The generalized Legendre transform and its applications to inverse spectral problems (Q2786435)

Let \((M,ds^2)\) be a Riemannian manifold, let \(V:M\rightarrow\mathbb{R}\) be a potential function, let \(\Delta_M\) be the Laplace operator, and let \(P:=h^2\Delta_M+V\) be the semi-classical Schrödinger operator. Assume either that \(M\) is compact or that \(V\) is proper and tends to \(+\infty\) as \(x\rightarrow\infty\) to ensure that the spectrum of \(P\) is discrete. Let \(\mathfrak{g}\) be the Lie algebra of a torus \(G\). Assume that the Riemannian metric \(ds^2\) and the potential \(V\) are invariant under an action of \(G\) on \(M\) so that \(G\) preserves the eigenspaces of \(P\) and restricts to a self-adjoint operator \(P_{\alpha,h}\) on \(L^2(M)_{\alpha/h}\) where \(\alpha\) is a weight of \(G\) and \(\frac1h\) is a large positive integer. Let \([c_\alpha,\infty)\) be the asymptotic support of the spectrum of \(P_{\alpha,h}\). The authors show \(c_\alpha\) extends to a function from \(\mathfrak{g}^*\) to \(\mathbb{R}\) and modulo suitable assumptions one can recover the potential \(V\) from this function and thus \(V\) is spectrally determined. The first section contains an introduction to the problem and outlines the structure of the paper. Section 2 treats toric manifolds and their canonical reduced Riemannian metrics. Section 3 deals with the inversion formula for the Legendre transform. Section 4 involves inverse spectral results on toric manifolds. Section 5 presents local inverse and spectral rigidity results. The final section treats `reduction in stages' which involves the spectral invariants of a Schr\"doing operator gotten by reduction with respect to the semi-classical weights \(\frac\alpha h\).