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Communications in Mathematical Physics

, Volume 131, Issue 3, pp 571–603 | Cite as

Zonal Schrödinger operators on then-sphere: Inverse spectral problem and rigidity

  • David Gurarie
Article

Abstract

We study the Direct and Inverse Spectral Problems for a class of Schrödinger operatorsH=−Δ+V onSn withzonal (axisymmetric) potentials. Spectrum ofH is known to consist of clusters of eigenvalues {λkm=k(k+n-1)+μkm:m≦k}. The main result of the work is to derive asymptotic expansion of spectral shifts {μkm} in powers ofk−1, and to link coefficients of the expansion to certain transforms ofV. As a corollary we solve the Inverse Problem, get explicit formulae for the Weinsteinband-invariants of cluster distribution measures, and establishlocal spectral rigidity for zonal potential. The latter provides a partial answer to a long standing Spectral Rigidity Hypothesis of V. Guillemin.

Keywords

Neural Network Statistical Physic Complex System Inverse Problem Nonlinear Dynamics 
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 1990

Authors and Affiliations

  • David Gurarie
    • 1
  1. 1.Department of Mathematics and StatisticsCase Western UniversityClevelandUSA

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