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


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.


Neural Network Statistical Physic Complex System Inverse Problem Nonlinear Dynamics 
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© Springer-Verlag 1990

Authors and Affiliations

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

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