Abstract
We consider a class of one-dimensional nonselfadjoint semiclassical pseudo-differential operators, subject to small random perturbations, and study the statistical properties of their (discrete) spectra, in the semiclassical limit h → 0. We compare two types of random perturbation: a random potential vs. a random matrix. Hager and Sjostrand had shown that, with high probability, the local spectral density of the perturbed operator follows a semiclassical form of Weyl's law, depending on the value distribution of the principal symbol of our pseudodifferential operator. Beyond the spectral density, we investigate the full local statistics of the perturbed spectrum, and show that it satisfies a form of universality: the statistical only depends on the local spectral density, and of the type of random perturbation, but it is independent of the precise law of the perturbation. This local statistics can be described in terms of the Gaussian Analytic Function, a classical ensemble of random entire functions.
Highlights
The spectral analysis of linear operators acting on a Hilbert space is much developed in the case of selfadjoint operators: one can use powerful tools, like the spectral theorem, or variational methods
The evolution of the system may be described by a linear operator, which is often nonselfadjoint: the Fokker-Planck, or the linearized Boltzmann equation typically contain convective as well as dissipative terms, leading to nonselfadjoint operators
Let us mention a model studied recently by Capitaine and Bordenave [6], namely the case of a large Jordan block perturbed by a Ginibre random matrix: the authors prove that most eigenvalues of the perturbed matrix lie close to the unit circle, but they show that the “outliers” are statistically distributed like the zeros of a “hyperbolic” Gaussian analytic function (GAF)
Summary
The spectral analysis of linear operators acting on a Hilbert space is much developed in the case of selfadjoint operators: one can use powerful tools, like the spectral theorem, or variational methods. In the nonselfadjoint case, the connection between the long time evolution and a spectrum of complex eigenvalues is not so obvious as in the selfadjoint case, since eigenstates do not form an orthonormal family This difficulty of relating spectrum and dynamics is linked with a characteristics of nonselfadjoint operators, namely the possible strong instability of their spectrum with respect to small perturbations, a phenomenon nowadays commonly called pseudospectral effect. Let us mention a model studied recently by Capitaine and Bordenave [6] (see [10]), namely the case of a large Jordan block perturbed by a Ginibre random matrix: the authors prove that most eigenvalues of the perturbed matrix lie close to the unit circle, but they show that the “outliers” (the relatively few eigenvalues away from the unit circle) are statistically distributed like the zeros of a “hyperbolic” Gaussian analytic function (GAF). These GAFs will describe the bulk of the spectrum, as opposed to a few outliers; in our case Gaussian functions appear in the limit, eventhough the perturbation operator or potential is not necessarily Gaussian distributed
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