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Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. (English) Zbl 1151.35063
The authors consider square integrable order functions \(0<\widetilde m, \widehat m\leq 1\) on \(\mathbb R^{2n}\) such that \(\widetilde m\) or \(\widehat m\) is integrable and they let \(\widetilde S\in S(\widetilde m)\), \(\widehat S\in S(\widehat m)\) be elliptic symbols which also denote the \(h\)-Weyl quantizations. The operators \(\widehat S\), \(\widetilde S\) are then Hilbert-Schmidt with \(| | \widetilde S| | _{\text{HS}}, | | \widehat S| | _{\text{HS}} \sim h^{-n/2}\), where \(\sim\) indicate the same order of magnitude. If \(\widetilde e_1,\widetilde e_2,\ldots\) and \(\widehat e_1,\widehat e_2,\ldots\) are orthonormal bases for \(L^2(\mathbb R^n)\), then their random perturbation is \(Q_\omega=\widehat S\circ\sum_{j,k}\alpha_{j,k}(\omega)\widehat e_j \widetilde e_k^*\circ\widetilde S\), where \(\alpha_{j,k}\) are independent complex \({\mathcal N}(0,1)\) random variables, and \(\widehat e_j \widetilde e_k^*u=(u| \widetilde e_k)\widehat e_j\), \(u\in L^2\). If \(M=C_1 h^{-n}\), for some \(C_1\gg 1\), it is shown that the following estimate holds on the probability that \(Q\) is large in the Hilbert-Schmidt norm, \(P(\|Q\|_{\text{HS}}\geq M^2)\leq C\exp(-h^{-2n} /C)\) for some new constant \(C>0\). Several results are derived under the following assumption: let \(\Gamma\subset \Omega\) be open with \(C^2\) boundary and assume that for every \(z\in \partial \Gamma\), \(\Sigma_z:=p^{-1}(z)\) is a smooth sub-manifold of \(T^*\mathbb R^n\) on which \(dp\), \(d\bar p\) are linearly independent at every point.

MSC:
35P20 Asymptotic distributions of eigenvalues in context of PDEs
30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
35S05 Pseudodifferential operators as generalizations of partial differential operators
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