Local trace formulae and scaling asymptotics in Toeplitz quantization, II

TL;DR

This paper investigates local asymptotics of smoothing kernels in Toeplitz quantization, revealing how they concentrate on fixed points of the linearized Hamiltonian flow, extending spectral analysis techniques.

Contribution

It introduces analogues of spectral kernels in Toeplitz quantization and analyzes their asymptotic behavior near fixed points of the dynamics.

Findings

Smoothing kernels concentrate on fixed loci of the linearized flow.

Asymptotic behavior of kernels is characterized in the Toeplitz setting.

Results extend spectral theory methods to Toeplitz quantization.

Abstract

In the spectral theory of positive elliptic operators, an important role is played by certain smoothing kernels, related to the Fourier transform of the trace of a wave operator, which may be heuristically interpreted as smoothed spectral projectors asymptotically drifting to the right of the spectrum. In the setting of Toeplitz quantization, we consider analogues of these, where the wave operator is replaced by the Hardy space compression of a linearized Hamiltonian flow, possibly composed with a family of zeroth order Toeplitz operators. We study the local asymptotics of these smoothing kernels, and specifically how they concentrate on the fixed loci of the linearized dynamics.