  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.
