Student Harmonic Analysis and PDE Seminar (HADES): Random perturbations of nonselfadjoint operators, and the Gaussian Analytic Function

Seminar | September 10 | 3:40-5 p.m. | 748 Evans Hall

 Stephane Nonnenmacher, Univ. Paris-Sud & MSRI

 Department of Mathematics

The spectrum of a nonselfadjoint linear operator can be very unstable, that is sensitive to perturbations, an phenomenon usually referred to as the "pseudospectral effect". In order to quantify this phenomenon, we investigate a simple class of nonselfadjoint 1-dimensional semiclassical (pseudo-)differential operators, submitted to small random perturbations. The spectrum of this randomly perturbed operator is then viewed as a random point process on the complex plane, whose statistical properties we wish to analyze.

Hager & Sjöstrand have shown that, in the semiclassical limit, the randomly perturbed eigenvalues satisfies a probabilistic form of Weyl's law, at the macroscopic scale. We in turn investigate the statistical distribution of the eigenvalues at the microscopic scale (scale of the distance between nearby eigenvalues). We show that at this scale, the spectral statistics satisfy a partial form of universality: spectral correlations can be expressed in terms of a universal object, the Gaussian Analytic Function (GAF), and a few parameters depending on the initial operator, and of the type of random disorder.

A central tool in our analysis is a well-posed Grushin problem, which turns our spectral problem on $L^2(\mathbb R)$ into an effective nonlinear spectral problem on a finite dimensional subspace ("effective Hamiltonian"). This Grushin problem is set up by studying the "classical spectrum" of our initial semiclassical operator (a region in the complex plane), constructing quasimodes of this operator, and analyzing the (complex-)energy-dependence of these quasimodes.

This is joint work with Martin Vogel.