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

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


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.


Weyl law random 


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© 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

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