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Diophantine Tori and Weyl Laws for Non-selfadjoint Operators in Dimension Two

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Abstract

We study the distribution of eigenvalues for non-selfadjoint perturbations of selfadjoint semiclassical analytic pseudodifferential operators in dimension two, assuming that the classical flow of the unperturbed part is completely integrable. An asymptotic formula of Weyl type for the number of eigenvalues in a spectral band, bounded from above and from below by levels corresponding to Diophantine invariant Lagrangian tori, is established. The Weyl law is given in terms of the long time averages of the leading non-selfadjoint perturbation along the classical flow of the unperturbed part.

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Authors and Affiliations

  1. Department of Mathematics, University of California, Los Angeles, CA, 90095-1555, USA

    Michael Hitrik

  2. IMB, Université de Bourgogne, 9, Av. A. Savary, BP 47870, 21078, Dijon Cédex, France

    Johannes Sjöstrand

  3. UMR 5584 CNRS, Dijon, France

    Johannes Sjöstrand

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  1. Michael Hitrik
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  2. Johannes Sjöstrand
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Correspondence to Michael Hitrik.

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Communicated by S. Zelditch

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Hitrik, M., Sjöstrand, J. Diophantine Tori and Weyl Laws for Non-selfadjoint Operators in Dimension Two. Commun. Math. Phys. 314, 373–417 (2012). https://doi.org/10.1007/s00220-012-1530-z

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