Inverse spectral results for non-abelian group actions

Abstract

In this paper we will extend to non-abelian groups inverse spectral results, proved by us in an earlier paper (Guillemin and Wang, 2016), for compact abelian groups, i.e. tori. More precisely, Let G be a compact Lie group acting isometrically on a compact Riemannian manifold X. We will show that for the Schrödinger operator −ħ2Δ+V with V∈C∞(X)G, the potential function V is, in some interesting examples, determined by the G-equivariant spectrum. The key ingredient in this proof is a generalized Legendrian relation between the Lagrangian manifolds Graph(dV) and Graph(dF), where F is a spectral invariant defined on an open subset of the positive Weyl chamber.