Semi-classical weights and equivariant spectral theory (Q2629105)

Let \(X\) be a Riemannian manifold admitting an isometric action of a torus \(\mathbb T^n\). The authors show the asymptotic expansion of certain spectral measure associated to the \(\mathbb T^n\)-equivariant spectrum of semi-classical differential operators on \(X\) invariant by \(\mathbb T^n\). The resulting leading term allows them to prove some spectral inverse results. The most important one is that a generic toric orbifold endowed with a toric Kähler metric is determined by the asymptotic equivariant spectrum of the Laplacian up to two choices up to symplectomorphism. Similarly, the asymptotic equivariant spectrum of a \(\mathbb T^n\)-invariant Schrödinger operator on \(\mathbb R^n\) determines its potential in some suitably convex cases. Furthermore, the asymptotic equivariant spectrum of a \(S^1\)-invariant metric on \(S^2\) determines the metric in many cases. Finally, the asymptotic equivariant spectrum of a weighted projective space determines the product of the weights, in particular it determines the weights in the case they are different primes.