Rational invariant tori and spectral asymptotics for non-selfadjoint operators in dimension $2$

We study spectra for non-selfadjoint perturbations of selfadjoint semiclassical operators in dimension $2$, assuming that the classical flow of the unperturbed part is completely integrable. Complete asymptotic expansions are established for all individual eigenvalues in suitable regions of the complex spectral plane, close to the edges of the spectral band. The eigenvalues have the form of the « legs in a spectral centipede » and are produced by suitable rational flow-invariant Lagrangian tori.

This is joint work with Johannes Sjöstrand.