Spectra for non-self-adjoint operators and integrable dynamics

03/06/2019, 15:00 - 16:00 in MAP/0G/017

Hitrik, Michael (UCLA)


Abstract:

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 give a broad discussion of 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 Sjöstrand.

Mathematical Sciences Research Centre