Quantum symbolic dynamics
|
|
- Kathryn Bennett
- 5 years ago
- Views:
Transcription
1 Quantum symbolic dynamics Stéphane Nonnenmacher Institut de Physique Théorique, Saclay Quantum chaos: routes to RMT and beyond Banff, 26 Feb. 2008
2 What do we know about chaotic eigenstates? Hamiltonian H(q, p), such that the dynamics on Σ E is chaotic. H = Op (H) has discrete spectrum (E,n, ψ,n ) near the energy E. Laplace operator on a chaotic cavity, or on a surface of negative curvature. What do the eigenstates ψ look like in the semiclassical limit 1.?
3 Chaotic quantum maps chaotic map Φ on a compact phase space propagators U N (Φ), N 1 (advantage: easy numerics, some models are partially solvable ). Husimi densities of some eigenstates.
4 Phase space localization Interesting to study the localization of ψ in both position and momentum: phase space description. Ex: for any bounded test function (observable) f(q, p), study the matrix elements f(ψ ) = ψ, Op (f) ψ = dqdp f(q, p) ρ ψ (q, p) Depending on the quantization, the function ρ ψ can be the Wigner function, the Husimi function. Def: from any sequence (ψ ) 0, one can always extract a subsequence (ψ ) such that for any f, lim f(ψ ) = µ(f) 0 µ is a measure on phase space, called the semiclassical measure of the sequence (ψ ). µ takes the macroscopic features of ρ ψ into account. Fine details (e.g. oscillations, correlations, nodal lines) have disappeared.
5 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics.
6 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics. If Φ is ergodic w.r.to the Liouville measure, one can show (again, using Egorov) that almost all eigenstates ψ,n become equidistributed when : N f, N 1 f(ψ,n ) f dµ L 2 h 0 0. n=1 Quantum ergodicity [Shnirelman 74, Zelditch 87, Colin de Verdière 85]
7 Quantum-classical correspondence For the eigenstates of H, µ is supported on Σ E. Let us use the flow Φ t generated by H, and call U t = e ith / the quantum propagator. Egorov s theorem: for any observable f, U t Op (f) U t = Op (f Φ t ) + O( e Λt ) Idem for a quantum map U = U N (Φ). Breaks down at the Ehrenfest time T E = log /Λ (cf. R.Whitney s talk). the semiclassical measure µ is thus invariant through the classical dynamics. If Φ is ergodic w.r.to the Liouville measure, one can show (again, using Egorov) that almost all eigenstates ψ,n become equidistributed when : f, N 1 N f(ψ,n ) f dµ L 2 h 0 0. n=1 Quantum ergodicity [Shnirelman 74, Zelditch 87, Colin de Verdière 85] do ALL eigenstates become equidistributed [Rudnick-Sarnak 93]? exceptional sequences of eigenstates? Or are there
8 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06].
9 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06]. explicit exceptional semiclassical measures for the quantum cat map [Faure-N- DeBièvre] and Walsh-quantized baker s map [Anantharaman-N 06] (cf. Kelmer s talk). B
10 Some counter-examples No exceptional sequences for arithmetic eigenstates of (2D) cat maps [Rudnick- Sarnak 00] and for arithmetic surfaces [Lindenstrauss 06]. explicit exceptional semiclassical measures for the quantum cat map [Faure-N- DeBièvre] and Walsh-quantized baker s map [Anantharaman-N 06] (cf. Kelmer s talk). B in general, can ANY invariant measures occur as a semiclassical measure? In particular, can one have strong scars µ sc = δ P O?
11 Symbolic dynamics of the classical flow To classify Φ-invariant measures, one may use a phase space partition; each trajectory will be represented by a symbolic sequence ɛ 1 ɛ 0 ɛ 1 denoting its history. 1 t=0 2 [ ε i ] Φ 1 [ ε ] i Φ 2 [ ] εi t=2 t=1 At each time n, the rectangle [ɛ 0 ɛ n ] contains all points sharing the same history between times 0 and n (ex: [121]). Let µ be an invariant proba. measure. The time-n entropy H n (µ) = ɛ 0,...,ɛ n µ([ɛ 0 ɛ n ]) log µ([ɛ 0 ɛ n ]) measures the distribution of the weights µ([ɛ 0 ɛ n ]).
12 KS entropy of semiclassical measures The Kolmogorov-Sinai entropy H KS (µ) = lim n n 1 H n (µ) represents the information complexity of µ w.r.to the flow. Related to localization: H KS (δ P O ) = 0, H KS (µ L ) = log J u dµ L. Affine function of µ.
13 KS entropy of semiclassical measures The Kolmogorov-Sinai entropy H KS (µ) = lim n n 1 H n (µ) represents the information complexity of µ w.r.to the flow. Related to localization: H KS (δ P O ) = 0, H KS (µ L ) = log J u dµ L. Affine function of µ. t=0 1 2 t=2 t=1 What can be the entropy of a semiclassical measure for an Anosov system?
14 Semiclassical measures are at least half-delocalized Theorem [Anantharaman-Koch-N 07]: For any quantized Anosov system, any semiclassical measure µ satisfies H KS (µ) log J u dµ 1 2 Λ max(d 1) full scars are forbidden. bound. Some of the exceptional measures saturate this lower p q
15 Quantum partition of unity Using quasi-projectors P j = Op (χ j ) on the components of the partition, we construct a quantum partition of unity Id = J j=1 P j. Egorov thm for n < T E the operator P ɛ0 ɛ n = U n Pɛ0 ɛ n def = U n P ɛn U P ɛ1 UP ɛ0 is a quasi-projector on the rectangle [ɛ 0 ɛ n ]. t=0 1 2 t=2 t=1 Can we get some information on the distribution of the weights P ɛ0 ɛ n ψ 2 h 0 µ([ɛ 0 ɛ n ])?
16 Quantum partition of unity Using quasi-projectors P j = Op (χ j ) on the components of the partition, we construct a quantum partition of unity Id = J j=1 P j. Egorov thm for n < T E the operator P ɛ0 ɛ n = U n Pɛ0 ɛ n def = U n P ɛn U P ɛ1 UP ɛ0 is a quasi-projector on the rectangle [ɛ 0 ɛ n ]. t=0 1 2 t=2 t=1 Can we get some information on the distribution of the weights P ɛ0 ɛ n ψ 2 h 0 µ([ɛ 0 ɛ n ])? YES, provided we consider times n > T E (for which the quasi-projector interpretation breaks down).
17 Evolution of adapted elementary states To estimate P ɛ0 ɛ n ψ, we decompose ψ in a well-chosen family of states ψ Λ, and compute each P ɛ0 ɛ n ψ Λ separately. The Anosov dynamics is anisotropic (stable/unstable foliations). 0.5 p Λ 0 q use states adapted to these foliations We consider Lagrangian states associated with Lagrangian manifolds close to the unstable foliation: ψ Λ (q) = a(q) e is Λ(q)/ is localized on Λ = {(q, p = S Λ (q))}
18 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1).
19 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1). Summing over all [ɛ 0 ɛ n ] we recover U n ψ Λ : no breakdown of the semiclassical evolution at T E [Heller-Tomsovic 91].
20 Λ η Φ Φ( Λ η ) Through the sequence of stretching-and-cutting, the state P ɛ0 ɛ n ψ Λ remains a nice Lagrangian state up to large times (n C T E for any C > 1). Summing over all [ɛ 0 ɛ n ] we recover U n ψ Λ : no breakdown of the semiclassical evolution at T E [Heller-Tomsovic 91]. The amplitude of Pɛ0 ɛ n ψ Λ is governed by the unstable Jacobian along the path ɛ 0 ɛ t : P ɛ0 ɛ n ψ Λ J n u (ɛ 0 ɛ n ) 1/2 e Λn/2 From the decomposition ψ = 1/h η=1 c η ψ Λη, one obtains the bound [Anantharaman 06] P ɛ0 ɛ n h 1/2 J n u (ɛ 0 ɛ n ) 1/2 h 1/2 e Λn/2. This hyperbolic estimate is nontrivial for times t > T E (no more a quasi-projector).
21 What to do with this estimate? [Anantharaman-N 06 07]: rewrite the operator at time 2n as U n P ɛ0 ɛ 2n = P ɛn+1 ɛ 2n U n P ɛ0 ɛ n This can be seen as a block matrix element of the unitary propagator U n, expressed in the block-basis {P ɛ0 ɛ n }. Setting n = T E, the hyperbolic estimate at time 2n states that these block matrix elements are all 1/2. An entropic uncertainty principle then implies that the entropy constructed from the weights P ɛ0 ɛ n ψ 2 satisfies H n (ψ ) log 1/2 = n Λ 2. From this bound at n = T E, one uses subadditivity and Egorov to get a similar bound at finite time n, and then the bound for H KS (µ).
22 Where else to use this hyperbolic estimate? semiclassical resonance spectra of chaotic scattering systems. 0 E c E E+c gh Discrete model: open quantum baker B 8 subunitary propagator B N = U N Π N.
23 Gap in the resonance spectrum Use a quantum partition outside the hole: Π N = j P j. (B N ) n = Pɛ0 ɛ n 1 ɛ 0 ɛ n 1 = (B N ) n P ɛ0 ɛ n 1 ɛ 0 ɛ n 1 ɛ 0 ɛ n 1 h 1/2 J n u (ɛ 0 ɛ n 1 ) 1/2 For n T E, the RHS is approximately given by the topological pressure P( log J u /2) associated with the classical trapped set: (B N ) n exp ( n P( log J u /2) ) If P( log J u /2) < 0 ( thin trapped set ), this gives an upper bound on the quantum lifetimes [Ikawa 88,Gaspard-Rice 89,N-Zworski 07].
24 Perspectives To obtain nontrivial information on eigenstates, it was crucial to analyze the dynamics beyond the Ehrenfest time. The partition allows to control the evolution of Lagrangian states (also wavepackets) [Heller-Tomsovic 91,Schubert 08] The decomposition into P ɛ0 ɛ n could also be useful to: analyze the phase space structure of resonant states [Keating-Novaes-Prado- Sieber 06, N-Rubin 06] expand the validity of the Gutzwiller trace formula to times n T E [Faure 06] show that (some) resonances are close to the zeros of the Gutzwiller-Voros Zeta function analyze transport through chaotic cavities
Quantum decay rates in chaotic scattering
Quantum decay rates in chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski (Berkeley) National AMS Meeting, New Orleans, January 2007 A resonant state for the partially open stadium billiard, computed
More informationFractal uncertainty principle and quantum chaos
Fractal uncertainty principle and quantum chaos Semyon Dyatlov (UC Berkeley/MIT) July 23, 2018 Semyon Dyatlov FUP and eigenfunctions July 23, 2018 1 / 11 Overview This talk presents two recent results
More informationQuantum ergodicity. Nalini Anantharaman. 22 août Université de Strasbourg
Quantum ergodicity Nalini Anantharaman Université de Strasbourg 22 août 2016 I. Quantum ergodicity on manifolds. II. QE on (discrete) graphs : regular graphs. III. QE on graphs : other models, perspectives.
More informationQuantum chaotic scattering
Quantum chaotic scattering S. Nonnenmacher (Saclay) + M. Zworski, M. Rubin Toulouse, 7/4/26 A resonant state for a dielectric cavity, computed by Ch.Schmit. Outline Quantum chaotic scattering, resonances
More informationChaos, Quantum Mechanics and Number Theory
Chaos, Quantum Mechanics and Number Theory Peter Sarnak Mahler Lectures 2011 Hamiltonian Mechanics (x, ξ) generalized coordinates: x space coordinate, ξ phase coordinate. H(x, ξ), Hamiltonian Hamilton
More informationRecent developments in mathematical Quantum Chaos, I
Recent developments in mathematical Quantum Chaos, I Steve Zelditch Johns Hopkins and Northwestern Harvard, November 21, 2009 Quantum chaos of eigenfunction Let {ϕ j } be an orthonormal basis of eigenfunctions
More informationQuantum Billiards. Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications
Quantum Billiards Martin Sieber (Bristol) Postgraduate Research Conference: Mathematical Billiard and their Applications University of Bristol, June 21-24 2010 Most pictures are courtesy of Arnd Bäcker
More informationHALF-DELOCALIZATION OF EIGENFUNCTIONS FOR THE LAPLACIAN ON AN ANOSOV MANIFOLD
HALF-DELOCALIZATION OF EIGENFUNCTIONS FOR THE LAPLACIAN ON AN ANOSOV MANIFOLD NALINI ANANTHARAMAN AND STÉPHANE NONNENMACHER Abstract We study the high-energy eigenfunctions of the Laplacian on a compact
More informationErgodicity of quantum eigenfunctions in classically chaotic systems
Ergodicity of quantum eigenfunctions in classically chaotic systems Mar 1, 24 Alex Barnett barnett@cims.nyu.edu Courant Institute work in collaboration with Peter Sarnak, Courant/Princeton p.1 Classical
More informationArithmetic quantum chaos and random wave conjecture. 9th Mathematical Physics Meeting. Goran Djankovi
Arithmetic quantum chaos and random wave conjecture 9th Mathematical Physics Meeting Goran Djankovi University of Belgrade Faculty of Mathematics 18. 9. 2017. Goran Djankovi Random wave conjecture 18.
More informationarxiv: v1 [nlin.cd] 27 May 2008
SOME OPEN QUESTIONS IN WAVE CHAOS arxiv:0805.4137v1 [nlin.cd] 27 May 2008 The subject area referred to as wave chaos, quantum chaos or quantum chaology has been investigated mostly by the theoretical physics
More informationMicrolocal limits of plane waves
Microlocal limits of plane waves Semyon Dyatlov University of California, Berkeley July 4, 2012 joint work with Colin Guillarmou (ENS) Semyon Dyatlov (UC Berkeley) Microlocal limits of plane waves July
More informationANNALESDEL INSTITUTFOURIER
ANNALESDEL INSTITUTFOURIER ANNALES DE L INSTITUT FOURIER Nalini ANANTHARAMAN & Stéphane NONNENMACHER Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold Tome 57, n o 7 (007),
More informationQuantum Ergodicity for a Class of Mixed Systems
Quantum Ergodicity for a Class of Mixed Systems Jeffrey Galkowski University of California, Berkeley February 13, 2013 General Setup Let (M, g) be a smooth compact manifold with or without boundary. We
More informationBouncing Ball Modes and Quantum Chaos
SIAM EVIEW Vol. 47,No. 1,pp. 43 49 c 25 Society for Industrial and Applied Mathematics Bouncing Ball Modes and Quantum Chaos Nicolas Burq Maciej Zworski Abstract. Quantum ergodicity for classically chaotic
More informationSpectra, dynamical systems, and geometry. Fields Medal Symposium, Fields Institute, Toronto. Tuesday, October 1, 2013
Spectra, dynamical systems, and geometry Fields Medal Symposium, Fields Institute, Toronto Tuesday, October 1, 2013 Dmitry Jakobson April 16, 2014 M = S 1 = R/(2πZ) - a circle. f (x) - a periodic function,
More informationWhat is quantum unique ergodicity?
What is quantum unique ergodicity? Andrew Hassell 1 Abstract A (somewhat) nontechnical presentation of the topic of quantum unique ergodicity (QUE) is attempted. I define classical ergodicity and unique
More informationEigenvalues and eigenfunctions of the Laplacian. Andrew Hassell
Eigenvalues and eigenfunctions of the Laplacian Andrew Hassell 1 2 The setting In this talk I will consider the Laplace operator,, on various geometric spaces M. Here, M will be either a bounded Euclidean
More informationVariations on Quantum Ergodic Theorems. Michael Taylor
Notes available on my website, under Downloadable Lecture Notes 8. Seminar talks and AMS talks See also 4. Spectral theory 7. Quantum mechanics connections Basic quantization: a function on phase space
More informationFluctuation statistics for quantum star graphs
Contemporary Mathematics Fluctuation statistics for quantum star graphs J.P. Keating Abstract. Star graphs are examples of quantum graphs in which the spectral and eigenfunction statistics can be determined
More informationEntropy and the localization of eigenfunctions
Annals of Mathematics, 168 008, 435 475 Entropy and the localization of eigenfunctions By Nalini Anantharaman Abstract We study the large eigenvalue limit for the eigenfunctions of the Laplacian, on a
More informationStrichartz estimates for the Schrödinger equation on polygonal domains
estimates for the Schrödinger equation on Joint work with Matt Blair (UNM), G. Austin Ford (Northwestern U) and Sebastian Herr (U Bonn and U Düsseldorf)... With a discussion of previous work with Andrew
More informationUniversality. Why? (Bohigas, Giannoni, Schmit 84; see also Casati, Vals-Gris, Guarneri; Berry, Tabor)
Universality Many quantum properties of chaotic systems are universal and agree with predictions from random matrix theory, in particular the statistics of energy levels. (Bohigas, Giannoni, Schmit 84;
More informationBOUNCING BALL MODES AND QUANTUM CHAOS
BOUNCING BALL MODES AND QUANTUM CHAOS NICOLAS BUQ AND MACIEJ ZWOSKI Abstract. Quantum ergodicity of classically chaotic systems has been studied extensively both theoretically and experimentally, in mathematics,
More informationUniversity of Washington. Quantum Chaos in Scattering Theory
University of Washington Quantum Chaos in Scattering Theory Maciej Zworski UC Berkeley 10 April 2007 In this talk I want to relate objects of classical chaotic dynamics such as trapped sets and topological
More informationA gentle introduction to Quantum Ergodicity
A gentle introduction to Quantum Ergodicity Tuomas Sahlsten University of Bristol, UK 7 March 2017 Arithmetic study group, Durham, UK Joint work with Etienne Le Masson (Bristol) Supported by the EU: Horizon
More informationOn the sup-norm problem for arithmetic hyperbolic 3-manifolds
On the sup-norm problem for arithmetic hyperbolic 3-manifolds Gergely Harcos Alfréd Rényi Institute of Mathematics http://www.renyi.hu/ gharcos/ 29 September 2014 Analytic Number Theory Workshop University
More informationCentre de recherches mathématiques
Centre de recherches mathématiques Université de Montréal Atelier «Aspects mathématiques du chaos quantique» Du 2 au 6 juin 2008 Workshop Mathematical aspects of quantum chaos June 2-6, 2008 HORAIRE /
More informationIs Quantum Mechanics Chaotic? Steven Anlage
Is Quantum Mechanics Chaotic? Steven Anlage Physics 40 0.5 Simple Chaos 1-Dimensional Iterated Maps The Logistic Map: x = 4 x (1 x ) n+ 1 μ n n Parameter: μ Initial condition: 0 = 0.5 μ 0.8 x 0 = 0.100
More informationQUANTUM ERGODICITY AND BEYOND. WITH A GALLERY OF PICTURES.
QUANTUM ERGODICITY AND BEYOND. WITH A GALLERY OF PICTURES. NALINI ANANTHARAMAN AND ARND BÄCKER Nalini Anantharaman completed her PhD in 2000 at Université Paris 6, under the supervision of François Ledrappier,
More informationMATHEMATICS: CONCEPTS, AND FOUNDATIONS - Introduction To Mathematical Aspects of Quantum Chaos - Dieter Mayer
ITRODUCTIO TO MATHEMATICAL ASPECTS OF QUATUM CHAOS Dieter University of Clausthal, Clausthal-Zellerfeld, Germany Keywords: arithmetic quantum chaos, Anosov system, Berry-Tabor conjecture, Bohigas-Giannoni-Schmit
More informationarxiv: v2 [math.sp] 2 Sep 2015
SEMICLASSICAL ANALYSIS AND SYMMETRY REDUCTION II. EQUIVARIANT QUANTUM ERGODICITY FOR INVARIANT SCHRÖDINGER OPERATORS ON COMPACT MANIFOLDS BENJAMIN KÜSTER AND PABLO RAMACHER arxiv:508.0738v2 [math.sp] 2
More informationWave Packet Representation of Semiclassical Time Evolution in Quantum Mechanics
Wave Packet Representation of Semiclassical Time Evolution in Quantum Mechanics Panos Karageorge, George Makrakis ACMAC Dept. of Applied Mathematics, University of Crete Semiclassical & Multiscale Aspects
More informationApplications of measure rigidity: from number theory to statistical mechanics
Applications of measure rigidity: from number theory to statistical mechanics Simons Lectures, Stony Brook October 2013 Jens Marklof University of Bristol http://www.maths.bristol.ac.uk supported by Royal
More informationQuantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions
p.1 Quantummushrooms,scars,andthe high-frequencylimitofchaoticeigenfunctions ICIAM07, July 18, 2007 Alex Barnett Mathematics Department, Dartmouth College p.2 PlanarDirichleteigenproblem Normalmodesofelasticmembraneor
More informationRecent Progress on QUE
Recent Progress on QUE by Peter Sarnak Princeton University & Institute for Advanced Study In Figure 1 the domains Ω E, Ω S, Ω B are an ellipse, a stadium and a Barnett billiard table respectively. Super-imposed
More informationParticipants. 94 Participants
94 Participants Participants 1 Arians, Silke Aachen silke@iram.rwth-aachen.de 2 Backer, Arnd Ulm baec@physik.uni-ulm.de 3 Baladi, Viviane Genf baladi@sc2a.unige.ch 4 Bauer, Gernot Miinchen bauer@rz.mathematik.uni-muenchen.de
More informationHomoclinic and Heteroclinic Motions in Quantum Dynamics
Homoclinic and Heteroclinic Motions in Quantum Dynamics F. Borondo Dep. de Química; Universidad Autónoma de Madrid, Instituto Mixto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM Stability and Instability in
More informationQuantum chaos on graphs
Baylor University Graduate seminar 6th November 07 Outline 1 What is quantum? 2 Everything you always wanted to know about quantum but were afraid to ask. 3 The trace formula. 4 The of Bohigas, Giannoni
More informationArithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus
Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus Thesis submitted for the degree Doctor of Philosophy by Dubi Kelmer Prepared under the supervision of Professor
More informationarxiv:nlin/ v2 [nlin.cd] 8 Feb 2005
Signatures of homoclinic motion in uantum chaos D. A. Wisniacki,,2 E. Vergini,, 3 R. M. Benito, 4 and F. Borondo, Departamento de Química C IX, Universidad Autónoma de Madrid, Cantoblanco, 2849 Madrid,
More informationRECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS PRELIMINARY VERSION/COMMENTS APPRECIATED
RECENT DEVELOPMENTS IN MATHEMATICAL QUANTUM CHAOS PRELIMINARY VERSION/COMMENTS APPRECIATED STEVE ZELDITCH Abstract. This is a survey of recent results on quantum ergodicity, specifically on the large energy
More informationQuantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces
Quantum Ergodicity and Benjamini-Schramm convergence of hyperbolic surfaces Etienne Le Masson (Joint work with Tuomas Sahlsten) School of Mathematics University of Bristol, UK August 26, 2016 Hyperbolic
More information(5) N #{j N : X j+1 X j [a, b]}
ARITHETIC QUANTU CHAOS JENS ARKLOF. Introduction The central objective in the study of quantum chaos is to characterize universal properties of quantum systems that reflect the regular or chaotic features
More informationChapter 29. Quantum Chaos
Chapter 29 Quantum Chaos What happens to a Hamiltonian system that for classical mechanics is chaotic when we include a nonzero h? There is no problem in principle to answering this question: given a classical
More informationFractal Weyl Laws and Wave Decay for General Trapping
Fractal Weyl Laws and Wave Decay for General Trapping Jeffrey Galkowski McGill University July 26, 2017 Joint w/ Semyon Dyatlov The Plan The setting and a brief review of scattering resonances Heuristic
More informationThe Transition to Chaos
Linda E. Reichl The Transition to Chaos Conservative Classical Systems and Quantum Manifestations Second Edition With 180 Illustrations v I.,,-,,t,...,* ', Springer Dedication Acknowledgements v vii 1
More informationApplications of homogeneous dynamics: from number theory to statistical mechanics
Applications of homogeneous dynamics: from number theory to statistical mechanics A course in ten lectures ICTP Trieste, 27-31 July 2015 Jens Marklof and Andreas Strömbergsson Universities of Bristol and
More informationAsymptotic rate of quantum ergodicity in chaotic Euclidean billiards
Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards Alex H. Barnett Courant Institute, New York University, 251 Mercer St, New York, NY 10012 barnett@cims.nyu.edu July 3, 2004 Abstract
More information0. Introduction. Suppose that M is a compact Riemannian manifold. The Laplace operator is given in local coordinates by f = g 1/2
ASIAN J. MATH. c 2006 International Press Vol. 10, No. 1, pp. 115 126, March 2006 004 EIGENFUNCTIONS OF THE LAPLACIAN ON COMPACT RIEMANNIAN MANIFOLDS HAROLD DONNELLY Key words. Mathematical physics, quantum
More informationLocal smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping
Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North
More informationANNALESDEL INSTITUTFOURIER
ANNALESDEL INSTITUTFOURIER ANNALES DE L INSTITUT FOURIER Suresh ESWARATHASAN & Stéphane NONNENMACHER Strong scarring of logarithmic quasimodes Tome 67, n o 6 (2017), p. 2307-2347.
More informationCounting stationary modes: a discrete view of geometry and dynamics
Counting stationary modes: a discrete view of geometry and dynamics Stéphane Nonnenmacher Institut de Physique Théorique, CEA Saclay September 20th, 2012 Weyl Law at 100, Fields Institute Outline A (sketchy)
More informationUnique equilibrium states for geodesic flows in nonpositive curvature
Unique equilibrium states for geodesic flows in nonpositive curvature Todd Fisher Department of Mathematics Brigham Young University Fractal Geometry, Hyperbolic Dynamics and Thermodynamical Formalism
More informationarxiv: v1 [quant-ph] 6 Apr 2016
Short periodic orbits theory for partially open quantum maps arxiv:1604.01817v1 [quant-ph] 6 Apr 2016 Gabriel G. Carlo, 1, R. M. Benito, 2 and F. Borondo 3 1 Comisión Nacional de Energía Atómica, CONICET,
More informationESI Lectures on Semiclassical Analysis and Quantum Ergodicity 2012
ESI Lectures on Semiclassical Analysis and Quantum Ergodicity 2012 Andrew Hassell Department of Mathematics Australian National University ESI, Vienna, July August 2012 Outline Introduction Semiclassical
More informationarxiv: v1 [quant-ph] 11 Feb 2015
Local quantum ergodic conjecture Eduardo Zambrano Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 87, Dresden, Germany W.P. Karel Zapfe and Alfredo M. Ozorio de Almeida arxiv:5.335v
More informationErgodic billiards that are not quantum unique ergodic
ANNALS OF MATHEMATICS Ergodic billiards that are not quantum unique ergodic By Andrew Hassell With an Appendix by Andrew Hassell and Luc Hillairet SECOND SERIES, VOL. 171, NO. 1 January, 2010 anmaah Annals
More informationThe Berry-Tabor conjecture
The Berry-Tabor conjecture Jens Marklof Abstract. One of the central observations of quantum chaology is that statistical properties of quantum spectra exhibit surprisingly universal features, which seem
More informationRecent Progress on the Quantum Unique Ergodicity Conjecture
Recent Progress on the Quantum Unique Ergodicity Conjecture by Peter Sarnak Abstract We report on some recent striking advances on the quantum unique ergodicity or QUE conjecture, concerning the distribution
More information3 Symmetry Protected Topological Phase
Physics 3b Lecture 16 Caltech, 05/30/18 3 Symmetry Protected Topological Phase 3.1 Breakdown of noninteracting SPT phases with interaction Building on our previous discussion of the Majorana chain and
More informationAbstracts. Furstenberg The Dynamics of Some Arithmetically Generated Sequences
CHAOS AND DISORDER IN MATHEMATICS AND PHYSICS Monday 10:00-11:00 Okounkov Algebraic geometry of random surfaces 11:30-12:30 Furstenberg Dynamics of Arithmetically Generated Sequences 12:30-14:30 lunch
More informationSymmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators. Philippe Jacquod. U of Arizona
Symmetries in Quantum Transport : From Random Matrix Theory to Topological Insulators Philippe Jacquod U of Arizona UA Phys colloquium - feb 1, 2013 Continuous symmetries and conservation laws Noether
More informationThermalization in Quantum Systems
Thermalization in Quantum Systems Jonas Larson Stockholm University and Universität zu Köln Dresden 18/4-2014 Motivation Long time evolution of closed quantum systems not fully understood. Cold atom system
More informationWeak chaos, infinite ergodic theory, and anomalous diffusion
Weak chaos, infinite ergodic theory, and anomalous diffusion Rainer Klages Queen Mary University of London, School of Mathematical Sciences Marseille, CCT11, 24 May 2011 Weak chaos, infinite ergodic theory,
More informationPersistent Chaos in High-Dimensional Neural Networks
Persistent Chaos in High-Dimensional Neural Networks D. J. Albers with J. C. Sprott and James P. Crutchfield February 20, 2005 1 Outline: Introduction and motivation Mathematical versus computational dynamics
More informationSymbolic dynamics and non-uniform hyperbolicity
Symbolic dynamics and non-uniform hyperbolicity Yuri Lima UFC and Orsay June, 2017 Yuri Lima (UFC and Orsay) June, 2017 1 / 83 Lecture 1 Yuri Lima (UFC and Orsay) June, 2017 2 / 83 Part 1: Introduction
More informationTitleQuantum Chaos in Generic Systems.
TitleQuantum Chaos in Generic Systems Author(s) Robnik, Marko Citation 物性研究 (2004), 82(5): 662-665 Issue Date 2004-08-20 URL http://hdl.handle.net/2433/97885 Right Type Departmental Bulletin Paper Textversion
More informationTheory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004
Preprint CAMTP/03-8 August 2003 Theory of Adiabatic Invariants A SOCRATES Lecture Course at the Physics Department, University of Marburg, Germany, February 2004 Marko Robnik CAMTP - Center for Applied
More informationNotes on fractal uncertainty principle version 0.5 (October 4, 2017) Semyon Dyatlov
Notes on fractal uncertainty principle version 0.5 (October 4, 2017) Semyon Dyatlov E-mail address: dyatlov@math.mit.edu Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts
More informationSuperiority of semiclassical over quantum mechanical calculations for a three-dimensional system
Physics Letters A 305 (2002) 176 182 www.elsevier.com/locate/pla Superiority of semiclassical over quantum mechanical calculations for a three-dimensional system Jörg Main a,, Günter Wunner a, Erdinç Atılgan
More informationGeometrical theory of diffraction and spectral statistics
mpi-pks/9907008 Geometrical theory of diffraction and spectral statistics arxiv:chao-dyn/990006v 6 Oct 999 Martin Sieber Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Str. 38, 087 Dresden,
More informationUlam method and fractal Weyl law for Perron Frobenius operators
EPJ manuscript No. (will be inserted by the editor) Ulam method and fractal Weyl law for Perron Frobenius operators L.Ermann and D.L.Shepelyansky 1 Laboratoire de Physique Théorique (IRSAMC), Université
More informationRandom Wave Model in theory and experiment
Random Wave Model in theory and experiment Ulrich Kuhl Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Deutschland Maribor, 23-27 February 2009 ulrich.kuhl@physik.uni-marburg.de Literature
More informationGeneral Exam Part II, Fall 1998 Quantum Mechanics Solutions
General Exam Part II, Fall 1998 Quantum Mechanics Solutions Leo C. Stein Problem 1 Consider a particle of charge q and mass m confined to the x-y plane and subject to a harmonic oscillator potential V
More informationThe Structure of Hyperbolic Sets
The Structure of Hyperbolic Sets p. 1/35 The Structure of Hyperbolic Sets Todd Fisher tfisher@math.umd.edu Department of Mathematics University of Maryland, College Park The Structure of Hyperbolic Sets
More informationMARKOV PARTITIONS FOR HYPERBOLIC SETS
MARKOV PARTITIONS FOR HYPERBOLIC SETS TODD FISHER, HIMAL RATHNAKUMARA Abstract. We show that if f is a diffeomorphism of a manifold to itself, Λ is a mixing (or transitive) hyperbolic set, and V is a neighborhood
More informationAsymptotics of polynomials and eigenfunctions
ICM 2002 Vol. III 1 3 Asymptotics of polynomials and eigenfunctions S. Zelditch Abstract We review some recent results on asymptotic properties of polynomials of large degree, of general holomorphic sections
More informationOn Sinai-Bowen-Ruelle measures on horocycles of 3-D Anosov flows
On Sinai-Bowen-Ruelle measures on horocycles of 3-D Anosov flows N.I. Chernov Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294 September 14, 2006 Abstract Let φ t be a
More informationOPEN PROBLEMS IN DYNAMICS AND RELATED FIELDS
OPEN PROBLEMS IN DYNAMICS AND RELATED FIELDS ALEXANDER GORODNIK Contents 1. Local rigidity 2 2. Global rigidity 3 3. Measure rigidity 5 4. Equidistribution 8 5. Divergent trajectories 12 6. Symbolic coding
More information2000 Mathematics Subject Classification. Primary: 37D25, 37C40. Abstract. This book provides a systematic introduction to smooth ergodic theory, inclu
Lyapunov Exponents and Smooth Ergodic Theory Luis Barreira and Yakov B. Pesin 2000 Mathematics Subject Classification. Primary: 37D25, 37C40. Abstract. This book provides a systematic introduction to smooth
More informationHYPERBOLIC DYNAMICAL SYSTEMS AND THE NONCOMMUTATIVE INTEGRATION THEORY OF CONNES ABSTRACT
HYPERBOLIC DYNAMICAL SYSTEMS AND THE NONCOMMUTATIVE INTEGRATION THEORY OF CONNES Jan Segert ABSTRACT We examine hyperbolic differentiable dynamical systems in the context of Connes noncommutative integration
More informationDispersive Equations and Hyperbolic Orbits
Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University Outline 1 Introduction 3 Applications 2 Main
More informationSpectral theory, geometry and dynamical systems
Spectral theory, geometry and dynamical systems Dmitry Jakobson 8th January 2010 M is n-dimensional compact connected manifold, n 2. g is a Riemannian metric on M: for any U, V T x M, their inner product
More informationSRB measures for non-uniformly hyperbolic systems
SRB measures for non-uniformly hyperbolic systems Vaughn Climenhaga University of Maryland October 21, 2010 Joint work with Dmitry Dolgopyat and Yakov Pesin 1 and classical results Definition of SRB measure
More informationQuantum Chaos as a Practical Tool in Many-Body Physics
Quantum Chaos as a Practical Tool in Many-Body Physics Vladimir Zelevinsky NSCL/ Michigan State University Supported by NSF Statistical Nuclear Physics SNP2008 Athens, Ohio July 8, 2008 THANKS B. Alex
More information31 Lecture 31 Information loss rate
338 31 Lecture 31 nformation loss rate 31.1 Observation with finite resolution When a dynamical system is given, we wish to describe its state at each instant with a constant precision. Suppose the data
More informationORIGINS. E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956
ORIGINS E.P. Wigner, Conference on Neutron Physics by Time of Flight, November 1956 P.W. Anderson, Absence of Diffusion in Certain Random Lattices ; Phys.Rev., 1958, v.109, p.1492 L.D. Landau, Fermi-Liquid
More informationSymbolic extensions for partially hyperbolic diffeomorphisms
for partially hyperbolic diffeomorphisms Todd Fisher tfisher@math.byu.edu Department of Mathematics Brigham Young University International Workshop on Global Dynamics Beyond Uniform Hyperbolicity Joint
More informationQUANTUM DECAY RATES IN CHAOTIC SCATTERING
QUANTUM DECAY RATES IN CHAOTIC SCATTERING STÉPHANE NONNENMACHER AND MACIEJ ZWORSKI 1. Statement of Results In this article we prove that for a large class of operators, including Schrödinger operators,
More informationSYMBOLIC DYNAMICS FOR HYPERBOLIC SYSTEMS. 1. Introduction (30min) We want to find simple models for uniformly hyperbolic systems, such as for:
SYMBOLIC DYNAMICS FOR HYPERBOLIC SYSTEMS YURI LIMA 1. Introduction (30min) We want to find simple models for uniformly hyperbolic systems, such as for: [ ] 2 1 Hyperbolic toral automorphisms, e.g. f A
More informationarxiv:nlin/ v1 [nlin.cd] 27 Feb 2007
Quantum Baker Maps for Spiraling Chaotic Motion Pedro R. del Santoro, Raúl O. Vallejos and Alfredo M. Ozorio de Almeida, Centro Brasileiro de Pesquisas Físicas (CBPF), arxiv:nlin/0702056v1 [nlin.cd] 27
More informationQuantum chaos. University of Ljubljana Faculty of Mathematics and Physics. Seminar - 1st year, 2nd cycle degree. Author: Ana Flack
University of Ljubljana Faculty of Mathematics and Physics Seminar - 1st year, 2nd cycle degree Quantum chaos Author: Ana Flack Mentor: izred. prof. dr. Marko šnidari Ljubljana, May 2018 Abstract The basic
More informationEstimates for the resolvent and spectral gaps for non self-adjoint operators. University Bordeaux
for the resolvent and spectral gaps for non self-adjoint operators 1 / 29 Estimates for the resolvent and spectral gaps for non self-adjoint operators Vesselin Petkov University Bordeaux Mathematics Days
More informationAbteilung Theoretische Physik Universitat Ulm Semiclassical approximations and periodic orbit expansions in chaotic and mixed systems Habilitationsschrift vorgelegt an der Fakultat fur Naturwissenschaften
More informationGluing semiclassical resolvent estimates via propagation of singularities
Gluing semiclassical resolvent estimates via propagation of singularities Kiril Datchev (MIT) and András Vasy (Stanford) Postdoc Seminar MSRI Kiril Datchev (MIT) Semiclassical resolvent estimates 7 December
More informationHypersensitivity and chaos signatures in the quantum baker s maps
Hypersensitivity and chaos signatures in the quantum baker s maps Author Scott, A, Brun, Todd, Caves, Carlton, Schack, Rüdiger Published 6 Journal Title Journal of Physics A: Mathematical and General DOI
More informationHamiltonian Chaos and the standard map
Hamiltonian Chaos and the standard map Outline: What happens for small perturbation? Questions of long time stability? Poincare section and twist maps. Area preserving mappings. Standard map as time sections
More informationDecoherence and the Classical Limit
Chapter 26 Decoherence and the Classical Limit 26.1 Introduction Classical mechanics deals with objects which have a precise location and move in a deterministic way as a function of time. By contrast,
More informationSeismology and wave chaos in rapidly rotating stars
Seismology and wave chaos in rapidly rotating stars F. Lignières Institut de Recherche en Astrophysique et Planétologie - Toulouse In collaboration with B. Georgeot - LPT, M. Pasek -LPT/IRAP, D. Reese
More information