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An exotic calculus of Berezin-Toeplitz operators

Izak Oltman

TL;DR

We extend Berezin-Toeplitz quantization to exotic symbol classes Sδ(m) with δ ∈ [0,1/2) on quantizable Kähler manifolds and prove kernel asymptotics and a sharp composition formula T_{N,f} ∘ T_{N,g} = T_{N,h} + O(N^{−∞}) with h ∈ Sδ(m1 m2). The framework employs almost analytic extensions and the Melin-Sjöstrand complex stationary phase to handle unbounded amplitudes, enabling a full calculus with a parametrix, functional calculus, and a trace formula; these tools underpin probabilistic Weyl laws for randomly perturbed Toeplitz operators. Key contributions include explicit star-product expansions with h_0 = f g and h_1 given by differential operators, plus a robust trace formula and elliptic parametrices. The results broaden the scope of geometric quantization and have potential impact on PDEs and quantum chaos on Kähler manifolds by enabling spectral analysis for a wider class of observables.

Abstract

We develop a calculus of Berezin-Toeplitz operators quantizing exotic classes of smooth functions on compact Kähler manifolds and acting on holomorphic sections of powers of positive line bundles. These functions (classical observables) are exotic in the sense that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sjöstrand's method of complex stationary phase. We obtain a functional calculus result, a trace formula, and a parametrix construction for this larger class of functions. These results are crucially used in proving a probabilistic Weyl-law for randomly perturbed (standard) Berezin-Toeplitz operators.

An exotic calculus of Berezin-Toeplitz operators

TL;DR

We extend Berezin-Toeplitz quantization to exotic symbol classes Sδ(m) with δ ∈ [0,1/2) on quantizable Kähler manifolds and prove kernel asymptotics and a sharp composition formula T_{N,f} ∘ T_{N,g} = T_{N,h} + O(N^{−∞}) with h ∈ Sδ(m1 m2). The framework employs almost analytic extensions and the Melin-Sjöstrand complex stationary phase to handle unbounded amplitudes, enabling a full calculus with a parametrix, functional calculus, and a trace formula; these tools underpin probabilistic Weyl laws for randomly perturbed Toeplitz operators. Key contributions include explicit star-product expansions with h_0 = f g and h_1 given by differential operators, plus a robust trace formula and elliptic parametrices. The results broaden the scope of geometric quantization and have potential impact on PDEs and quantum chaos on Kähler manifolds by enabling spectral analysis for a wider class of observables.

Abstract

We develop a calculus of Berezin-Toeplitz operators quantizing exotic classes of smooth functions on compact Kähler manifolds and acting on holomorphic sections of powers of positive line bundles. These functions (classical observables) are exotic in the sense that their derivatives are allowed to grow in ways controlled by local geometry and the power of the line bundle. The properties of this quantization are obtained via careful analysis of the kernels of the operators using Melin and Sjöstrand's method of complex stationary phase. We obtain a functional calculus result, a trace formula, and a parametrix construction for this larger class of functions. These results are crucially used in proving a probabilistic Weyl-law for randomly perturbed (standard) Berezin-Toeplitz operators.
Paper Structure (17 sections, 27 theorems, 213 equations, 2 figures)

This paper contains 17 sections, 27 theorems, 213 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A new symbol class
  3. Almost Analytic Extension
  4. Composition of Toeplitz operators with symbols in Sδ(m)
  5. Holomorphic Sections
  6. Bergman Kernels
  7. An Asymptotic Expansion of the Kernel of a Toeplitz Operator
  8. Composition of Toeplitz Operators
  9. Applications of the Exotic Calculus
  10. Parametrix Construction
  11. Functional Calculus
  12. Trace Formula
  13. Computation of the second term in the star product
  14. Simple Example Worked Out
  15. Controlling Error Terms
  16. ...and 2 more sections

Key Result

Theorem 1

There exists $h\in S_\delta (m_1m_2)$ such that Moreover $h$ can be locally written as where $(\partial \overline{\partial} \varphi(x))^{j,k}$ is such that $\sum _k (\partial\overline{\partial} \varphi(x) )^{j,k} (\partial_k \overline{\partial} _\ell \varphi(x)) = \delta _{j,\ell}$ for $j,\ell = 1,\dots ,d$.

Figures (2)

  • Figure 1: Schematic of the contour deformation.
  • Figure 2: Diagram of the restrictions of $\tilde{\Psi}$ to totally real vector spaces. First, we have $\tilde{\Psi}(p,x,\bar{y}) \in C^\infty (\mathbb{C}^{2d}_p \times \mathbb{C}^d_x \times \mathbb{C}^d_y)$ . This is almost analytic off of $p\in \mathbb{R}^{2d}$, whose restriction to $p\in \mathbb{R}^{2d}$ is $\Psi (p,x,\bar{y}) \in C^\infty (\mathbb{R}^{2d}_p \times \mathbb{C}^d_x \times \mathbb{C}^d_y)$. We can restrict $\Psi$ to either $p = ({\rm{Re}} \left ( x\right ) , {\rm{Im}} \left ( x \right ) )$ or $p = ({\rm{Re}} \left ( y\right ) , {\rm{Im}} \left ( y \right ) )$ to get the same function as shown. When either of these functions are restricted to $x = y$ we get the zero function. Understanding various $\overline{\partial}$ estimates on $\tilde{\Psi}$ is the core part of this step of the proof.

Theorems & Definitions (58)

  • Theorem : Composition Formula
  • Theorem : Trace Formula
  • Theorem : Existence of a Parametrix
  • Theorem : Functional Calculus
  • Definition 2.1: $\bm{\delta}$-order Function on $\bm{X}$
  • Definition 2.2: $\bm{\delta}$-order Function on $\mathbb{R}^d$ and $\mathbb{C}^d$
  • Example 2.1
  • proof
  • Definition 2.3: $\bm{S_\delta (m)}$
  • Definition 2.4: Asymptotic Expansion of Symbols
  • ...and 48 more