Abstract:
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.
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