On the spectrum of the differential operators of odd order with $\mathcal{PT}$-symmetric coefficients

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 periodic $\mathcal{PT}$-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 $L$ is purely real and coincides with the real line.