Diophantine tori and spectral asymptotics for nonselfadjoint operators

We study spectral asymptotics for small nonselfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2, assuming that the classical flow of the unperturbed part possesses several invariant Lagrangian tori enjoying a Diophantine property. We get complete asymptotic expansions for all eigenvalues in certain rectangles in the complex plane in two different cases: in the first case, we assume that the strength ε of the perturbation is O(hδ) for some δ > 0 and is bounded from below by a fixed positive power of h. In the second case, ε is assumed to be sufficiently small but independent of h, and we describe the eigenvalues completely in a fixed h-independent domain in the complex spectral plane.