Topics in Quantum Chaos - An Infosys Prize Lecture by Nalini Anantharaman
International Centre for Theoretical Sciences via YouTube
Overview
Syllabus
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
Taught by
International Centre for Theoretical Sciences