The precise shape of the eigenvalue intensity for a class of non-self-adjoint operators under random perturbations. (English. French summary) Zbl 1368.81076
Author’s abstract: We consider a non-self-adjoint \(h\)-differential model operator \(P_h\) in the semiclassical limit (\(h\to 0\)) subject to small random perturbations. Furthermore, we let the coupling constant \(\delta\) be \(e^{-\frac{1}{C h}} \leq \delta \ll h^\kappa\) for constants \(C, \kappa >0\) suitably large. Let \(\Sigma\) 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 \(\delta \gg e^{-\frac{1}{C h}}\) 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 \(\gg (-h \ln (\delta h) )^{\frac23}\) to the boundary of \(\Sigma\). 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 \(\Sigma\): the interior 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.
Reviewer: Antonio M. Peralta (Granada)
MSC:
| 81Q12 | Nonselfadjoint operator theory in quantum theory including creation and destruction operators |
| 81Q15 | Perturbation theories for operators and differential equations in quantum theory |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
non-self-adjoint operators; \(\varepsilon\)-pseudoespectrum; semicalssical differential operators; random perturbations; principal symbol; Weyl lawSoftware:
EigtoolReferences:
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