Inverse spectral results for non-abelian group actions.

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 − ħ 2 Δ + 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 (d V) and Graph (d F) , where F is a spectral invariant defined on an open subset of the positive Weyl chamber. [ABSTRACT FROM AUTHOR]