  In this paper we investigate the non-self-adjoint operator H generated in all
real line 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.
