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We obtain uniform, with respect to t asymptotic formulas for the eigenvaluesof the operators generated in (0,1) by the Mathieu-Hill equation with acomplex-valued potential and by the t-periodic boundary conditions. Then usingit we investigate the non-self-adjoint Mathieu-Hill operator H generated in theallreal line by the same equation and establish the necessary and sufficientconditions for the potential for which H has no spectral singularity atinfinity and it is an asymptotically spectral operator.

Spectral Analysis of the Non-self-adjoint Mathieu-Hill Operator