### Abstract:

It is found that an essential singularity is introduced at the origin of the complex k-plane in
the S-matrix element in addition to the infinite number of zeros and poles apparently
introduced due to the Coulomb potential. It can be shown that the essential singularity at the
origin is a mathematical artifice and hence is unphysical. Necessary conditions for poles
corresponding to decaying resonance states, capture states and closed states can also be
derived using standard mathematical techniques in the presence of the Coulomb potential.
The partial wave radial wave equation of angular momentum l corresponding to elastic
scattering is given by,
[ ( ) ( ) ( )] ( , )
2
( , )
( 1)
2 2
2
2
2
u k r V r V r iw r u k r
r
l l
k
dr
d
l c l + + =
+
+ −
h
μ
where V(r) is the real part of the potential containing spin – orbit potential and volume
term, w(r) the imaginary part of the optical potential ,V (r) c the Coulomb potential and k is
the incident wave number The S-matrix element S (k) n
l is now can be written as
( 2 ) ( , ) ( 2 )
(2 ) ( , ) (2 )
( 1 )
( 1 )
( ) ( 1)
2
1
,
'
2
1
,
2
1
,
'
2
1
,
W ikr P k r W ikr
W ikr P k r W ikr
l i
l i
S k
i l
l
i l
i l
l
i l
n l
l − − − −
−
G + +
G + −
= −
− + − +
+ +
h h
h h
h
h
where
( , )
( , )
( , )
u k r
u k r
P k r
l
l
l
¢
= and m r ³ R , cutting off the potential tails at m R . It is clear that
there are infinite number of zeros and poles of the S-matrix element due to the fact that
k
z z e
2
2
1 2
h
μ
h = and the structure of the Gamma function. We have found that all salient
features of physically meaningful zeros and poles of S (k) n
l can be derived from the above
functional form of S (k) n
l .