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.