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Relation between the Singularities of the *S* Matrix and the *L* ^{2} Class of Solutions in Potential Theory

### Abstract

Ma has shown that, in the case of an exponential potential, in the *s*‐wave *S* matrix there exist poles that do not contribute to the completeness, even though they appear in the same part of the *k* plane as the bound state poles. These poles were called ``redundant'' poles. Subsequently, other examples of redundant poles have appeared in the literature. Recently their importance with regard to the concept of ``shadow'' states has been stressed by Sudarshan. If we construct the Green's function for the Schrödinger equation with the boundary conditions of regularity at the origin and the outgoing spherical waves at large distances, the singularities of it in the *k* plane completely determine the *L* ^{2} class of eigensolutions. From this Green's function, the *T* matrix (and hence the *S* matrix) is constructed explicitly. This *S* matrix is found to be the same as the one defined through the usual Jost solution, and it is shown that the singularities of the *S* matrix, besides corresponding to those appearing in the Green's function, also contain the ``redundant'' singularities. Thus it is shown that the wavefunctions associated with the ``redundant'' singularities do not belong to the *L* ^{2} class. By a careful derivation, we resolve the Ma paradox concerning the Heisenberg identity. In the particular cases of the exponential potential and the Eckart potential, the redundant solutions correspond to the vanishing of a Wronskian pointing to the breakdown of the linear independence of the starting wavefunctions in defining the *S* matrix through the Jost functions. It is stressed that when singularities appear in the *S* matrix other than those corresponding to the bound and scattering states of the given problem, the ``redundant'' singularities can only be understood by a ``dynamical'' equation such as the Schrödinger equation.

© 1971 The American Institute of Physics

Received 15 October 1970
Published online 28 October 2003

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2003-10-28

2016-10-22

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