The Generalized Legendre transform and its applications to inverse spectral problems

TL;DR

This paper introduces a generalized Legendre transform to connect spectral data of a Schrödinger operator on a Riemannian manifold with a torus symmetry to the potential function, enabling its recovery from spectral information.

Contribution

It develops a new generalized Legendre transform framework that allows reconstructing the potential function from spectral asymptotics in inverse spectral problems.

Findings

Spectral support function extends to a map on the dual Lie algebra.

Potential function can be recovered from spectral data under certain conditions.

The generalized Legendre transform maps the graph of the derivative of the spectral support to the derivative of the potential.

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 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 $dV$.