The precise shape of the eigenvalue intensity for a class of non-selfadjoint operators under random perturbations

2014/01/31 by Martin Vogel, Vogel, Martin
Mathematics · Physics and Astronomy · #Quantum chaos and dynamical systems #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.SP

paper · pdf · doi:10.48550/arxiv.1401.8134

75 pages, 13 figures

arxiv created 2015/12/20 · arxiv updated 2015/12/22

Abstract

We consider a non-selfadjoint h-differential model operator P_h in the semiclassical limit (h→ 0) subject to small random perturbations. Furthermore, we let the coupling constant δ be exp\-(1)/(Ch)\≤ δ≪ h^κ for constants C,κ>0 suitably large. Let Σ be the closure of the range of the principal symbol. Previous results on the same model by Hager, Bordeaux-Montrieux and Sjöstrand show that if δ≫exp\-(1)/(Ch)\ there is, with a probability close to 1, a Weyl law for the eigenvalues in the interior of the of the pseudospectrum up to a distance ≫(-hlnδh)^(2)/(3) to the boundary of Σ. We study the intensity measure of the random point process of eigenvalues and prove an h-asymptotic formula for the average density of eigenvalues. With this we show that there are three distinct regions of different spectral behavior in Σ: The interior of the of the pseudospectrum is solely governed by a Weyl law, close to its boundary there is a strong spectral accumulation given by a tunneling effect followed by a region where the density decays rapidly.

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