Diophantine tori and spectral asymptotics for non-selfadjoint operators

We study spectral asymptotics for small non-selfadjoint 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 $\epsilon$ of the perturbation is ${\cal O}(h^{\delta})$ for some $\delta>0$ and is bounded from below by a fixed positive power of $h$. In the second case, $\epsilon$ 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.