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2018 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

A Brief Introduction to Berezin–Toeplitz Operators on Compact Kähler Manifolds

verfasst von: Yohann Le Floch

Verlag: Springer International Publishing

Buchreihe : CRM Short Courses

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This text provides a comprehensive introduction to Berezin–Toeplitz operators on compact Kähler manifolds. The heart of the book is devoted to a proof of the main properties of these operators which have been playing a significant role in various areas of mathematics such as complex geometry, topological quantum field theory, integrable systems, and the study of links between symplectic topology and quantum mechanics. The book is carefully designed to supply graduate students with a unique accessibility to the subject. The first part contains a review of relevant material from complex geometry. Examples are presented with explicit detail and computation; prerequisites have been kept to a minimum. Readers are encouraged to enhance their understanding of the material by working through the many straightforward exercises.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Berezin–Toeplitz operators appear in the study of the semiclassical limit of the quantisation of compact symplectic manifolds. They were introduced by Berezin [5], their microlocal analysis was initiated by Boutet de Monvel and Guillemin [33], and they have been studied by many authors since, see for instance [8, 9, 14, 23, 30, 49]. This list is of course far from exhaustive, and the very nice survey paper by Schlichenmaier [43] gives a review of the Kähler case and contains a lot of additional useful references.
Yohann Le Floch
Chapter 2. A Short Introduction to Kähler Manifolds
Abstract
In this chapter, we recall some general facts about complex and Kähler manifolds. It is not an exhaustive list of such facts, but rather an introduction of objects and properties that we will need in the rest of the notes. The interested reader might want to take a look at some standard textbooks, such as [24, 35] for instance.
Yohann Le Floch
Chapter 3. Complex Line Bundles with Connections
Abstract
Let us now recall some facts about complex line bundles. A certain number of definitions and properties could be stated for general vector bundles, but we prefer to focus on the one-dimensional case, since this is the case that will be encountered in the following sections.
Yohann Le Floch
Chapter 4. Geometric Quantisation of Compact Kähler Manifolds
Abstract
Let \((M, \omega )\) be a compact, connected, Kähler manifold. The aim of this chapter is to construct a Hilbert space (or rather a family of Hilbert spaces) which will serve as the state space of quantum mechanics associated with the classical phase space M.
Yohann Le Floch
Chapter 5. Berezin–Toeplitz Operators
Abstract
As before, let \((M, \omega )\) be a prequantizable, compact, connected Kähler manifold, let \(L \rightarrow M\) be a prequantum line bundle, and let \(\mathcal {H}_k\) be the associated Hilbert spaces. Let \(L^2(M, L^k)\) be the completion of the space of smooth sections of \(L^k \rightarrow M\) with respect to the inner product \(\langle \,\cdot \,,\cdot \, \rangle _k\) introduced earlier, and let \(\varPi _k\) be the orthogonal projector from \(L^2(M, L^k)\) to \(\mathcal {H}_k\). This projector is often called the Szegő projector.
Yohann Le Floch
Chapter 6. Schwartz Kernels
Abstract
In this section we give a quick review of the notion of section distributions and Schwartz kernels of operators acting on spaces of sections of vector bundles. A good reference for this material is the classical textbook by Hörmander [26].
Yohann Le Floch
Chapter 7. Asymptotics of the Projector
Abstract
The goal of this chapter is to describe the asymptotic properties of the Schwartz kernel of the Szegő projector \(\varPi _k:L^2(M, L^k) \rightarrow \mathcal {H}_k\), called the Bergman kernel.
Yohann Le Floch
Chapter 8. Proof of Product and Commutator Estimates
Abstract
The aim of this chapter is to prove Theorems 5.​2.​2 and 5.​2.​3.
Yohann Le Floch
Chapter 9. Coherent States and Norm Correspondence
Abstract
Finally, we prove the lower bound for the operator norm of a Berezin–Toeplitz operator. In order to do so, we use the so-called coherent states.
Yohann Le Floch
Backmatter
Metadaten
Titel
A Brief Introduction to Berezin–Toeplitz Operators on Compact Kähler Manifolds
verfasst von
Yohann Le Floch
Copyright-Jahr
2018
Electronic ISBN
978-3-319-94682-5
Print ISBN
978-3-319-94681-8
DOI
https://doi.org/10.1007/978-3-319-94682-5

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