In this chapter, we investigate the non-self-adjoint operatorH generated in \(L_{2}(-\infty ,\infty )\) by the Mathieu-Schrödinger equation with a complex-valued potentialComplex-valued potential. First, we investigate the asymptotic formulasAsymptotic formulas for the isolated Bloch eigenvaluesBloch eigenvalue and find a condition for the isospectrality. Then we consider the asymptotic formulasAsymptotic formulas for the pair of the Bloch eigenvaluesBloch eigenvalue and find a necessary and sufficient condition on the potential for which H has no spectral singularity at infinity and is an asymptotically spectral operatorAsymptotically spectral operator. Besides, we find necessary and sufficient conditions for which the non-self-adjoint operatorNon-self-adjoint operator H has no ESS (essential spectral singularitiesSpectral singularities) at infinity, has at most finite number of ESSEssential spectral singularity and the corresponding spectral expansion has the asymptotically elegant form. Moreover, we investigate the simplicity of the periodic and antiperiodic eigenvaluesAntiperiodic eigenvalues and find a condition on potential for which the operator H has no ESSEssential spectral singularity and ESS at infinity and the corresponding spectral expansion has the elegant form. Finally, 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 ESSEssential spectral singularity.