t i n g e n e r a l a c o n s t a n t , t o g e t h e r w i t h t h e f a c t t h a t t h e e x t r a i n t e g r a l w h i c h accommodates t h e n o n l o c a l p o t e n t i a l c o n v e r g e s o n l y f o r a r e s t r i c t e d c l a s s o f s o l u t i o n s . Thus, i t i s a g e n e r a l p r o p e r t y o f n o n l o c a l p o t e n t i a l s t h a t t h e s c a t t e r i n g a m p l i t u d e has a b r a n c h l i n e a l o n g t h e n e g a t i v e r e a l a x i s o f b o t h s h e e t s o f t h e c o m p l e x - e n e r g y p l a n e . 57 l o l A p p e n d i x 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 ) a n d Vk)-I n t h i s a p p e n d i x , i t i s shown t h a t t h e i n t e g r a l s CO V ^ k ) = J d r r 2 j ^ k r ) V ^ r ) ( I . l ) 0 a n d r - o ^ U ) D j k ) = / d q q ^ - \ j ( 1 . 2 ) Q k - q c o n v e r g e i f V j r ) = * ( r ." 5/2) a s r - 0 ( 1 . 3 ) a n d a s r -» » , ( l 0 4 ) The r e l e v a n t p r o p e r t i e s o f t h e s p h e r i c a l B e s s e l f u n c t i o n s a c c o r d i n g t o M o r s e ( 1 9 5 3 ) p a g e 1 5 7 3 a r e hM ~ 2 T ( W / 2 ) ( f > " a S Z " ° ( I ' 5 ) ~ — c o s ( z - (-t+1)) a s z -• <=°. ( 1 . 6 ) The c o n v e r g e n c e o f t h e i n t e g r a l i n ( I . l ) i s f i r s t c o n s i d e r e d . T h e w o r s t c a s e f o r c o n v e r g e n c e o f t h e l o w e r e n d i s w h e n t = 0; I n t h i s c a s e ( f o r s m a l l r ) , j ( k r ) a p p r o a c h e s a f i n i t e c o n s t a n t . T h u s , t h e r e q u i r e m e n t f o r c o n v e r g e n c e a t t h e l o w e r l i m i t i s r 2 V ^ ( r ) = «f(r°"^) a s r -» 0 , or 58 1.2 V.(p) = * ( r ~ 3 ) as r -* 0, (1.7) Since j.(kr) = 0(r~ 1) , _ r \ - i as r -» », the requirement for convergence at the upper l i m i t of ( I . l ) i s r V (r) =s «(r~^) as r •* <», V or V j r ) = #(r~ 2) as r «. (1.8) Convergence of the in t e g r a l i n (1.2) w i l l now be studied. It w i l l be found that somewhat stronger r e s t r i c t i o n s on the potential w i l l be necessary to ensure convergence. Prom (1,5) and (1.6), J , U ) < ^ ^ 0 < z < oo for I > 0, where C i s an arbitrary, f i n i t e constant and 0 < b < 1. Thus, CO l v k ) l 2 If dr r 2 C 0 (kr) b 'l V.(r) k (1.9) and D t(k)| < 1 + 0 f^j£ ( i ) 2b 0 ™ 2b Thus, the integrand i n (1.2) is" bounded by q , provided the integral i n (1,9) converges; that i s , provided V.(r) = * ( r b ~ 3 ) as r - 0 ( i . i o ) and as r -» °°. 59 1.3 F o r c o n v e r g e n c e a t t h e l o w e r l i m i t o f t h e i n t e g r a l i n ( l „ 2 ) , w h e r e t h e i n t e g r a n d may b e s i n g u l a r , t h e m o s t u n f a v o u r a b l e v a l u e o f 2 k i s z e r o , a n d t h i s v a l u e i s a s s u m e d i n t h i s p a r a g r a p h . T h e 2 b i n t e g r a n d i s t h e n 0 ( q ~ ) a s q 0, a n d t h e r e f o r e c o n v e r g e s f o r b < 1/2. T h u s , t h e i n t e g r a l i n ( 1 . 9 ) m u s t c o n v e r g e f o r some b s u c h t h a t 0 < b < 1 / 2 . W i t h t h e s e b o u n d s o n b , ( 1 . 1 0 ) l e a d s t o f"«(r° ) a s r -» 0 V ( r ) = ) - / 9 ( I . 1 1 ) * 0 ( r " 3 / i i ) a s 1 r - c o . C o n v e r g e n c e o f t h e i n f i n i t e t a i l o f ( 1 . 2 ) i s now s t u d i e d . / \ «=• 2b U s i n g a g a i n ( 1 . 9 ) , a s q « t h e i n t e g r a n d b e c o m e s b o u n d e d b y q , a n d t h e i n t e g r a l c o n v e r g e s f o r b > 1/2. T h u s , i n o r d e r t h a t t h e t a i l of ( 1 . 2 ) c o n v e r g e , i t i s s u f f i c i e n t t h a t ( 1 . 9 ) c o n v e r g e f o r some b sue h t h a t 1/2 < b <L 1. ( I . 1 0 ) t h u s l e a d s t o V , ( r ) = ^ ( r " 5 / / 2 ) a s r - 0 _ 2 ( 1 . 1 2 ) = jer(r~" ) a s r - * » . I n o r d e r t o s a t i s f y a l l t h e r e s t r i c t i o n s i m p o s e d i n ( 1 . 7 ) , ( 1 . 8 ) , ( I . 1 0 ) , a n d ( I . l l ) , V ^ ( r ) m u s t s a t i s f y V, ( r ) = < r ( r ~ 5 / 2 ) a s r - 0 1 ( 1 . 1 3 ) a n d a s r -» », a s g i v e n b y M c M i l l a n ( 1 9 6 3 ) . I n t h e l o c a l c a s e , b y a s i m i l a r a r g u m e n t , t h e r e q u i r e m e n t t u r n s o u t t o b e ™ 2 VA T ) = « ( r ~ ) a s r -» 0 a n d a s r -* <=. 60 I I c l Appendix I I . The Asymptotic Behaviour of P ^ ( t ) as Jt| -» o= 0 E I 3 o 9 , 2 ( 1 9 ) and E I 3 . 9 . 2 ( 2 0 ) y i e l d P ( t ) = max (0(tl)t ^ i i T 1 " 1 ) ) as It! - «. ( l i d ) when Belt) ?i 1 / 2 . In t h i s appendix, i t i s shown that t h i s l a t t e r c o n d i t i o n may be relaxed, EI 3 . 3 . 1 (8) says = t a r ^ u l (Q t(t) - Q_^_ 1(t)) » (II.2 ) a n d E l 3 « 9 < > 2 ( 2 1 ) y i e l d s Q ^ t ) - 0 ( t ^ " 1 ) as Itl - « ( I I . 3 ) f o r a l l - to Combining the two confirms equation ( 9 . 8 ) f o r Re{l)~ - l / ; a n d Im(-t) / 0 „ For the case I = - 1 / 2 , E I 3 . 1 4 ( 5 ) may be useds P ~ l / 2 ^ c o s ^ = n K ( s i n ( © - / 2 ) ) ( I I . 4 ) where K i s the complete e l l i p t i c i n t e g r a l of the f i r s t k i n d . S u b s t i t u t i n g i n t o E I I 1 3 , 8 ( 1 ) y i e l d s P - l / 2 ( l + 2 z ) = | y J i + z ' s i n 2 , ' ( I I » 5 ) 0 s p e c i a l i z i n g to the case z r e a l and p o s i t i v e . P , / o ( l + 2 z ) = !• 6 1 I I . 2 a s i n ( z = , l / 2 ) . 2 + _ / — i / dcp(~sin(cp) + — vising) ^"(^ - 0 ( z - 3 / 2 ) ) . (II.6) A l t h o u g h t h a t l a s t i n t e g r a n d i s s i n g u l a r a t cp = a s i n ( z ~ 1 ) b e c a u s e t h e b i n o m i a l e x p a n s i o n u s e d d o e s n ' t c o n v e r g e t h e r e , t h e s i n g u l a r i t y "1/2 i s r e m o v a b l e b e c a u s e t h e i n t e g r a n d a p p r o a c h e s 2 ' a t t h i s p o i n t . Thus, b o t h i n t e g r a l s a r e bounded and, u s i n g a s i n ( x ) x as x -* 0 , e q u a t i o n ( l l 0 l ) i s p r o v e n f o r t h i s c a s e a l s o . 62. I I I . l B i b l i o g r a p h y 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 T r a n s c e n d e n t a l F u n c t i o n s , M c G r a w - H i l l , 1 9 5 3 . B a t e m a n M a n u s c r i p t P r o j e c t , T a b l e s o f I n t e g r a l T r a n s f o r m s , M c G r a w - H i l l , 1 9 5 4 . B l a t t , J . M . , a n d V . J . V e i s s k o p f , T h e o r e t i c a l N u c l e a r P h y s i c s , W i l e y , 1952.* B o t t i n o , A., A.M. L o n g o n i , T. R e g g e , N u o v o C i m e n t o , 2 3 , 9 5 4 , 1 9 6 2 . C h u r c h i l l , R.V., C o m p l e x V a r i a b l e s a n d A p p l i c a t i o n s , M c G r a w - H i l l , 1 9 6 0 . C u s h i n g , J . T . , N u o v o C i m e n t o , 2 7 , 2 3 6 4 , 1 9 6 3 . D i c k e , R.H., a n d I . R . W i t t k e , I n t r o d u c t i o n t o Q u a n t u m M e c h a n i c s , A d d i s o n - W e s l e y , 1 9 6 1 . L a n z , L , a n d G.M, P r o s p e r i , N u o v o C i m e n t o , 3_3, 3 4 8 1 , 1 9 6 4 . L o m o n , E . a n d M. M c M i l l a n , A n n a l s o f P h y s i c s , 2 3 , 4 3 9 , 1 9 6 3 . M c M i l l a n , M., P h . D . T h e s i s , M c G i l l U n i v e r s i t y , 1 9 6 1 . M c M i l l a n , M. , N u o v o C i m e n t o J 29., 4 1 5 3 , 1 9 6 3 . M i t r a , A.N. , P h y s i c a l R e v i e w , 1 3 0 . 2 ^ 7 . 1 9 6 3 . M o r s e , P.M., a n d H. F e s h b a c h , M e t h o d s o f T h e o r e t i c a l P h y s i c s , M c G r a w - H i l l , 1 9 5 3 . 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 t h e o r y a n d R e g g e P o l e s , B e n j a m i n , 1 9 6 3 . S a l a m , A., D i r e c t o r , T h e o r e t i c a l P h y s i c s , I n t e r n a t i o n a l A t o m i c E n e r g y A g e n c y , V i e n n a , 1 9 6 3 . S q u i r e s , E . J . , N u o v o C i m e n t o , 2_5, 2 4 2 , . 1 9 6 2 , T a b a k i n , F. , A n n a l s o f P h y s i c s , 30_, 5 1 , 1 9 6 4 . 

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