The Generalized Legendre transform and its applications to inverse spectral problems

Abstract

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 Schrödinger operator, $P=-{\hslash }^2{\Delta }_M+V$, restricts to a self-adjoint operator on $L^2(M{)}_{\alpha /\hslash }$, $\alpha$ being a weight of $G$ and $1/\hslash$ 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, $W:{\mathfrak{g}}^{\ast }\to \mathbb{R}$ 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 $dV$.

23 pages