Nous étudions l’asymptotique de Weyl de la distribution des valeurs propres d’opérateurs (pseudo-)différentiels avec des petites perturbations aléatoires multiplicatives en dimension quelconque. Nous avons été amenés à faire des améliorations essentielles des aspects probabilistes.
We study the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.
@article{SEDP_2007-2008____A20_0,
author = {Sj\"ostrand, Johannes},
title = {Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:20},
pages = {1--16},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2007-2008},
mrnumber = {2532954},
language = {en},
url = {http://archive.numdam.org/item/SEDP_2007-2008____A20_0/}
}
TY - JOUR AU - Sjöstrand, Johannes TI - Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:20 PY - 2007-2008 SP - 1 EP - 16 PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://archive.numdam.org/item/SEDP_2007-2008____A20_0/ LA - en ID - SEDP_2007-2008____A20_0 ER -
%0 Journal Article %A Sjöstrand, Johannes %T Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations %J Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" %Z talk:20 %D 2007-2008 %P 1-16 %I Centre de mathématiques Laurent Schwartz, École polytechnique %U http://archive.numdam.org/item/SEDP_2007-2008____A20_0/ %G en %F SEDP_2007-2008____A20_0
Sjöstrand, Johannes. Weyl asymptotics for non-self-adjoint operators with small multiplicative random perturbations. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2007-2008), Exposé no. 20, 16 p. http://archive.numdam.org/item/SEDP_2007-2008____A20_0/
[1] W. Bordeaux-Montrieux, thèse, en préparation
[2] E.B. Davies, Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200(1999), 35–41. | MR | Zbl
[3] N.Dencker, J.Sjöstrand, M.Zworski, Pseudospectra of semiclassical (pseudo-) differential operators, Comm. Pure Appl. Math., 57(3)(2004), 384–415. | MR | Zbl
[4] M. Dimassi, J. Sjöstrand, Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Ser., 268, Cambridge Univ. Press, (1999). | MR | Zbl
[5] I.C. Gohberg, M.G. Krein, Introduction to the theory of linear non-selfadjoint operators, Translations of mathematical monographs, Vol 18, AMS, Providence, R.I. (1969). | MR | Zbl
[6] M. Hager, Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle, Ann. Fac. Sci. Toulouse Math. (6)15(2)(2006), 243–280. | Numdam | MR | Zbl
[7] M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré, 7(6)(2006), 1035–1064. | Zbl
[8] M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, to appear, http://arxiv.org/abs/math/0601381,
[9] A. Melin, J. Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods and Applications of Analysis, 9(2)(2002), 177-238. http://xxx.lanl.gov/abs/math.SP/0111292 | MR | Zbl
[10] K. Pravda Starov Etude du pseudo-spectre d’opérateurs non auto-adjoints, thèse Rennes 2006, http://tel.archives-ouvertes.fr/tel-00109895
[11] R.T. Seeley, Complex powers of an elliptic operator. 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) pp. 288–307, Amer. Math. Soc., Providence, R.I. | MR | Zbl
[12] J. Sjöstrand, Resonances for bottles and trace formulae, Math. Nachr., 221(2001), 95–149. | MR | Zbl
[13] J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations, http://arxiv.org/abs/0802.3584 .
[14] J. Sjöstrand, M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier, 57(7)(2007), 2095-2141. | Numdam
[15] M. Zworski, A remark on a paper of E.B. Davies, Proc. A.M.S. 129(2001), 2955–2957. | MR | Zbl
