Scattering and bound states for nonselfadjoint Schr\"odinger operator

TL;DR

This paper investigates the spectral properties of one-dimensional Schrödinger operators with complex potentials, providing bounds on eigenvalues and conditions for spectral decomposition, especially in dissipative cases.

Contribution

It introduces an effective upper bound on eigenvalues and spectral singularities and offers conditions for the absence of singular components in the continuous spectrum.

Findings

Established an upper bound for eigenvalues and spectral singularities.

Provided sufficient conditions for the absence of spectral singularities in dissipative operators.

Detailed spectral decomposition for specific Schrödinger operators.

Abstract

Spectral components of one-dimensional Schr\"odinger operator with complex potential are investigated. An effective upper bound for the total number of eigenvalues and spectral singularities is established. For dissipative Schr\"odinger operator sufficient condition is found which guarantees the absence of singular component in the continuous spectrum and spectral decomposition is exposed.