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Eigenfunctions in the high frequency limit
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Topics in Quantum Chaos - An Infosys Prize Lecture by Nalini Anantharaman
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- 1 Date: 03 January 2019, 16:00 to
- 2 Introduction
- 3 Topics in quantum chaos
- 4 1. Some history
- 5 1913 : Bohr's model of the hydrogen atom
- 6 1917 : A paper of Einstein
- 7 1925 : operators wave mechanics
- 8 Wigner 1950' Random Matrix model for heavy nuclei
- 9 Spectral statistics for hydrogen atom in strong magnetic field
- 10 Billiard tables
- 11 Spectral statistics for several billiard tables
- 12 A list of questions and conjectures
- 13 II. Quantum ergodicity
- 14 Disk
- 15 Sphere
- 16 Square / torus
- 17 Eigenfunctions in a mushroom-shaped billiard. Source A. Backer
- 18 Figure: Propagation of a gaussian wave packet in a cardioid. Source A. Backer.
- 19 Eigenfunctions in the high frequency limit
- 20 QE Theorem simplified: Shnirelman 74, Zelditch 85, Colin de Verdiere 85
- 21 Equivalently, there exists a subset S c N of density 1, such that
- 22 The full statement uses analysis on phase space, i.e.
- 23 Let 2kKEN be an orthonormal basis of L2M, with
- 24 Figure: Ergodic billiards. Source A. Backer
- 25 Quantum Unique Ergodicity conjecture: Rudnick, Sarnak 94
- 26 Theorem: Let M have negative curvature and dimension d. Assume
- 27 lll. Toy models
- 28 Regular graphs
- 29 Why do they seem relevant
- 30 A major difference
- 31 Some advantages
- 32 A geometric assumption
- 33 Numerical simulations on Random Regular Graphs RRG
- 34 Recent results : deterministic
- 35 Examples
- 36 Recent results : random
- 37 Open questions and suggestions
- 38 Q&A
