On the shape of spectra for non-self-adjoint periodic Schr\"odinger operators

TL;DR

This paper investigates the spectral structure of periodic Schr"odinger operators, revealing how complex potentials influence the spectrum's shape and the emergence or disappearance of energy bands, especially in non-self-adjoint cases.

Contribution

It provides a general description of the local spectral shape for non-self-adjoint periodic Schr"odinger operators, linking band appearance/disappearance to nonreal spectra.

Findings

Complex potentials lead to spectra with analytic arcs in the complex plane.

Appearance/disappearance of energy bands correlates with nonreal spectral components.

The paper establishes a general result on the local spectral shape for these operators.

Abstract

The spectra of the Schr\"odinger operators with periodic potentials are studied. When the potential is real and periodic, the spectrum consists of at most countably many line segments (energy bands) on the real line, while when the potential is complex and periodic, the spectrum consists of at most countably many analytic arcs in the complex plane. In some recent papers, such operators with complex $\mathcal{PT}$-symmetric periodic potentials are studied. In particular, the authors argued that some energy bands would appear and disappear under perturbations. Here, we show that appearance and disappearance of such energy bands imply existence of nonreal spectra. This is a consequence of a more general result, describing the local shape of the spectrum.