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The Journal of Geometric Analysis

, Volume 13, Issue 2, pp 301–328 | Cite as

Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups

  • Carolyn S. Gordon
  • Dorothee Schueth
Article

Abstract

We construct pairs of conformally equivalent isospectral Riemannian metrics ϕ1g and ϕ2g on spheres Sn and balls Bn+1 for certain dimensions n, the smallest of which is n=7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions ϕ1 and ϕ2 are isospectral potentials for the Schrödinger operator ħ2\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.

Math Subject Classifications

58J53 58J50 

Key Words and Phrases

Laplace operator spectrum Schrödinger operator conformally equivalent metrics isospectral potentials 

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Copyright information

© Mathematica Josephina, Inc. 2003

Authors and Affiliations

  1. 1.Dartmouth CollegeHanover
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany

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