Mathematics – Spectral Theory
Scientific paper
2012-02-21
Mathematics
Spectral Theory
Scientific paper
We obtain uniform, with respect to t\in(-{\pi},{\pi}], asymptotic formulas for the operators generated in [0,1] by Mathieu-Hill equation with complex-valued potential and the t-periodic boundary conditions. Using these formulas, we prove that there exists a bounded set S which is independent of t such that all the eigenvalues of these operators, lying out of the set S, are simple. These results imply the following consequences for the non-self-adjoint Mathieu-Hill operator H generated in (-\infty,\infty) by the same Mathieu-Hill equation: (i) The spectrum of H in a neighborhood of \infty consists of the separated simple analytic arcs. (ii) The distance between the end points of the neighboring arcs of the spectrum of H satisfies an asymptotic formula, that results in the Avron-Simon-Harrell formula for the widths of the instability intervals of the self-adjoint Mathieu-Hill operator. (iii) The number of the spectral singularities in the spectrum of the operator H is finite. Furthermore, we establish the necessary and sufficient conditions for the potential, for which the operator H has no spectral singularity at infinity and H is an asymptotically spectral operator.
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