Spectral Problems of a Class of Non-self-adjoint One-dimensional Schrodinger Operators

TL;DR

This paper analyzes the spectral properties of a class of non-self-adjoint one-dimensional Schrödinger operators with complex periodic potentials, deriving explicit formulas for Bloch eigenvalues and functions, and addressing the inverse spectral problem.

Contribution

It provides explicit formulas for Bloch eigenvalues and functions for a specific class of non-self-adjoint Schrödinger operators and investigates the inverse spectral problem.

Findings

Bloch eigenvalues are (2πn+t)^2 for n in Z, t in C

Explicit formulas for Bloch functions are derived

Inverse problem for the operator is considered

Abstract

In this paper we investigate the one-dimensional Schrodinger operator L(q) with complex-valued periodic potential q when q\inL_{1}[0,1] and q_{n}=0 for n=0,-1,-2,..., where q_{n} are the Fourier coefficients of q with respect to the system {e^{i2{\pi}nx}}. We prove that the Bloch eigenvalues are (2{\pi}n+t)^{2} for n\inZ, t\inC and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.