Equivariant local scaling asymptotics for smoothed T\"{o}plitz spectral projectors

TL;DR

This paper investigates the local asymptotic behavior of smoothed spectral projectors for a class of elliptic Toeplitz operators on circle bundles, extending understanding of spectral asymptotics in the presence of symmetries.

Contribution

It provides new local scaling asymptotics for spectral projectors of Toeplitz operators, without requiring invariance under the circle action, and derives an equivariant Weyl law.

Findings

Derived explicit local asymptotic expansions for spectral projectors.

Extended spectral asymptotics to operators with Hamiltonian symmetries.

Established an equivariant Weyl law from local asymptotics.

Abstract

Let $X$ be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated to a first order elliptic T\"{o}plitz operator $T$ on $X$, possibly in the presence of Hamiltonian symmetries. The resulting expansion is then used to give a local derivation of an equivariant Weyl law. It is not required that $T$ be invariant under the structure circle action, that is, $T$ needn't be a Berezin-T\"{o}plitz operator.