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Stéphane Nonnenmacher

Spectral correlations for randomly perturbed nonselfadjoint operators

Abstract: We are interested in the spectrum of semiclassical nonselfadjoint operators. Due to a strong pseudospectral effect, a tiny perturbation can dramatically modify the spectrum of such an operator. Hager & Sjöstrand have thus considered adding small random pertubations, and proved that the eigenvalues of the perturbed operator typically spread over the classical spectrum, satisfying a probabilistic Weyl's law in the semiclassical limit.

Beyond this Weyl's law, we investigate the correlations between the eigenvalues, at microscopic distances. In the case of 1-dimensional operators, these correlations depend on the structure of the energy shell of the unperturbed operator (a finite set of points), and of the type of  perburbation (random matrix vs. random potential), but otherwise enjoy a form of universality, where the central object is the Gaussian Analytic Function (GAF), a family of random entire functions. The GAF was originally introduced in the context of Quantum Chaos in the 1990s, in order to describe the statistical properties of 1D chaotic eigenfunctions. In the present model the GAF (and its variants) rather arise through the spectral determinant of our randomly perturbed operator.

This is a joint work with Martin Vogel (Orsay).