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

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

International Centre for Theoretical Sciences
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