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Non-self-adjoint operators appear in many settings, from kinetic theory

and quantum mechanics to linearizations of equations of mathematical

physics. The spectral analysis of such operators, while often notoriously

difficult, reveals a wealth of new phenomena, compared with their

self-adjoint counterparts. Spectra for non-self-adjoint operators display

fascinating features, such as lattices of eigenvalues for operators of

Kramers-Fokker-Planck type, say, and eigenvalues for operators with

analytic coefficients in dimension one, concentrated to unions of curves

in the complex spectral plane. In this talk, after a general introduction,

we shall discuss spectra for non-self-adjoint perturbations of

self-adjoint operators in dimension two, under the assumption that the

classical flow of the unperturbed part is completely integrable.

The role played by the flow-invariant Lagrangian tori of the completely

integrable system, both Diophantine and rational, in the spectral analysis

of the non-self-adjoint operators will be described. In particular, we

shall discuss the spectral contributions of rational tori, leading to

eigenvalues having the form of the "legs in a spectral centipede". This

talk is based on joint work with Johannes Sj\"ostrand.