On manifolds of negative curvature with isospectral potentials
Let M be a manifold of negative curvature and let \(\Delta\) be the Laplacian. Let \(\Phi\) be a potential. One wishes to known when it is possible to recover \(\Phi\) from \(spec(\Delta +\Phi)\). M has simple length spectrum if no two closed geodesics on M have the same length. Guillemin and Kazhdan show that if M has simple length spectrum and if the curvature of M is suitably pinched, then \(\Phi\) is determined by \(spec(\Delta +\Phi)\). The author uses the method of Sunada to construct a manifold M of negative curvature (which does not have simple length spectrum) and two potentials \(\Phi_ 1\) and \(\Phi_ 2\) (which are not equivalent under the action of the group of isometries) so that \(spec(\Delta +\Phi_ 1)=spec(\Delta +\Phi_ 2)\). The author also constructs isospectral potentials on manifolds of all ranks with nonpositive curvature.
scientific article; zbMATH DE number 4000926