The generalized Legendre transform and its applications to inverse spectral problems

Abstract

Let M be a Riemannian manifold, τ : G x M --> M an isometric action on M of an n-torus G and V : M --> R a bounded G-invariant smooth function. By G-invariance the Schrödinger operator, P = -h[superscript 2][Delta]M + V, restricts to a self-adjoint operator on L[superscript 2](M)[subscript alpha over h], alpha being a weight of G and 1[over h] a large positive integer. Let [c[subscript alpha], [infinity]] be the asymptotic support of the spectrum of this operator. We will show that c[subscript alpha] extend to a function, W : g*-->R and that, modulo assumptions on τ and V one can recover V from W, i.e. prove that V is spectrally determined. The main ingredient in the proof of this result is the existence of a 'generalized Legendre transform' mapping the graph of dW onto the graph of dV.