On the spectrality and spectral expansion of the non-self-adjoint mathieu-hill operator in $ L_{2}(-\infty, \infty) $

Abstract

In this paper we investigate the non-self-adjoint operator \begin{document}$ H $\end{document} generated in \begin{document}$ L_{2}(-\infty, \infty) $\end{document} by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which \begin{document}$ H $\end{document} has no spectral singularity at infinity and it is an asymptotically spectral operator. Moreover, we give a detailed classification, stated in term of the potential, for the form of the spectral decomposition of the operator \begin{document}$ H $\end{document} by investigating the essential spectral singularities.