Spectra for non-selfadjoint differential operators
Time and place
1 PM on Thursday, March 27th, 2014; NAC 6113
Michael Hitrik (UCLA)
Abstract
For selfadjoint differential operators, the spectral asymptotics in the high energy and semiclassical limits, are given by the celebrated Weyl law, going back to the classical works of H. Weyl around 1912. According to the Weyl law, the density of eigenvalues can be described in terms of the direct image of the symplectic volume element on the phase space under a symbol map. Turning the attention to the non-selfadjoint case, it turns out that for operators with analytic coefficients, the situation becomes quite different. Here the eigenvalue asymptotics may no longer be governed by volumes of subsets of the real phase space, and the spectrum is often determined by the behavior of the holomorphic continuation of the symbol along suitable complex deformations of the phase space. A radically different spectral behavior occurs if the analyticity is no longer maintained, and for a randomly perturbed non-selfadjoint elliptic operator, one recovers a bidimensional Weyl law, holding in the semiclassical limit with an overwhelming probability. In this talk, I would like to describe some of the recent progress in understanding the semiclassical asymptotics of spectra for non-selfadjoint operators, with an emphasis on the analytic case and the ideas of complex phase space deformations.
