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


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.


Neural Network Statistical Physic Detailed Analysis Energy Surface Complex System 
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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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