Semiclassical spectral estimates for Toeplitz operators
type: Item
instance of: scholarly article
title: Semiclassical spectral estimates for Toeplitz operators
zbMATH Open document ID: 0920.58059
author: David Borthwick
author: Thierry Paul
author: A. Uribe
publication date: 1998-11-09T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75314
review text: Let \(X\) be a compact Kähler manifold of real dimension \(2n\) and \(L\to X\) a holomorphic Hermitian line bundle such that the curvature of its natural connection is a Kähler form of \(X\). The quantization of \(X\) with Planck's constant \(1/k\) is the space \(H^0(x,L^{\otimes k})\), and the Toeplitz quantization of a classical Hamiltonian function \(H\in C^\infty(X)\) is \(S_k= \pi_k M_H\pi_k\), where \(\pi_k: L^2(X,L^{\otimes k})\to H^0(X, L^{\otimes k})\) is the orthogonal projector and \(M_H\) is the operator of multiplication by \(H\). Let \(P\subset L^*\) be the unit circle bundle in the dual of \(L\). By the condition that the curvature of \(L\) be the Kähler form, \(P\) is strictly pseudoconvex and the contact form \(\alpha\) is the connection form on \(P\). Let \(\partial_\theta\) denote the contact vector field on \(P\), uniquely determined by \(\alpha\) (\(\partial_\theta\) generates a free \(S^1\) \((=\mathbb{R}/2\pi \mathbb{Z})\) action of automorphisms of \(P\)). The operator \(\partial_\theta\) determines the decomposition of the Hardy space of \(P\) into eigenspaces of \(\partial_\theta: {\mathcal H}=\oplus{\mathcal H}_k\). In this decomposition one can naturally make the identification: \({\mathcal H}_k= H^0(X, L^{\otimes k})\). Under this identification \(S_k\) gets identified with the operator on \({\mathcal H}_k\) with eigenvalues and eigenfunctions \((E_i^k, \psi_i^k)\). Let \(\phi\) be Hamiltonian flow of \(H\) on \(X\) and \(\pi:P\to X\) the obvious projection. The first main result (Theorem 1.1) is concerned with the integral kernel of \(\varphi(k(T_k- E))\), where \(T\) is the Toeplitz operator on \({\mathcal H}\), \(T_k= T|{\mathcal H}_k\) and \(\varphi\) is a smooth test function with compactly supported Fourier transform, and the second one (Theorem 1.2) concerns the trace of \(\varphi (k(T_k-E))\). Theorem 1.1. Let \(p_1,p_2\in P\), \(x_i= \pi(p_i)\) and in case \(p_1=p_2\) the point \(x_1= x_2\) is not fixed of \(\phi\). Then: (1) if…
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J40
Mathematics Subject Classification ID: 81Q20
zbMATH DE Number: 1220358
zbMATH Keywords: Toeplitz operators
zbMATH Keywords: semiclassical analysis
zbMATH Keywords: spectral theory
zbMATH Keywords: geometric quantization
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
OpenAlex ID: W2330884496
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DOI: 10.5802/AIF.1654
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1220358