Topics in quantum chaos (An Infosys Prize Lecture) by Nalini Anantharaman

Описание: Distinguished Lectures

Topics in quantum chaos (An Infosys Prize Lecture)

Speaker: Nalini Anantharaman (Institute for Advanced Study, University of Strasbourg, France)

Date: 03 January 2019, 16:00 to 17:00

Venue: Ramanujan Lecture Hall, ICTS-TIFR

A hundred years ago, Einstein wondered about a good description of the spectrum of disordered systems in the emerging quantum theory. Although a full mathematical answer is still missing, a lot of progress has been made to describe chaotic behavior of waves in quantum mechanics

Table of Contents (powered by https://videoken.com)
0:00:00 Introduction
0:05:45 Topics in quantum chaos
0:06:09 1. Some history
0:07:25 1913 : Bohr's model of the hydrogen atom
0:08:45 1917 : A paper of Einstein
0:11:17 1925 : operators wave mechanics
0:15:16 Wigner 1950' Random Matrix model for heavy nuclei
0:17:01 Spectral statistics for hydrogen atom in strong magnetic field
0:18:02 Billiard tables
0:19:31 Spectral statistics for several billiard tables
0:21:27 A list of questions and conjectures
0:25:43 II. Quantum ergodicity
0:28:07 Disk
0:28:43 Sphere
0:29:00 Square / torus
0:31:11 Eigenfunctions in a mushroom-shaped billiard. Source A. Backer
0:31:27 Figure: Propagation of a gaussian wave packet in a cardioid. Source A. Backer.
0:32:36 Eigenfunctions in the high frequency limit
0:33:40 QE Theorem (simplified): Shnirelman 74, Zelditch 85, Colin de Verdiere 85
0:35:52 Equivalently, there exists a subset S c N of density 1, such that
0:37:29 The full statement uses analysis on phase space, i.e.
0:38:58 Let (2)k)KEN be an orthonormal basis of L2(M), with
0:39:37 Figure: Ergodic billiards. Source A. Backer
0:41:28 Quantum Unique Ergodicity conjecture: Rudnick, Sarnak 94
0:43:50 Theorem: Let M have negative curvature and dimension d. Assume
0:50:23 lll. Toy models
0:51:36 Regular graphs
0:53:18 Why do they seem relevant
0:55:22 A major difference
0:56:03 Some advantages
0:57:37 A geometric assumption
0:58:42 Numerical simulations on Random Regular Graphs (RRG)
0:59:42 Recent results : deterministic
1:04:55 Examples
1:06:35 Recent results : random
1:08:55 Open questions and suggestions
1:11:24 Q&A