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Weyl Law for Semi-Classical Resonances with Randomly Perturbed Potentials
Johannes Sjöstrand, Université de Bourgogne, Dijon, France
A publication of the Société Mathématique de France.
Mémoires de la Société Mathématique de France
2014; 144 pp; softcover
Number: 136
ISBN-10: 2-85629-780-3
ISBN-13: 978-2-85629-780-3
List Price: US$52
Member Price: US$41.60
Order Code: SMFMEM/136
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The author considers semi-classical Schrödinger operators with potentials supported in a bounded strictly convex subset \(\mathcal{O}\) of \(\mathbb{R}^n\) with smooth boundary. Letting \(h\) denote the semi-classical parameter, the author considers classes of small random perturbations and shows that with probability very close to 1, the number of resonances in rectangles \([a,b]-i[0,ch^{\frac 23}\)] is equal to the number of eigenvalues in \([a,b]\) of the Dirichlet realization of the unperturbed operator in \(\mathcal{O}\) up to a small remainder.

A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.


Graduate students and research mathematicians.

Table of Contents

  • Introduction
  • The result
  • Some elements of the proof
  • Grushin problems and determinants
  • Complex dilations
  • Semi-classical Sobolev spaces
  • Reductions to \(\mathcal{O}\) and to \(\partial\mathcal{O}\)
  • Some ODE preparations
  • Parametrix for the exterior Dirichlet problem
  • Exterior Poisson operator and DN map
  • The interior DN map
  • Some determinants
  • Upper bounds on the basic determinant
  • Some estimates for \(P_\mathrm{out}\)
  • Perturbation matrices and their singular values
  • End of the construction
  • End of the proof of Theorem 2.2 and proof of Proposition 2.4
  • Appendix. WKB estimates on an interval
  • Bibliography
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