Szeg\"o Limit Theorems for Singular Berezin-Toeplitz Operators

TL;DR

This paper establishes Szeg"o limit theorems and asymptotic spectral properties for Berezin-Toeplitz operators supported on submanifolds in complex space, linking geometric conditions to quantum semi-classical limits.

Contribution

It provides new asymptotic formulas for spectral measures and Schatten norms of Berezin-Toeplitz operators on isotropic or co-isotropic submanifolds, advancing understanding of their semi-classical behavior.

Findings

Szeg"o limit theorems for specific submanifold cases

Asymptotic spectral measure moments computed

Semi-classical entropy limits derived

Abstract

We consider Berezin-Toeplitz operators whose multipliers are compactly supported densities carried by a submanifold of ${\mathbb C}^N$ . We compute asymptotically the moments of their spectral measures, and we prove Szeg\"o limit theorems in cases when the submanifold is isotropic or co-isotropic, from which Weyl estimates follow. We also obtain asymptotics of the Schatten norms of such operators. Rescaled versions of these operators can be thought of as quantum mechanical mixed states, and our results give the semi-classical limit of their entropy.