On a Class of Non-self-adjoint Multidimensional Periodic Schrodinger Operators

TL;DR

This paper studies a class of multidimensional periodic Schrödinger operators with complex potentials, showing that under certain conditions, their spectral properties match those of the free operator, with implications for PT symmetric quantum theory.

Contribution

It proves the equality of Bloch eigenvalues and Fermi surfaces for a broad class of non-self-adjoint operators with specific Fourier coefficient conditions.

Findings

Bloch eigenvalues of L(q) and L(0) are identical

Fermi surfaces of L(q) and L(0) coincide

explicit formulas for Bloch functions are derived

Abstract

We investigate the multidimensional Schrodinger operator L(q) with complex-valued periodic, with respect to a lattice, potential q when the Fourier coefficients of q with respect to the orthogonal system {exp(i(a,x))}, where a changes in the dual lattice, vanish if a belong to a half-space We prove that the Bloch eigenvalues of L(q) and of the free operator L(0) are the same and find explicit formulas for the Bloch functions. It implies that the Fermi surfaces of L(q) and L(0) are the same. The considered set of operators includes a large class of PT symmetric operators used in the PT symmetric quantum theory.