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

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.