Abstract
We consider a simple model operator P in dimension 1 and show how random perturbations give rise to Weyl asymptotics in the interior of the range of P. We follow rather closely the work of Hager (Ann Henri Poincaré 7(6):1035–1064, 2006) with some input also from Bordeaux Montrieux (Loi de Weyl presque sûreet résolvante pour des opérateurs différentiels nonautoadjoints, thèse, CMLS, Ecole Polytechnique, 2008) and Hager–Sjöstrand (Math Ann 342(1):177–243, 2008). Some of the general ideas appear perhaps more clearly in this special situation.
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Notes
- 1.
The choice of Gaussian random variables is convenient in order to formulate a less technical result. In Chap. 15 more general classes of perturbations will be treated, also in higher dimensions.
- 2.
We write a ≍ b, when a, b are real numbers with the same sign, such that \(|a|={\mathcal {O}}(|b|)\) and \(|b|={\mathcal {O}}(|a|)\). This notion has a natural extension to the case when a, b are positive Radon measures on the same set in R n.
- 3.
If \({\mathcal {F}}\) is a seminormed space, we write \(b(h)\sim b_0+hb_1+\cdots \mbox{ in }{\mathcal {F}}\), if for every seminorm q and every N ∈N, we have \(q(b-(b_0+hb_1+\cdots +b_Nh^N))={\mathcal {O}}(h^{N+1})\).
References
L. Ahlfors, Complex analysis. An Introduction to the Theory of Analytic Functions of One Complex Variable, 3rd edn. International Series in Pure and Applied Mathematics (McGraw-Hill Book Co., New York, 1978)
W. Bordeaux Montrieux, Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, thèse, CMLS, Ecole Polytechnique, 2008. https://pastel.archives-ouvertes.fr/pastel-00005367
A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators. London Mathematical Society Lecture Notes Series, vol. 196 (Cambridge University Press, Cambridge, 1994)
M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle. Ann. Fac. Sci. Toulouse Math. 15(2), 243–280 (2006)
M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré 7(6), 1035–1064 (2006)
M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342(1), 177–243 (2008). http://arxiv.org/abs/math/0601381
O. Kallenberg, Foundations of Modern Probability. Probability and Its Applications (New York) (Springer, New York, 1997)
J. Sjöstrand, Resonances for bottles and trace formulae. Math. Nachr. 221, 95–149 (2001)
M. Vogel, The precise shape of the eigenvalue intensity for a class of non-selfadjoint operators under random perturbations. Ann. Henri Poincaré 18(2), 435–517 (2017). http://arxiv.org/abs/1401.8134
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Sjöstrand, J. (2019). Weyl Asymptotics and Random Perturbations in a One-Dimensional Semi-classical Case. In: Non-Self-Adjoint Differential Operators, Spectral Asymptotics and Random Perturbations. Pseudo-Differential Operators, vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-10819-9_3
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