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

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