Abstract
We consider quite general h-pseudodifferential operators on R n with small random perturbations and show that in the limit h → 0 the eigenvalues are distributed according to a Weyl law with a probabality that tends to 1. The first author has previously obtained a similar result in dimension 1. Our class of perturbations is different.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Davies E.B. (2005). Semi-classical analysis and pseudospectra. J. Diff. Equ. 216(1): 153–187
Dimassi, M., Sjöstrand, J.: Spectral asymptotics in the semi-classical limit, London Mathematical Society, Lecture Note Series, vol. 268 (1999)
Dynkin E.M. (1975). An operator calculus based upon the Cauchy–Green formula. J. Soviet Math. 4(4): 329–334
Embree M. and Trefethen L.N. (2005). Spectra and Pseudospectra. The Behaviour of Non-normal Matrices and operator. Princeton University Press, Princeton
Girko V.I. (1990). Theory of Random Determinants. Mathematics and its Applications. Kluwer, Dordrecht
Hager M. (2006). Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints I: un modèle. Ann. Fac. Sci. Toulouse Math. (6) 15(2): 243–280
Hager M. (2006). Instabilité spectrale semiclassique d’opérateurs non-autoadjoints II. Ann. Henri Poincaré 7(6): 1035–1064
Hager M. (2007). Bound on the number of eigenvalues near the boundary of the pseudospectrum, Proc. Amer. Math. Soc. 135(12): 3867–3873
Helffer B. and Sjöstrand J. (1989). équation de Schrödinger avec champs magnétique et équation de Harper. Springer Lect. Notes Phys. 345: 118–197
Melin A. and Sjöstrand J. (2002). Determinants of pseudodifferential operators and complex deformations of phase space. Methods Appl. Anal. 9(2): 177–237
Sjöstrand J. (1974). Parametrices for pseudodifferential operators with multiple characteristics. Ark. f. Mat. 12(1): 85–130
Sjöstrand J. and Zworski M. (2007). Elementary linear algebra for advanced spectral problems. Ann. Inst. Fourier 57(7): 2095–2141
Trefethen L.N. and Chapman S.J. (2004). Wave packet pseudomodes of twisted Toeplitz matrices. Comm. Pure Appl. Math. 57: 1233–1264
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
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hager, M., Sjöstrand, J. Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342, 177–243 (2008). https://doi.org/10.1007/s00208-008-0230-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-008-0230-7