The Generalized Legendre transform and its applications to inverse spectral problems

Let $M$ be a Riemannian manifold, $\tau: G \times M \to M$ an isometric action on $M$ of an $n$-torus $G$ and $V: M \to \mathbb R$ a bounded $G$-invariant smooth function. By $G$-invariance the Schr\"odinger operator, $P=-\hbar^2 \Delta_M+V$, restricts to a self-adjoint operator on $L^2(M)_{\alpha/\hbar}$, $\alpha$ being a weight of $G$ and $1/\hbar$ a large positive integer. Let $[c_\alpha, \infty)$ be the asymptotic support of the spectrum of this operator. We will show that $c_\alpha$ extend to a function, $W: \mathfrak g^* \to \mathbb R$ and that, modulo assumptions on $\tau$ 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 $dV$.