Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. (English) Zbl 1151.35063

Hager, Mildred; Sjöstrand, Johannes

Math. Ann. 342, No. 1, 177-243 (2008).

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.

Reviewer: Rémi Vaillancourt (Ottawa)

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

Keywords:

eigenvalue asymptotics; randomly perturbed non-selfadjoint operators, Grushin problem; counting zeros of holomorphic functions

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