# Toeplitz operators with analytic symbols

## TL;DR

This paper develops an asymptotic analysis and calculus for Toeplitz operators with real-analytic symbols on compact Kähler manifolds, providing exponential error control and applications to eigenfunction concentration.

## Contribution

It introduces a calculus for Toeplitz operators with real-analytic symbols and establishes exponential accuracy for their composition, inversion, and kernel estimates on real-analytic Kähler manifolds.

## Key findings

- Bergman kernel controlled up to O(e^{-cN})
- Toeplitz operators with analytic symbols can be composed and inverted with exponential accuracy
- Eigenfunction concentration exhibits exponential decay in classically forbidden regions

## Abstract

We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact K{\"a}hler manifold. These objects depend on an integer N and we study, in the limit N $\rightarrow$ +$\infty$, situations in which one can control them up to an error O(e^{-cN}) for some c > 0. We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to K{\"a}hler man-ifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to O(e^{-cN}) on any real-analytic K{\"a}hler manifold as N $\rightarrow$ +$\infty$. We also prove that Toeplitz operators with analytic symbols can be composed and inverted up to O(e^{-cN}). As an application, we study eigenfunction concentration for Toeplitz operators if both the manifold and the symbol are real-analytic. In this case we prove exponential decay in the classically forbidden region.