Local trace formulae and scaling asymptotics in Toeplitz quantization

TL;DR

This paper develops a local trace formula for Toeplitz operators on symplectic manifolds, revealing scaling asymptotics related to Hamiltonian flows and symmetries, extending previous global results.

Contribution

It introduces a local version of the trace formula for Toeplitz operators associated with symmetries, focusing on scaling asymptotics along fixed loci.

Findings

Derived local trace formula involving scaling asymptotics

Connected trace formula to Hamiltonian flow fixed points

Extended global trace results to local symplectic settings

Abstract

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hamiltonian flow of the symbol, reminiscent of the scaling asymptotics of the equivariant components of the Szeg\"o kernel along the diagonal.