## 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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### References:

 [1] Davies E.B. (2005). Semi-classical analysis and pseudospectra. J. Diff. Equ. 216(1): 153–187 · Zbl 1088.81042 · doi:10.1016/j.jde.2005.03.005 [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 · doi:10.1007/BF01084915 [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 · doi:10.1007/s00023-006-0275-7 [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 · doi:10.1090/S0002-9939-07-08914-9 [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 · doi:10.1007/3-540-51783-9_19 [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 · doi:10.1007/BF02384749 [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 · doi:10.1002/cpa.20034 [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 · doi:10.1007/BF01456804
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