On the Spectrum of the Differential Operators of Odd Order with \(\mathcal{PT}\)-Symmetric Coefficients

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 periodic \(\mathcal{PT}\)-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 \(L\) is purely real and coincides with the real line.