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

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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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