Advertisement

Eigenvalue distributions and Weyl laws for semiclassical non-self-adjoint operators in 2 dimensions

Chapter
Part of the Progress in Mathematics book series (PM, volume 292)

Abstract

In this note we compare two recent results about the distribution of eigenvalues for semiclassical pseudodifferential operators in two dimensions. For classes of analytic operators A. Melin and the author [6] obtained a complex Bohr–Sommerfeld rule, showing that the eigenvalues are situated on a distorted lattice. On the other hand, with M. Hager [4] we showed in any dimension that Weyl asymptotics holds with probability close to 1 for small random perturbations of the operator. In both cases the eigenvalues distribute to leading order according to smooth densities, and we show here that the two densities are in general different.

Keywords

Weyl law random 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J.J. Duistermaat, L. Hörmander, Fourier integral operators. II, Acta Math. (3–4)128(1972), 183–269.Google Scholar
  2. 2.
    J.J. Duistermaat, J. Sjöstrand, A global construction for pseudo-differential operators with non-involutive characteristics, Invent. Math. 20(1973), 209–225.Google Scholar
  3. 3.
    M. Hager, Instabilité spectrale semiclassique d’opérateurs non-autoadjoints. II, Ann. Henri Poincaré 7(6)(2006), 1035–1064.Google Scholar
  4. 4.
    M. Hager, J. Sjöstrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators, Math. Annalen, 342(1)(2008), 177–243.Google Scholar
  5. 5.
    A. Melin,J.Sjöstrand, Determinants of pseudodifferential operators and complex deformations of phase space, Methods and Applications of Analysis, 9(2)(2002), 177–238. Preprint: http://xxx.lanl.gov/abs/math.SP/0111292
  6. 6.
    A. Melin,J.Sjöstrand, Bohr–Sommerfeld quantization condition for non-selfadjoint operators in dimension 2, Astérique 284(2003), 181–244. Preprint: http://xxx.lanl.gov/abs/math.SP/0111293
  7. 7.
    J. Sjöstrand, Function spaces associated to global I-Lagrangian manifolds, pages 369–423 in Structure of solutions of differential equations, Katata/Kyoto, 1995, World Scientific 1996.Google Scholar
  8. 8.
    J. Sjöstrand, Spectral instability for non-selfadjoint operators, Algebraic analysis of differential equations, Springer 2008, 265–273.Google Scholar
  9. 9.
    J. Sjöstrand, Some results on non-self-adjoint operators, a survey, Further progress in analysis, 45–74, World Sci. Publ., Hackensack, NJ, 2009. Preprint: http://arxiv.org/abs/0804.3701
  10. 10.
    J. Sjöstrand, Eigenvalue distribution for non-self-adjoint operators with small multiplicative random perturbations, Annales Fac. Sci. Toulouse 18(4)(2009), 739–795. Preprint: http://arxiv.org/abs/0802.3584.
  11. 11.
    S. Vũ Ngoc, Systèmes intégrables semi-classiques: du local au global, Panoramas et Synthèses, 22. SociétéMathématique de France, Paris, 2006.Google Scholar

Copyright information

© Springer Science+Buisness Media, LLC 2011

Authors and Affiliations

  1. 1.UMR 7640, CNRSCMLS, École PolytechniquePalaiseauFrance
  2. 2.Institut de Mathématiques de Bourgogne, UMR 5584 du CNRSUniversité de BourgogneDijon cedexFrance

Personalised recommendations