Spectral monodromy of small non-selfadjoint perturbed operators: completely integrable or quasi-integrable case

We build a combinatorial invariant, called the spectral monodromy from the spectrum of a non-selfadjoint h -pseudodifferential operator with two degrees of freedom in the semi-classical limit. We treat small non-selfadjoint perturbation of selfadjoint h-pseudodifferential operators in two case: in the first, we assume that the classical flow of the unperturbed part is integrable; the second case, more interesting, when this flow is assumed to be quasi-integrable.