Singularities of the elastic S-matrix element

Abstract

It is well known that the standard conventional method of integral equations is not able to explain the analyticity of the elastic S-matrix element for the nuclear optical potential including the Coulomb potential. It has been shown[1],[2] that the cutting down of the potential at a large distance is essential to get rid of the redundant poles of the S-matrix element in case of an attractive exponentially decaying potential. This method has been found [3] to be quite general and it does not change the physics of the problem. Using this method , analiticity and the singularities of the S-matrix element is discussed. Singularities of the elastic S-matrix element Partial wave radial wave equation of angular momentum l corresponding to elastic scattering is given by, [ d2 - 2 /(/+ 1)] 2p [ . ] 2 + k - 2 u1(k,r)=-2 V(r)+ Vc(r)+ zW(r) u1(k,r) M r n (1) where V (r) is the real part of nuclear potential, W (r) is the imaginary part of the optical potential ��· (r) is the Coulomb potential, and k is the incident wave number. Energy dependence of the optical potential is usually through laboratory energy E1ah and hence it depend on k2 and therefore k2- 2 �� [V(r) + Vc(r) + i W(r)] is depending on k n through e . It is Well known that ��. (r) is independent Of k Jn Order tO make U 1 ( k, r) an entire function of k , we impose k independent boundary condition at the origin . Now, we can make use of a well known theorem of Poincare to deduce that the wave function is an entire function of k2 and hence it is an entire function of k as well. We cut off the exponential tails of the optical potential at sufficiently large Rm and use the relation -1 --d u1 =.:u ;<--l -(k,-r)--s--'-r (--k,R--m----') u-'-;(+l--- ( k,-r) u, dr u/-l(k ,r)-sr(k,Rm) uj+l(k,r) (2) to define St(k,Rm),where u,<-l(k,r) and u,<+l(k,r) stand for incoming and outgoing Coulomb wave functions respectively which are given by I (±l(k )-+· [r(/+ l+i 17) ]2 [J[2"+iU+n��J w (-2 'u1 ,r - _l . e 1 1k r ) r(Z+I-z17) +i,,/+2 (3) where Ware the Whittaker functions. In the limit Rm �� oo St (k,Rm) ,the nuclear part of the S-matrix element , becomes St { k) and the redundant poles removed[1 ],[2].Now, the nuclear S-matrix element , in terms of the Whittaker functions is given by where w' 1 (2ikr)-��(k,r) W I (2ikr) IIJ, 1+-2 in'" 1+-2 , ,r 2 Rm W. 1 (-2ikr)-��(-k,r)W 1 (-2ikr) -IIJ, I+-2 -ill ' 1+-2 P1(k,r) = u;(k,r) ,and St (k) has an essential singularity at k = 0, which u1(k,r) , (4) is apparent from the Wister's definition of the Gamma function l(z) smce z= l+ 1 ±i lJ .However, this singularity has no any physical meaning and is an outcome of treating 21Jk as well defined quantity for all k including k = 0 in the corresponding r Schrodinger equation .The infinite number of zeros and poles of S- matrix element due to the Gamma functions associated with S - matrix element have to be interpreted 1 carefully. S;'(k)=O at the zeros of ----­ f(l+1+i1J ) and then the total wave function reduces to [ . ]�� I J( '7 +i(/+I)ZZ"j uj-l(k,r)=-i f(l+1-��'7) e l 2 2 W (2ikr) f(l+1+zlJ) i11,1+l2 which is also zero. Even though the corresponding energies of these states are negative since the corresponding wave number is given by ? k= z · z,z2 e- 2 n= 0' 1' 2 , ... 11 (n+l+1) they are not physically meaningful bound states as found in[1],[2] long ago. These states are unphysical since poles are redundant poles. This fact is clearly understood by the fact that all these poles are absent in the physically meaningful total S - matrix element. For large 1k1, Sin' (k) �� (- ) { e-21k r S(k), where S(k) = [-ik+��(k)] • +2k [ik-��(-k)]. smce W = e- " for large k. Therefore the S-matrix element has an essential singularity at infinity, which is on the imaginary axis. It is clear that there are no redundant poles in the total S-matrix element is free from redundant poles sinceSJ (k) =SeS t , where Se = f(l + 1 + i7J) f(l+1-i1]) .