Order and chaos in wave propagation
|
|
- Dortha Stone
- 5 years ago
- Views:
Transcription
1 Order and chaos in wave propagation Nalini Anantharaman Université Paris 11, Orsay January 8, 2012
2 A physical phenomenon : harmonics (stationary vibration modes). A mathematical theorem : the spectral decomposition of the laplacian.
3 The wave equation : u(t, x, y)
4 The wave equation : u(t, x, y) 2 ( ) 2 t 2 u = c2 x y 2 u 2 t 2 u = c2 u
5 An abstract language allowing unified treatment of all wave phenomena, regardless of their physical origin (sound, electromagnetic waves, seismic waves,...)
6 The wave equation has special stationary solutions, i.e. of the form u(t, x, y) = e iωt ψ(x, y), also called eigenmodes, characteristic modes. The function ψ must satisfy ψ = ω2 c 2 ψ.
7 Figure: Ernst Chladni ( )
8 Figure: Sophie Germain ( )
9 Doing research in math? Observing the world... answering questions... always leads to new questions!
10 Can we calculate explicitly the eigenfrequencies and eigenmodes? it depends on the shape of the membrane, but in general the answer is NO.
11 Nodal lines for a guitar table :
12 Nodal lines on a sphere : [Eric Heller Gallery]
13 (R. Courant s theorem) For the n-th eigenmode, the nodal lines cut the membrane into at most n pieces (= nodal domains). (Pleijel s theorem) In dimension 2. For n large enough, the nodal lines of the n-th eigenmode cut the membrane into at most αn pieces with α = 0, 54.
14 The nodal lines seem to invade the domain, to form a denser and denser family of curves, as the frequency increases. Mathematical proof?
15 (Donnelly-Fefferman 1988) The total length of nodal lines grows proportionnally to frequency : Z ψ = {x, ψ(x) = 0}. C 1 ω n length(z ψn ) C 2 ω n. (proven if the boundary is analytic. Conjectured by Yau to hold always).
16 (Egorov-Kondratiev 1996, Nazarov-Polterovich-Sodin 2005) The spacing between nodal lines decreases like the inverse of frequency. Let r ψn be the inner radius (radius of a ball included in a nodal domain). Then C 1 ω n r ψn C 2 ω n
17 Billard régulier n = 100 n = 1000 n = 1500 n = 2000 Billard chaotique
18 How to explain the following patterns? [Eric Heller Gallery]
19 [Pär Kurlberg]
20 The Weyl law : N(λ) := {n, ω n λ}
21 The Weyl law : N(λ) := {n, ω n λ} λ + 1 4π Area(Ω)λ2
22 The Weyl law : N(λ) := {n, ω n λ} λ + 1 4π Area(Ω)λ2 and more generally, for a C 0 (Ω), a(x, y) ψ n (x, y) 2 1 dx dy λ + 4π λ2 n,ω n λ Ω (Here (ψ n ) is an orthonormal basis of L 2 (Ω) such that ψ n = ω 2 nψ n ) Ω a(x, y)dx dy.
23 1 N(λ) n,ω n λ a(x, y) ψ n (x, y) 2 dx dy λ + 1 a(x, y)dx dy Area(Ω) Ω
24 1 N(λ) n,ω n λ a(x, y) ψ n (x, y) 2 dx dy λ + Individual behaviour of ψ n (x, y) 2 as n +? 1 a(x, y)dx dy Area(Ω) Ω
25 Do we have a(x, y) ψ n (x, y) 2 dx dy λ + 1 a(x, y)dx dy? Area(Ω) Ω If this is true, this means that the eigenfunctions are becoming uniformly spread out for high frequencies, and is interpreted as their having a disordered behaviour. The answer seems to depend on the geometry of the domain.
26 For negatively curved manifolds, it is believed that a(x, y) ψ n (x, y) 2 1 dx dy a(x, y)dx dy. λ + Area(Ω) Quantum Unique Ergodicity conjecture 1992, proven in some cases by E. Lindenstrauss (2002) Ω
27 For negatively curved manifolds, it is believed that a(x, y) ψ n (x, y) 2 1 dx dy a(x, y)dx dy. λ + Area(Ω) Quantum Unique Ergodicity conjecture 1992, proven in some cases by E. Lindenstrauss (2002) For the stadium billiard, Hassell has proved in 2008 that QUE is false : there are families of eigenfunctions that concentrate inside the rectangle. Ω
28 [Pär Kurlberg]
29 Can one hear the shape of a drum?
30 If one hears the harmonics produced by a drum, can one guess its shape? Is it possible for two membranes with different shapes to produce the same harmonics?
31 One can prove mathematically that : if two membranes have the same harmonics, they must have the same area ; this comes from the Weyl law : {n, ω n λ} λ + 1 4π Area(Ω)λ2. if two membranes have the same harmonics, they must have the same perimeter ; if a membrane has the same harmonics as a circular membrane, it must be circular ; if two rectangular membranes have the same harmonics, the rectangles must be the same.
32 BUT...
33
Spectral 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 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 informationGaussian Fields and Percolation
Gaussian Fields and Percolation Dmitry Beliaev Mathematical Institute University of Oxford RANDOM WAVES IN OXFORD 18 June 2018 Berry s conjecture In 1977 M. Berry conjectured that high energy eigenfunctions
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 informationThe nodal count mystery
The nodal count mystery Peter Kuchment, Texas A&M University Partially supported by NSF 21st Midwest Geometry Conference Wichita State University, KS March 11 13 2016 Table of contents 1 Nodal patterns
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 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 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 informationNodal lines of Laplace eigenfunctions
Nodal lines of Laplace eigenfunctions Spectral Analysis in Geometry and Number Theory on the occasion of Toshikazu Sunada s 60th birthday Friday, August 10, 2007 Steve Zelditch Department of Mathematics
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 informationINFLUENCE OF ROOM GEOMETRY ON MODAL DENSITY, SPATIAL DISTRIBUTION OF MODES AND THEIR DAMPING M. MEISSNER
ARCHIVES OF ACOUSTICS 30, 4 (Supplement), 203 208 (2005) INFLUENCE OF ROOM GEOMETRY ON MODAL DENSITY, SPATIAL DISTRIBUTION OF MODES AND THEIR DAMPING M. MEISSNER Institute of Fundamental Technological
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 informationMAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 7: Nodal Sets
MAT 280: Laplacian Eigenfunctions: Theory, Applications, and Computations Lecture 7: Nodal Sets Lecturer: Naoki Saito Scribe: Bradley Marchand/Allen Xue April 20, 2007 Consider the eigenfunctions v(x)
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 informationLocal Asymmetry and the Inner Radius of Nodal Domains
Local Asymmetry and the Inner Radius of Nodal Domains Dan MANGOUBI Institut des Hautes Études Scientifiques 35, route de Chartres 91440 Bures-sur-Yvette (France) Avril 2007 IHES/M/07/14 Local Asymmetry
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 informationNON-UNIVERSALITY OF THE NAZAROV-SODIN CONSTANT
NON-UNIVERSALITY OF THE NAZAROV-SODIN CONSTANT Abstract. We prove that the Nazarov-Sodin constant, which up to a natural scaling gives the leading order growth for the expected number of nodal components
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 informationLecture 1 Introduction
Lecture 1 Introduction 1 The Laplace operator On the 3-dimensional Euclidean space, the Laplace operator (or Laplacian) is the linear differential operator { C 2 (R 3 ) C 0 (R 3 ) : f 2 f + 2 f + 2 f,
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 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 informationOn the Inner Radius of Nodal Domains
arxiv:math/0511329v1 [math.sp] 14 Nov 2005 On the Inner Radius of Nodal Domains Dan Mangoubi Abstract Let M be a closed Riemannian manifold. We consider the inner radius of a nodal domain for a large eigenvalue
More informationStadium Billiards. Charles Kankelborg. March 11, 2016
Stadium Billiards Charles Kankelborg March 11, 2016 Contents 1 Introduction 1 2 The Bunimovich Stadium 1 3 Classical Stadium Billiards 2 3.1 Ray Tracing............................ 2 3.2 Stability of Periodic
More informationSpectral theory of first order elliptic systems
Spectral theory of first order elliptic systems Dmitri Vassiliev (University College London) 24 May 2013 Conference Complex Analysis & Dynamical Systems VI Nahariya, Israel 1 Typical problem in my subject
More informationOptimal shape and placement of actuators or sensors for PDE models
Optimal shape and placement of actuators or sensors for PDE models Y. Privat, E. Trélat, E. uazua Univ. Paris 6 (Labo. J.-L. Lions) et Institut Universitaire de France Rutgers University - Camden, March
More informationElliptic PDEs, vibrations of a plate, set discrepancy and packing problems
Elliptic PDEs, vibrations of a plate, set discrepancy and packing problems Stefan Steinerberger Mathematisches Institut Universität Bonn Oberwolfach Workshop 1340 Stefan Steinerberger (U Bonn) A Geometric
More informationOn the spectrum of the Laplacian
On the spectrum of the Laplacian S. Kesavan Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai - 600113. kesh@imsc.res.in July 1, 2013 S. Kesavan (IMSc) Spectrum of the Laplacian July 1,
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 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 informationNodal lines of random waves Many questions and few answers. M. Sodin (Tel Aviv) Ascona, May 2010
Nodal lines of random waves Many questions and few answers M. Sodin (Tel Aviv) Ascona, May 2010 1 Random 2D wave: random superposition of solutions to Examples: f + κ 2 f = 0 Random spherical harmonic
More informationKUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS
KUIPER S THEOREM ON CONFORMALLY FLAT MANIFOLDS RALPH HOWARD DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA COLUMBIA, S.C. 29208, USA HOWARD@MATH.SC.EDU 1. Introduction These are notes to that show
More informationNodal Sets of High-Energy Arithmetic Random Waves
Nodal Sets of High-Energy Arithmetic Random Waves Maurizia Rossi UR Mathématiques, Université du Luxembourg Probabilistic Methods in Spectral Geometry and PDE Montréal August 22-26, 2016 M. Rossi (Université
More informationNodal domains of Laplacian eigenfunctions
Nodal domains of Laplacian eigenfunctions Yoni Rozenshein May 20 & May 27, 2014 Contents I Motivation, denitions, Courant's theorem and the Faber-Krahn inequality 1 1 Motivation / entertainment 2 1.1 The
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 information24 Solving planar heat and wave equations in polar coordinates
24 Solving planar heat and wave equations in polar coordinates Now that all the preparations are done, I can return to solving the planar heat and wave equations in domains with rotational symmetry. 24.1
More informationCourse Description for Real Analysis, Math 156
Course Description for Real Analysis, Math 156 In this class, we will study elliptic PDE, Fourier analysis, and dispersive PDE. Here is a quick summary of the topics will study study. They re described
More informationOn the spectrum of the Hodge Laplacian and the John ellipsoid
Banff, July 203 On the spectrum of the Hodge Laplacian and the John ellipsoid Alessandro Savo, Sapienza Università di Roma We give upper and lower bounds for the first eigenvalue of the Hodge Laplacian
More informationLOCALIZED EIGENFUNCTIONS: HERE YOU SEE THEM, THERE YOU
LOCALIZED EIGENFUNCTIONS: HERE YOU SEE THEM, THERE YOU DON T STEVEN M. HEILMAN AND ROBERT S. STRICHARTZ The Laplacian Δ ( 2 x + 2 2 y in the plane) is one 2 of the most basic operators in all of mathematical
More informationIII. Quantum ergodicity on graphs, perspectives
III. Quantum ergodicity on graphs, perspectives Nalini Anantharaman Université de Strasbourg 24 août 2016 Yesterday we focussed on the case of large regular (discrete) graphs. Let G = (V, E) be a (q +
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
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 informationLecture 21. Section 6.1 More on Area Section 6.2 Volume by Parallel Cross Section. Jiwen He. Department of Mathematics, University of Houston
Section 6.1 Section 6.2 Lecture 21 Section 6.1 More on Area Section 6.2 Volume by Parallel Cross Section Jiwen He Department of Mathematics, University of Houston jiwenhe@math.uh.edu math.uh.edu/ jiwenhe/math1431
More informationn 2 xi = x i. x i 2. r r ; i r 2 + V ( r) V ( r) = 0 r > 0. ( 1 1 ) a r n 1 ( 1 2) V( r) = b ln r + c n = 2 b r n 2 + c n 3 ( 1 3)
Sep. 7 The L aplace/ P oisson Equations: Explicit Formulas In this lecture we study the properties of the Laplace equation and the Poisson equation with Dirichlet boundary conditions through explicit representations
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationMATH325 - QUANTUM MECHANICS - SOLUTION SHEET 11
MATH35 - QUANTUM MECHANICS - SOLUTION SHEET. The Hamiltonian for a particle of mass m moving in three dimensions under the influence of a three-dimensional harmonic oscillator potential is Ĥ = h m + mω
More informationAdvanced Electrodynamics Exercise 11 Guides
Advanced Electrodynamics Exercise 11 Guides Here we will calculate in a very general manner the modes of light in a waveguide with perfect conductor boundary-conditions. Our derivations are widely independent
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 informationThe Postulates. What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21
The Postulates What is a postulate? Jerry Gilfoyle The Rules of the Quantum Game 1 / 21 The Postulates What is a postulate? 1 suggest or assume the existence, fact, or truth of (something) as a basis for
More informationEXPLICIT SOLUTIONS OF THE WAVE EQUATION ON THREE DIMENSIONAL SPACE-TIMES: TWO EXAMPLES WITH DIRICHLET BOUNDARY CONDITIONS ON A DISK ABSTRACT
EXPLICIT SOLUTIONS OF THE WAVE EQUATION ON THREE DIMENSIONAL SPACE-TIMES: TWO EXAMPLES WITH DIRICHLET BOUNDARY CONDITIONS ON A DISK DANIIL BOYKIS, PATRICK MOYLAN Physics Department, The Pennsylvania State
More informationFeshbach-Schur RG for the Anderson Model
Feshbach-Schur RG for the Anderson Model John Z. Imbrie University of Virginia Isaac Newton Institute October 26, 2018 Overview Consider the localization problem for the Anderson model of a quantum particle
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 informationDifferential Geometry qualifying exam 562 January 2019 Show all your work for full credit
Differential Geometry qualifying exam 562 January 2019 Show all your work for full credit 1. (a) Show that the set M R 3 defined by the equation (1 z 2 )(x 2 + y 2 ) = 1 is a smooth submanifold of R 3.
More informationMicrowave studies of chaotic systems
Maribor, Let s face chaos, July 2008 p. Microwave studies of chaotic systems Lecture 1: Currents and vortices Hans-Jürgen Stöckmann stoeckmann@physik.uni-marburg.de Fachbereich Physik, Philipps-Universität
More informationLecture 4.6: Some special orthogonal functions
Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics
More informationIf electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle.
CHEM 2060 Lecture 18: Particle in a Box L18-1 Atomic Orbitals If electrons moved in simple orbits, p and x could be determined, but this violates the Heisenberg Uncertainty Principle. We can only talk
More informationThe Sound of Symmetry
Zhiqin Lu, University of California, Irvine February 28, 2014, California State Long Beach In 1966, M. Kac asked the the following question: Question If two planar domains have the same spectrum, are they
More informationIntroduction to spectral geometry
Introduction to spectral geometry Bruno Colbois, University of Neuchâtel Beirut, February 6-March 7, 018 Preamble. These notes correspond to a six hours lecture given in the context of the CIMPA School
More informationUnderstand the basic principles of spectroscopy using selection rules and the energy levels. Derive Hund s Rule from the symmetrization postulate.
CHEM 5314: Advanced Physical Chemistry Overall Goals: Use quantum mechanics to understand that molecules have quantized translational, rotational, vibrational, and electronic energy levels. In a large
More informationChemistry 432 Problem Set 4 Spring 2018 Solutions
Chemistry 4 Problem Set 4 Spring 18 Solutions 1. V I II III a b c A one-dimensional particle of mass m is confined to move under the influence of the potential x a V V (x) = a < x b b x c elsewhere and
More informationVibrations of Ideal Circular Membranes (e.g. Drums) and Circular Plates:
Vibrations of Ideal Circular Membranes (e.g. Drums) and Circular Plates: Solution(s) to the wave equation in 2 dimensions this problem has cylindrical symmetry Bessel function solutions for the radial
More informationQualifying Exam. Aug Part II. Please use blank paper for your work do not write on problems sheets!
Qualifying Exam Aug. 2015 Part II Please use blank paper for your work do not write on problems sheets! Solve only one problem from each of the four sections Mechanics, Quantum Mechanics, Statistical Physics
More informationarxiv:chao-dyn/ v1 2 Mar 1999
Vibrating soap films: An analog for quantum chaos on billiards E. Arcos, G. Báez, P. A. Cuatláyol, M. L. H. Prian, R. A. Méndez-Sánchez a and H. Hernández-Saldaña. Laboratorio de Cuernavaca, Instituto
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 informationarxiv: v1 [physics.comp-ph] 28 Jan 2008
A new method for studying the vibration of non-homogeneous membranes arxiv:0801.4369v1 [physics.comp-ph] 28 Jan 2008 Abstract Paolo Amore Facultad de Ciencias, Universidad de Colima, Bernal Díaz del Castillo
More informationIn chapter 3, when we discussed metallic bonding, the primary attribute was that electrons are delocalized.
Lecture 15: free electron gas Up to this point we have not discussed electrons explicitly, but electrons are the first offenders for most useful phenomena in materials. When we distinguish between metals,
More informationAsymptotic Statistics of Nodal Domains of Quantum Chaotic Billiards in the Semiclassical Limit
Asymptotic Statistics of Nodal Domains of Quantum Chaotic Billiards in the Semiclassical Limit Kyle Konrad Dartmouth College Computer Science Department Advisor: Alex Barnett May 31, 2012 Abstract Quantum
More informationQuantum symbolic dynamics
Quantum symbolic dynamics Stéphane Nonnenmacher Institut de Physique Théorique, Saclay Quantum chaos: routes to RMT and beyond Banff, 26 Feb. 2008 What do we know about chaotic eigenstates? Hamiltonian
More informationGlobal solutions for the cubic non linear wave equation
Global solutions for the cubic non linear wave equation Nicolas Burq Université Paris-Sud, Laboratoire de Mathématiques d Orsay, CNRS, UMR 8628, FRANCE and Ecole Normale Supérieure Oxford sept 12th, 2012
More informationSecond Order Results for Nodal Sets of Gaussian Random Waves
1 / 1 Second Order Results for Nodal Sets of Gaussian Random Waves Giovanni Peccati (Luxembourg University) Joint works with: F. Dalmao, G. Dierickx, D. Marinucci, I. Nourdin, M. Rossi and I. Wigman Random
More informationQuantum Mechanics Solutions. λ i λ j v j v j v i v i.
Quantum Mechanics Solutions 1. (a) If H has an orthonormal basis consisting of the eigenvectors { v i } of A with eigenvalues λ i C, then A can be written in terms of its spectral decomposition as A =
More informationGeneral Physics I. Lecture 14: Sinusoidal Waves. Prof. WAN, Xin ( 万歆 )
General Physics I Lecture 14: Sinusoidal Waves Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ Motivation When analyzing a linear medium that is, one in which the restoring force
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationOptimal shape and location of sensors or actuators in PDE models
Optimal shape and location of sensors or actuators in PDE models Y. Privat, E. Trélat 1, E. Zuazua 1 Univ. Paris 6 (Labo. J.-L. Lions) et Institut Universitaire de France SIAM Conference on Analysis of
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationYAO LIU. f(x t) + f(x + t)
DUNKL WAVE EQUATION & REPRESENTATION THEORY OF SL() YAO LIU The classical wave equation u tt = u in n + 1 dimensions, and its various modifications, have been studied for centuries, and one of the most
More informationShape optimisation for the Robin Laplacian. an overview
: an overview Group of Mathematical Physics University of Lisbon Geometric Spectral Theory Université de Neuchâtel Thursday, 22 June, 2017 The Robin eigenvalue problem Consider u = λu in Ω, u ν + αu =
More informationVibrations of string. Henna Tahvanainen. November 8, ELEC-E5610 Acoustics and the Physics of Sound, Lecture 4
Vibrations of string EEC-E5610 Acoustics and the Physics of Sound, ecture 4 Henna Tahvanainen Department of Signal Processing and Acoustics Aalto University School of Electrical Engineering November 8,
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
More informationLocalized Eigenfunctions: Here You See Them, There You Don t
Localized Eigenfunctions: Here You See Them, There You Don t Steven M. Heilman and Robert S. Strichartz 2 x 2 + 2 y 2 The Laplacian ( in the plane) is one of the most basic operators in all of mathematical
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 informationZeros and Nodal Lines of Modular Forms
Zeros and Nodal Lines of Modular Forms Peter Sarnak Mahler Lectures 2011 Zeros of Modular Forms Classical modular forms Γ = SL 2 (Z) acting on H. z γ az + b [ ] a b cz + d, γ = Γ. c d (i) f (z) holomorphic
More informationarxiv: v1 [quant-ph] 8 Sep 2010
Few-Body Systems, (8) Few- Body Systems c by Springer-Verlag 8 Printed in Austria arxiv:9.48v [quant-ph] 8 Sep Two-boson Correlations in Various One-dimensional Traps A. Okopińska, P. Kościk Institute
More informationGlobal Harmonic Analysis and the Concentration of Eigenfunctions, Part III:
Global Harmonic Analysis and the Concentration of Eigenfunctions, Part III: Improved eigenfunction estimates for the critical L p -space on manifolds of nonpositive curvature Christopher D. Sogge (Johns
More informationOptimal shape and location of sensors. actuators in PDE models
Modeling Optimal shape and location of sensors or actuators in PDE models Emmanuel Tre lat1 1 Sorbonne Universite (Paris 6), Laboratoire J.-L. Lions Fondation Sciences Mathe matiques de Paris Works with
More informationWave Phenomena Physics 15c
Wave Phenomena Phsics 15c Lecture 13 Multi-Dimensional Waves (H&L Chapter 7) Term Paper Topics! Have ou found a topic for the paper?! 2/3 of the class have, or have scheduled a meeting with me! If ou haven
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 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 informationLecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor
Lecture 5: Harmonic oscillator, Morse Oscillator, 1D Rigid Rotor It turns out that the boundary condition of the wavefunction going to zero at infinity is sufficient to quantize the value of energy that
More informationWaves and the Schroedinger Equation
Waves and the Schroedinger Equation 5 april 010 1 The Wave Equation We have seen from previous discussions that the wave-particle duality of matter requires we describe entities through some wave-form
More informationChapter 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: H = αδ(x a/2) (1)
Tor Kjellsson Stockholm University Chapter 6 6. Q. Suppose we put a delta-function bump in the center of the infinite square well: where α is a constant. H = αδ(x a/ ( a Find the first-order correction
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 informationSome Mathematical and Physical Background
Some Mathematical and Physical Background Linear partial differential operators Let H be a second-order, elliptic, self-adjoint PDO, on scalar functions, in a d-dimensional region Prototypical categories
More informationNotes for. Spectral Geometry. by Yaiza Canzani CRM Summer School in Quebec City. Laval University.
Notes for Spectral Geometry by Yaiza Canzani. 2016 CRM Summer School in Quebec City. Laval University. 3 Abstract I wrote these lectures using the material in set of notes Analysis on manifolds via the
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part IB Thursday 7 June 2007 9 to 12 PAPER 3 Before you begin read these instructions carefully. Each question in Section II carries twice the number of marks of each question in Section
More informationSharp bounds for the bottom of the spectrum of Laplacians (and their associated stability estimates)
Sharp bounds for the bottom of the spectrum of Laplacians (and their associated stability estimates) Lorenzo Brasco I2M Aix-Marseille Université lorenzo.brasco@univ-amu.fr http://www.latp.univ-mrs.fr/
More informationMATH 308 COURSE SUMMARY
MATH 308 COURSE SUMMARY Approximately a third of the exam cover the material from the first two midterms, that is, chapter 6 and the first six sections of chapter 7. The rest of the exam will cover the
More informationLECTURE 4 WAVE PACKETS
LECTURE 4 WAVE PACKETS. Comparison between QM and Classical Electrons Classical physics (particle) Quantum mechanics (wave) electron is a point particle electron is wavelike motion described by * * F ma
More informationEnergy levels. From Last Time. Emitting and absorbing light. Hydrogen atom. Energy conservation for Bohr atom. Summary of Hydrogen atom
From Last Time Hydrogen atom: One electron orbiting around one proton (nucleus) Electron can be in different quantum states Quantum states labeled by integer,2,3,4, In each different quantum state, electron
More informationTransient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation
Symmetry, Integrability and Geometry: Methods and Applications Vol. (5), Paper 3, 9 pages Transient Phenomena in Quantum Bound States Subjected to a Sudden Perturbation Marcos MOSHINSKY and Emerson SADURNÍ
More informationLecture 6: Differential Equations Describing Vibrations
Lecture 6: Differential Equations Describing Vibrations In Chapter 3 of the Benson textbook, we will look at how various types of musical instruments produce sound, focusing on issues like how the construction
More information