Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations

Johannes Sjöstrand[1]

  • [1] IMB, Université de Bourgogne, 9, av. A. Savary, BP 47870, FR-21078 Dijon cedex and UMR 5584 du CNRS

Annales de la faculté des sciences de Toulouse Mathématiques (2009)

  • Volume: 18, Issue: 4, page 739-795
  • ISSN: 0240-2963

Abstract

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In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.

How to cite

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Sjöstrand, Johannes. "Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations." Annales de la faculté des sciences de Toulouse Mathématiques 18.4 (2009): 739-795. <http://eudml.org/doc/10126>.

@article{Sjöstrand2009,
abstract = {In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.},
affiliation = {IMB, Université de Bourgogne, 9, av. A. Savary, BP 47870, FR-21078 Dijon cedex and UMR 5584 du CNRS},
author = {Sjöstrand, Johannes},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {eigenvalue distribution; multiplicative random perturbations; Weyl asymptotics; non-selfadjoint pseudodifferential operators},
language = {eng},
month = {10},
number = {4},
pages = {739-795},
publisher = {Université Paul Sabatier, Toulouse},
title = {Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations},
url = {http://eudml.org/doc/10126},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Sjöstrand, Johannes
TI - Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/10//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 4
SP - 739
EP - 795
AB - In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in arbitrary dimension. We were led to quite essential improvements of many of the probabilistic aspects.
LA - eng
KW - eigenvalue distribution; multiplicative random perturbations; Weyl asymptotics; non-selfadjoint pseudodifferential operators
UR - http://eudml.org/doc/10126
ER -

References

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  1. Bordeaux-Montrieux (W.).— Loi de Weyl presque sûre et résolvante pour des opérateurs différentiels non-autoadjoints, Thesis, CMLS, Ecole Polytechnique, 2008. http://pastel.paristech.org/5367/. 
  2. Dimassi (M.), Sjöstrand (J.).— Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Ser., 268, Cambridge Univ. Press, (1999). Zbl0926.35002MR1735654
  3. Gohberg (I.C.), Krein (M.G.).— Introduction to the theory of linear non-selfadjoint operators, Translations of mathematical monographs, Vol 18, AMS, Providence, R.I. (1969). Zbl0181.13504MR246142
  4. Grigis (A.), Sjöstrand (J.).— Microlocal analysis for differential operators, London Math. Soc. Lecture Notes Ser., 196, Cambridge Univ. Press, (1994). Zbl0804.35001MR1269107
  5. Hager (M.).— Instabilité spectrale semiclassique pour des opérateurs non-autoadjoints. I. Un modèle, Ann. Fac. Sci. Toulouse Math. (6)15(2), p. 243-280 (2006). Zbl1131.34057MR2244217
  6. Hager (M.).— Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré, 7(6), p. 1035-1064 (2006). Zbl1115.81032MR2267057
  7. Hager (M.), Sjöstrand (J.).— Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, 342(1), p. 177-243 (2008). Zbl1151.35063MR2415321
  8. Hörmander (L.).— Fourier integral operators I, Acta Math., 127, p. 79-183 (1971). Zbl0212.46601MR388463
  9. Iantchenko (A.), Sjöstrand (J.), Zworski (M.).— Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett. 9(2-3), p. 337-362 (2002). Zbl1258.35208MR1909649
  10. Seeley (R.T.).— Complex powers of an elliptic operator. 1967 Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966) p. 288-307 Amer. Math. Soc., Providence, R.I. Zbl0159.15504MR237943
  11. Sjöstrand (J.).— Resonances for bottles and trace formulae, Math. Nachr., 221, p. 95-149 (2001). Zbl0979.35109MR1806367
  12. Sjöstrand (J.), Vodev (G.).— Asymptotics of the number of Rayleigh resonances, Math. Ann. 309, p. 287-306 (1997). Zbl0890.35098MR1474193
  13. Sjöstrand (J.), Zworski (M.).— Fractal upper bounds on the density of semiclassical resonances, Duke Math J, 137(3), p. 381-459 (2007). Zbl1201.35189MR2309150
  14. Sjöstrand (J.), Zworski (M.).— Elementary linear algebra for advanced spectral problems, Annales Inst. Fourier, 57(7), p. 2095-2141 (2007). Zbl1140.15009MR2394537
  15. Wunsch (J.), Zworski (M.).— The FBI transform on compact C ∞ manifolds, Trans. A.M.S., 353(3), p. 1151-1167 (2001). Zbl0974.35005MR1804416

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