Exploring various aspects of quantum chaos, Anantharaman begins with a historical introduction, then focuses on the delocalization of eigenfunctions of Schrödinger operators for chaotic Hamiltonian systems. She also includes a short introduction to microlocal analysis necessary for proving the Shnirelman theorem, and presents her own work on the entropy of eigenfunctions on negatively curved manifolds and on quantum ergodicity of eigenfunctions on large graphs. She concludes with a survey of results on eigenfunctions on a round sphere, and a rather detailed exposition of the result by Backhousz and Szegedy on the Gaussian distribution of eigenfunctions on random regular graphs. Like the lecture series it is based on, the book is for mathematicians from the graduate level upward. Distributed in the US by the American Mathematical Society. Annotation ©2022 Ringgold, Inc., Portland, OR (protoview.com)
Author(s): Nalini Anantharaman
Series: Zurich Lectures in Advanced Mathematics, 27
Publisher: EMS
Year: 2022
Language: English
Pages: 134
1 Introduction
1.1 History of the problem
1.2 Conjectures, scope of the book
2 High frequency delocalisation
2.1 L^p-norms as measures of delocalisation?
2.2 The Shnirelman theorem
2.3 Overview of pseudo-differential operators
2.4 The geodesic flow and the ergodicity assumption
2.5 Proof of the Quantum Ergodicity theorem
3 Entropy and support of semiclassical measures
3.1 Quantum Unique Ergodicity conjecture
3.2 Statement of results
3.3 What is entropy?
3.4 Proof of the entropy result (Theorem 3.1)
3.5 A few words on Dyatlov and Jin's result
4 Quantum ergodicity on regular graphs
4.1 Regular graphs: quantum ergodicity
4.2 Harmonic analysis on the (q+1)-regular tree
4.3 Proof of Theorem 4.2
5 Quantum ergodicity on the sphere
6 Quantum ergodicity on non-regular graphs
6.1 Introduction
6.2 A proof based on the non-backtracking random walk
6.3 Adaptation to non-regular graphs
6.4 Trees of finite cone type and their stochastic perturbations
6.5 The measure 污渠gleaൡ渠gle_γ: one example
7 Backhausz and Szegedy's theorem
7.1 Gaussianity of eigenfunctions
7.2 Elements of the proof
References
