ON THE SPECTRALITY AND SPECTRAL EXPANSION OF THE NON-SELF-ADJOINT MATHIEU-HILL OPERATOR IN L-2 (-infinity, infinity)

Abstract

In this paper we investigate the non-self-adjoint operator H generated in L-2(-infinity, infinity) by the Mathieu-Hill equation with a complex-valued potential. We find a necessary and sufficient conditions on the potential for which H 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 H by investigating the essential spectral singularities.