Topics in Quantum Chaos - An Infosys Prize Lecture by Nalini Anantharaman

Explore the fascinating world of quantum chaos in this Infosys Prize Lecture delivered by Nalini Anantharaman. Delve into the historical context of quantum mechanics, starting from Bohr's model of the hydrogen atom and Einstein's early contributions. Examine the development of quantum ergodicity and its applications to various systems, including billiards and graphs. Investigate the Quantum Unique Ergodicity conjecture and recent theoretical advancements. Gain insights into toy models, regular graphs, and their relevance to quantum chaos. Discover numerical simulations, deterministic and random results, and engage with open questions in this cutting-edge field of physics. Conclude with a thought-provoking Q&A session to further expand your understanding of quantum chaos and its implications.

Date: 03 January 2019, 16:00 to
Introduction
Topics in quantum chaos
1. Some history
1913 : Bohr's model of the hydrogen atom
1917 : A paper of Einstein
1925 : operators wave mechanics
Wigner 1950' Random Matrix model for heavy nuclei
Spectral statistics for hydrogen atom in strong magnetic field
Billiard tables
Spectral statistics for several billiard tables
A list of questions and conjectures
II. Quantum ergodicity
Disk
Sphere
Square / torus
Eigenfunctions in a mushroom-shaped billiard. Source A. Backer
Figure: Propagation of a gaussian wave packet in a cardioid. Source A. Backer.
Eigenfunctions in the high frequency limit
QE Theorem simplified: Shnirelman 74, Zelditch 85, Colin de Verdiere 85
Equivalently, there exists a subset S c N of density 1, such that
The full statement uses analysis on phase space, i.e.
Let 2kKEN be an orthonormal basis of L2M, with
Figure: Ergodic billiards. Source A. Backer
Quantum Unique Ergodicity conjecture: Rudnick, Sarnak 94
Theorem: Let M have negative curvature and dimension d. Assume
lll. Toy models
Regular graphs
Why do they seem relevant
A major difference
Some advantages
A geometric assumption
Numerical simulations on Random Regular Graphs RRG
Recent results : deterministic
Examples
Recent results : random
Open questions and suggestions
Q&A