On the real spectrum of differential operators with PT-symmetric periodic matrix coefficients

Abstract

We study the spectrum of the operator $T$ generated by the differential expression of order $n>2$ with the $m\times m$ Parity-Time (PT)-symmetric periodic matrix coefficients. The case when $m$ and $n$ are the odd numbers was investigated in [18]. In this paper, we consider the all remained cases: (a) $n$ is an odd number and $m$ is an even number, (b) $n$ is an even number and $m$ is an arbitrary positive integer. We find conditions on the coefficients under which in the cases (a) and (b) the spectrum of $T$ contains the sets $(-\infty ,-H]$ $\cup [H,\infty )$ and $[H,\infty )$ respectively for some $H>0$.