Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds

William Bordeaux Montrieux[1]; Johannes Sjöstrand[2]

  • [1] Fakultät für Mathematik,Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria
  • [2] IMB, Université de Bourgogne, 9, Av. A. Savary, BP 47870, FR 21078 Dijon cédex, France, and UMR 5584 CNRS

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

  • Volume: 19, Issue: 3-4, page 567-587
  • ISSN: 0240-2963

Abstract

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In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.

How to cite

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Bordeaux Montrieux, William, and Sjöstrand, Johannes. "Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 567-587. <http://eudml.org/doc/115864>.

@article{BordeauxMontrieux2010,
abstract = {In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.},
affiliation = {Fakultät für Mathematik,Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria; IMB, Université de Bourgogne, 9, Av. A. Savary, BP 47870, FR 21078 Dijon cédex, France, and UMR 5584 CNRS},
author = {Bordeaux Montrieux, William, Sjöstrand, Johannes},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {non selfadjoint elliptic operator; Weyl asymptotics; large eigenvalues},
language = {eng},
number = {3-4},
pages = {567-587},
publisher = {Université Paul Sabatier, Toulouse},
title = {Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds},
url = {http://eudml.org/doc/115864},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Bordeaux Montrieux, William
AU - Sjöstrand, Johannes
TI - Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2010
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 567
EP - 587
AB - In this paper, we consider elliptic differential operators on compact manifolds with a random perturbation in the 0th order term and show under fairly weak additional assumptions that the large eigenvalues almost surely distribute according to the Weyl law, well-known in the self-adjoint case.
LA - eng
KW - non selfadjoint elliptic operator; Weyl asymptotics; large eigenvalues
UR - http://eudml.org/doc/115864
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). See also paper to appear in Annales Henri Poincaré. http://pastel.paristech.org/5367/ MR2605808
  2. Davies (E.B.).— Semi-classical states for non-self-adjoint Schrödinger operators, Comm. Math. Phys. 200(1), p. 35-41 (1999). Zbl0921.47060MR1671904
  3. Dimassi (M.), Sjöstrand (J.).— Spectral asymptotics in the semi-classical limit, London Math. Soc. Lecture Notes Ser., 268, Cambridge Univ. Press, (1999). Zbl0926.35002MR1735654
  4. Grigis (A.).— Estimations asymptotiques des intervalles d’instabilité pour l’équation de Hill, Ann. Sci. École Norm. Sup. (4) 20(4), p. 641-672 (1987). Zbl0644.34021MR932802
  5. Hager (M.).— Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II. Ann. Henri Poincaré, 7(6), p. 1035-1064 (2006). Zbl1115.81032MR2267057
  6. Hager (M.), Sjöstrand (J.).— Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, 342(1), p. 177-243 (2008). Zbl1151.35063MR2415321
  7. Seeley (R.).— A simple example of spectral pathology for differential operators, Comm. Partial Differential Equations 11(6), p. 595-598 (1986). Zbl0598.35013MR837277
  8. Sjöstrand (J.).— Eigenvalue distributions and Weyl laws for semi-classical non-selfadjoint operators in 2 dimensions, Proceedings of the Duistermaat conference 2007, Birkhäuser, Progress in Math., to appear. 
  9. Sjöstrand (J.).— Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations, Ann. Fac. Sci. Toulouse, (6) 18(4), p. 739-795 (2009). Zbl1194.47058MR2590387
  10. Sjöstrand (J.).— Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations, Ann. Fac. Sci. Toulouse (6) 19(2), p. 277-301 (2010). Zbl1206.35267MR2674764
  11. Trefethen (L.N.).— Pseudospectra of linear operators, SIAM Rev. 39(3), p. 383-406 (1997). Zbl0896.15006MR1469941
  12. Zworski (M.).— A remark on a paper of E. B. Davies: “Semi-classical states for non-self-adjoint Schrödinger operators”, Proc. Amer. Math. Soc. 129(10), p. 2955-2957 (2001). Zbl0981.35107MR1840099

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