关于Berezin-Toeplitz算子的伪谱

Szegö Limit Theorems for Singular Berezin-Toeplitz Operators

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 Szegö 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.

正则能量下 Berezin-Toeplitz 准模的半经典集中估计

Semiclassical concentration estimates for Berezin-Toeplitz quasimodes for regular energies

本文针对算子符号取正则值的 Berezin-Toeplitz 准模，在紧空间和 $\mathbb{C}^n$ 上证明尖锐 $L^p$ 界，并借助与伪微分算子的联系导出准模 FBI 变换的 $L^p$ 估计。

正Berezin liminf不意味着径向Toeplitz算子在Bergman和Fock空间上的本质正性

Positive Berezin liminf does not imply essential positivity for radial Toeplitz operators on Bergman and Fock spaces

研究径向Toeplitz算子在Bergman和Fock空间上的本质正性是否可通过Berezin变换的渐进行为检测。构造了具有正Berezin liminf但本质谱包含负点的径向符号，推翻Perälä-Virtanen猜想。

Spectral aspects of the Berezin transform

We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kahler manifolds with constant scalar curvature. Furthermore, viewing POVMs as …

Symbolic calculus for Toeplitz operators with half-forms

This paper is devoted to the use of half-form bundles in the symbolic calculus of Berezin-Toeplitz operators on Kahler manifolds. We state the Bohr-Sommerfeld conditions and relate them to the functional calculus of Toeplitz operators, a trace formula and the characteristic classes in deformation quantization. We also develop the symbolic calculus of Lagrangian sections, with the crucial estimate of the subprincipal terms.

复射影空间量子化中的最优余项估计

Optimal Remainder Estimates in the Quantization of Complex Projective Spaces

本文研究复射影空间的 Berezin-Toeplitz 量子化，通过完全渐近展开和算子不等式得到正则性最优且常数精确或渐近精确的余项估计。

Limit Operators, Compactness and Essential Spectra on Bounded Symmetric Domains

This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric domains and some cannot. The aim of this paper is to close the gaps to obtain comparable results for general bounded symmetric domains. In particular, we show that a Toeplitz operator on the Bergman space $A^p_ν$ is Fredholm if and only if all of its limit operators are invertible. Even more generally, we show that this is in fact true for all …

关于Toeplitz算子的半经典Szego型极限定理

A Semi-Classical Szegő-type Limit Theorem for Toeplitz Operators

本文研究了Toeplitz算子在加权Bergman空间和L²(G)上的Szego型极限定理，针对紧局部紧阿贝尔群分别给出形式化表达，并统一扩展了多个经典结果。

Toeplitz operators with analytic symbols

We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact K{ä}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{ä}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{ä}hler manifold as N $\rightarrow$ +$\infty$. We also prove that To…

Coburn's Lemma 和随机Jacobi算子的有限截断方法

Coburn's Lemma and the Finite Section Method for Random Jacobi Operators

本文研究了有限和无限三对角随机矩阵的谱和伪谱，证明了在伪ergodic条件下，有限和无限三对角随机矩阵的谱具有特定性质，并展示了有限截断方法在求解方程和估计谱时的有效性。

Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via parallel transport in the quantum spaces over the induced path of almost complex structures, and we establish various semi-classical estimates. In particular, we establish a Gutzwiller trace formula for the Kostant-Souriau operator and compute explicitly the leading term. We then desc…

Spectral limits of semiclassical commuting self-adjoint operators

Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of $\hbar$-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for q…