Optimal Remainder Estimates in the Quantization of Complex Projective Spaces

TL;DR

This paper provides detailed asymptotic expansions and optimal remainder estimates for Berezin-Toeplitz quantization on complex projective spaces, enhancing understanding of the accuracy and bounds of these quantization methods.

Contribution

It introduces full asymptotic expansions for Berezin transforms and Toeplitz operator products with optimal remainder control, improving precision in quantization of complex projective spaces.

Findings

Remainder controlled by the next term of the expansion

Bounds are optimal with sharp constants

Asymptotic expansions are fully characterized

Abstract

We study Berezin-Toeplitz quantization of complex projective spaces $\mathbb{CP}^{d-1}$ and obtain full asymptotic expansions of the Berezin transformation and of products of Toeplitz operators. In each case, the remainder is controlled by the next term of the expansion, either through a positivity-preserving transformation or via an operator inequality. This leads to bounds which are optimal in terms of the required regularity and feature sharp or asymptotically sharp constants.