Semiclassical concentration estimates for Berezin-Toeplitz quasimodes for regular energies

TL;DR

The article addresses sharp $L^p$ concentration bounds for semiclassical quasimodes of Berezin-Toeplitz operators on both flat and compact Kähler manifolds at regular energy levels. It develops a comprehensive framework that links Berezin-Toeplitz quantization to pseudodifferential operators through the FBI/Bargmann transform, and uses Lagrangian-state techniques to derive precise $L^p$ growth rates for quasimodes. The main result, $\|V_N\|_{L^p(M)} = O\left(N^{(n-1/2)(1/2-1/p)}\right)$ for $p\in[2,\infty]$, is shown to be sharp via explicit constructions on projective spaces, illustrating phase-space concentration without caustics. The paper also demonstrates how flat-space results can be inferred by adapting the Toeplitz calculus to $\mathbb{C}^n$ and relating them to pseudodifferential operators on $\mathbb{R}^n$ using the FBI transform, providing a versatile approach to spectral concentration questions in Berezin-Toeplitz quantization.

Abstract

The purpose of this article is to prove sharp $L^p$ bounds for quasimodes of Berezin-Toeplitz operators. We consider examples with explicit computations and a general situation on compact spaces and $\mathbb{C}^n$. In both cases the eigenvalue is a regular value of the operator symbol. We then use the link between pseudodifferential and Berezin-Toeplitz operators to obtain an $L^p$ bound of the FBI transform of quasimodes of pseudodifferential operators.

Figures (1)

• Figure 1: The exponent $\rho$ as a function of $\frac{1}{p}$.

Theorems & Definitions (80)

• Theorem 1.1
• Definition 2.1
• Proposition 2.1: barg61 Chapter 1.b
• proof
• Proposition 2.2
• Definition 2.2
• Lemma 2.1
• proof
• Proposition 2.3
• proof