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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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