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

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 Schrodinger 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.