On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential

Abstract

We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular $\delta$-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are studied by reduction to an integral equation, with the subsequent study of the spectral properties of the corresponding integral operator. We also construct an asymptotic formula for the eigenfunction at large distances. For this formula a physical interpretation from the point of view of wave scattering is given. Our approach can be used to deal with eigenfunctions in a broad class of related problems for the Schrödinger operator with singular potential. Bibliography: 17 titles.