On the Bloch eigenvalues and spectrum of the differential operators of odd order

TL;DR

This paper investigates the spectral properties of odd-order non-self-adjoint differential operators with PT-symmetric periodic coefficients, focusing on eigenvalue localization and conditions for real spectra.

Contribution

It provides new insights into the localization of Bloch eigenvalues and establishes conditions under which the spectrum is real for these operators.

Findings

Localization of Bloch eigenvalues analyzed

Conditions for spectrum to be real derived

Spectral structure characterized for PT-symmetric operators

Abstract

In this paper we consider the Bloch eigenvalues and spectrum of the non-self-adjoint differential operator L generated by the differential expression of odd order n with the periodic PT-symmetric coefficients, where n>1. We study the localizations of the Bloch eigenvalues and the structure of the spectrum. Moreover, we find conditions on the norm of the coefficients under which the spectrum of L coincides with the real line