Semi-classical weights and equivariant spectral theory

Abstract

We prove inverse spectral results for differential operators on manifolds and orbifolds invariant under a torus action. These inverse spectral results involve the asymptotic equivariant spectrum, which is the spectrum itself together with "very large" weights of the torus action on eigenspaces. More precisely, we show that the asymptotic equivariant spectrum of the Laplace operator of any toric metric on a generic toric orbifold determines the equivariant biholomorphism class of the orbifold; we also show that the asymptotic equivariant spectrum of a T n -invariant Schrödinger operator on R n determines its potential in some suitably convex cases. In addition, we prove that the asymptotic equivariant spectrum of an S 1 -invariant metric on S 2 determines the metric itself in many cases. Finally, we obtain an asymptotic equivariant inverse spectral result for weighted projective spaces. As a crucial ingredient in these inverse results, we derive a surprisingly simple formula for the asymptotic equivariant trace of a family of semi-classical differential operators invariant under a torus action.

Recovering U(n)-invariant toric Kähler metrics on CPn from the torus equivariant spectrum

“…The case corresponds to ‐invariant metric on . Inverse spectral results for ‐invariant metric on using the equivariant spectrum were previously obtained in and .…”

“…This sort of question is known to be hard and there are not many results in this direction. In , there are a few results concerning ‐invariant metrics on . The results in and concern the equivariant spectrum while those in concern the spectrum.…”

“…In , there are a few results concerning ‐invariant metrics on . The results in and concern the equivariant spectrum while those in concern the spectrum. In this note, we will use some of the general results derived in to prove a positive inverse equivariant spectral result for ‐invariant metrics on .…”

“…In , the authors define a family of equivariant spectral measures associated with any isometric torus action on a Riemannian manifold. More precisely, let be a Riemannian manifold with an isometric torus action.…”

“…There is a well‐defined spectral measure, , that ‘counts’ ‐equivariant eigenfunctions of the operator , where is a small real number and is the Laplacian of the Riemannian manifold. The main result in is an asymptotic expansion for as tends to zero. The leading order term of this asymptotic can be written down explicitly.…”