On the Differential Operators of Odd Order with \(\mathrm{PT}\)-Symmetric Periodic Matrix Coefficients

Abstract

In this paper, we investigate the spectrum of the differential operator \(T\) generated by an ordinary differential expression of order \(n\) with \(\mathrm{PT}\)-symmertic periodic \(m\times m\) matrix coefficients. We prove that if \(m\) and \(n\) are odd numbers, then the spectrum of \(T\) contains all the real line. Note that in standard quantum theory, observable systems must be Hermitian operators, so as to ensure that the spectrum is real. Research on \(\mathrm{PT}\)-symmetric quantum theory is based on the observation that the spectrum of a \(\mathrm{PT}\)-symmetric non-self-adjoint operator can contain real numbers. In this paper, we discover a large class of \(\mathrm{PT}\)-symmetric operators whose spectrum contains all real axes. Moreover, the proof is very short.