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Communications in Mathematical Physics

, Volume 136, Issue 3, pp 567–584 | Cite as

A semi-classical trace formula for Schrödinger operators

  • R. Brummelhuis
  • A. Uribe
Article

Abstract

LetS=−ℏΔ+V, withV smooth. If 0<E2<lim infV(x), the spectrum ofS nearE2 consists (for ℏ small) of finitely-many eigenvalues,λ j (ℏ). We study the asymptotic distribution of these eigenvalues aboutE2 as ℏ→0; we obtain semi-classical asymptotics for
$$\sum\limits_j {f\left( {\frac{{\sqrt {\lambda _j (\hbar )} - E}}{\hbar }} \right)} $$
with\(\hat f \in C_0^\infty \), in terms of the periodic classical trajectories on the energy surface\(B_E = \left\{ {\left| \xi \right|^2 + V(x) = E^2 } \right\}\). This in turn gives Weyl-type estimates for the counting function\(\# \left\{ {j;\left| {\sqrt {\lambda _j (\hbar )} - E} \right| \leqq c\hbar } \right\}\). We make a detailed analysis of the case when the flow onB E is periodic.

Keywords

Neural Network Statistical Physic Detailed Analysis Energy Surface Complex System 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • R. Brummelhuis
    • 1
  • A. Uribe
    • 2
    • 3
  1. 1.Mathematics DepartmentUniversity of WisconsinMadisonUSA
  2. 2.Mathematics DepartmentUniversity of MichiganAnn ArborUSA
  3. 3.Institute for Advanced StudyPrincetonUSA

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