Analysis of the Spectral Singularities of Schrödinger Operator with Complex Potential by Means of the SPPS Method

Abstract

In this work we present an effective way of finding the spectral singularities of the one-dimensional Schrödinger operator with a complex-valued potential defined in the half-axis [0, ∞). The spectral singularities are certain poles in the kernel of the resolvent, which are not eigenvalues of the operator. In this work, the spectral singularities are calculated from the real zeros of ϰ (ϱ) = 0, where ϰ (ϱ) is an analytic function of the complex variable ϱ, which is obtained by means of the Spectral Parameter Power Series Method. This representation is convenient from a numerical point of view since its numerical implementation implies truncating the series up to a M-th term. Hence, finding the approximate spectral singularities is equivalent to finding the real roots of a certain polynomial of degree 2M. In addition, we provide explicit formulas for calculating the eigenvalues of the operator, as well as the eigenfunctions and generalized eigenfunctions associated to both the continuous spectrum and the spectral singularities.