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.


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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[1] Davies E.B. (2005). Semi-classical analysis and pseudospectra. J. Diff. Equ. 216(1): 153–187 · Zbl 1088.81042
[2] Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit, London Mathematical Society, Lecture Note Series, vol. 268 (1999) · Zbl 0926.35002
[3] Dynkin E.M. (1975). An operator calculus based upon the Cauchy–Green formula. J. Soviet Math. 4(4): 329–334 · Zbl 0338.47011
[4] Embree M. and Trefethen L.N. (2005). Spectra and Pseudospectra. The Behaviour of Non-normal Matrices and operator. Princeton University Press, Princeton · Zbl 1085.15009
[5] Girko V.I. (1990). Theory of Random Determinants. Mathematics and its Applications. Kluwer, Dordrecht · Zbl 0717.60047
[6] Hager M. (2006). Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I: un modèle. Ann. Fac. Sci. Toulouse Math. (6) 15(2): 243–280 · Zbl 1131.34057
[7] Hager M. (2006). Instabilité spectrale semiclassique d’opérateurs non-autoadjoints II. Ann. Henri Poincaré 7(6): 1035–1064 · Zbl 1115.81032
[8] Hager M. (2007). Bound on the number of eigenvalues near the boundary of the pseudospectrum, Proc. Amer. Math. Soc. 135(12): 3867–3873 · Zbl 1130.47029
[9] Helffer B. and Sjöstrand J. (1989). équation de Schrödinger avec champs magnétique et équation de Harper. Springer Lect. Notes Phys. 345: 118–197 · Zbl 0699.35189
[10] Melin A. and Sjöstrand J. (2002). Determinants of pseudodifferential operators and complex deformations of phase space. Methods Appl. Anal. 9(2): 177–237 · Zbl 1082.35176
[11] Sjöstrand J. (1974). Parametrices for pseudodifferential operators with multiple characteristics. Ark. f. Mat. 12(1): 85–130 · Zbl 0317.35076
[12] Sjöstrand J. and Zworski M. (2007). Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier 57(7): 2095–2141 · Zbl 1140.15009
[13] Trefethen L.N. and Chapman S.J. (2004). Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57: 1233–1264 · Zbl 1055.15014
[14] Weyl H. (1912). Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung). Math. Ann. 71(4): 441–479 · JFM 43.0436.01
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