Non-selfadjoint perturbations of self-adjoint operators in 2 dimensions I

This is the first in a series of works devoted to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. In the present work we treat the case when the classical flow of the unperturbed part is periodic and the strength ǫ of the perturbation is ≫ h (or sometimes only ≫ h 2) and bounded from above by h δ for some δ> 0. We get a complete asymptotic description of all eigenvalues in certain rectangles [−1/C,1/C] + iǫ[F0 − 1/C,F0 + 1/C].

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