Asch / Joye Mathematical Physics of Quantum Mechanics

Selected and Refereed Lectures from QMath9
2006
ISBN: 978-3-540-34273-1
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 690, 485 Seiten

Reihe: Lecture Notes in Physics

QMath9 is a meeting for young scientists to learn about the state of the art in the Mathematical Physics of Quantum Systems. This selection of outstanding articles written in pedagocical style has six sections that cover new techniques and recent results on spectral theory, statistical mechanics, Bose-Einstein condensation, random operators, magnetic Schrödinger operators and much more. For postgraduate students this book can be used as a useful introduction to the research literature. For more expert researcher this book will be a concise and modern source of reference.

Written for: Postdocs and researchers in mathematical and quantum physics

Keywords:
quantum chaos
quantum dynamics
random operators
spectral theory

Autoren/Hrsg.

Asch, Joachim

Joye, Alain

Inhaltsverzeichnis

1;Preface;6
2;Contents;8
3;List of Contributors;18
4;Introduction;23
5;Part I Quantum Dynamics and Spectral Theory;25
5.1;Solving the Ten Martini Problem;27
5.1.1;1 Introduction;27
5.1.2;2 Analytic Extension;30
5.1.3;3 The Liouvillian Side;31
5.1.4;4 The Diophantine Side;32
5.1.5;5 A Localization Result;34
5.1.6;References;36
5.2;Swimming Lessons for Microbots;39
5.3;Landau-Zener Formulae from Adiabatic Transition Histories;41
5.3.1;1 Introduction;41
5.3.2;2 Exponentially Small Transitions;44
5.3.3;3 The Hamiltonian in the Super-Adiabatic Representation;47
5.3.4;4 The Scattering Regime;49
5.3.5;References;53
5.4;Scattering Theory of Dynamic Electrical Transport;55
5.4.1;1 From an Internal Response;55
5.4.2;to a Quantum Pump E.ect;55
5.4.3;2 Quantum Coherent Pumping: A Simple Picture;58
5.4.4;3 Beyond the Frozen Scatterer Approximation:;61
5.4.5;Instantaneous Currents;61
5.4.6;References;66
5.5;The Landauer-Büttiker Formula and Resonant Quantum Transport;67
5.5.1;1 The Landauer-Büttiker Formula;67
5.5.2;2 Resonant Transport in a Quantum Dot;69
5.5.3;3 A Numerical Example;70
5.5.4;References;75
5.6;Point Interaction Polygons: An Isoperimetric Problem;76
5.6.1;1 Introduction;76
5.6.2;2 The Local Result in Geometric Terms;77
5.6.3;3 Proof of Theorem 1;79
5.6.4;4 About the Global Maximizer;82
5.6.5;5 Some Extensions;83
5.6.6;Acknowledgments;85
5.6.7;References;85
5.7;Limit Cycles in Quantum Mechanics;87
5.7.1;1 Introduction;87
5.7.2;2 Definition of the Model;89
5.7.3;3 Renormalization Group;91
5.7.4;4 Limit Cycle;93
5.7.5;5 Marginal and Irrelevant Operators;95
5.7.6;6 Tuning to a Cycle;96
5.7.7;7 Generic Properties of Limit Cycles;97
5.7.8;8 Conclusion;98
5.7.9;Acknowledgments;98
5.7.10;References;98
5.8;Cantor Spectrum for Quasi-Periodic Schrödinger Operators;101
5.8.1;1 The Almost Mathieu Operator & the Ten Martini Problem;101
5.8.2;2 Extension to Real Analytic Potentials;109
5.8.3;3 Cantor Spectrum for Speci.c Models;110
5.8.4;References;112
6;Part II Quantum Field Theory and Statistical Mechanics;115
6.1;Adiabatic Theorems and Reversible Isothermal Processes;117
6.1.1;1 Introduction;117
6.1.2;2 A General Adiabatic Theorem ;119
6.1.3;3 The Isothermal Theorem ;121
6.1.4;4 (Reversible) Isothermal Processes;123
6.1.5;References;126
6.2;Quantum Massless Field in 1+1 Dimensions;129
6.2.1;1 Introduction;129
6.2.2;2 Fields;130
6.2.3;3 Poincar ´ e Covariance;133
6.2.4;4 Changing the Compensating Functions;134
6.2.5;5 Hilbert Space;135
6.2.6;6 Fields in Position Representation;137
6.2.7;7 The SL(2, R) × SL(2, R) Covariance;138
6.2.8;8 Normal Ordering;139
6.2.9;9 Classical Fields;140
6.2.10;10 Algebraic Approach;142
6.2.11;11 Vertex Operators;144
6.2.12;12 Fermions;145
6.2.13;13 Supersymmetry;147
6.2.14;Acknowledgement;148
6.2.15;References;148
6.3;Stability of Multi-Phase Equilibria;151
6.3.1;1 Stability of a Single-Phase Equilibrium;151
6.3.2;2 Stability of Multi-Phase Equilibria;159
6.3.3;3 Quantum Tweezers;160
6.3.4;References;170
6.4;Ordering of Energy Levels in Heisenberg Models and Applications;171
6.4.1;1 Introduction;171
6.4.2;2 Proof of the Main Result;174
6.4.3;3 The Temperley-Lieb Basis. Proof of Proposition 1;180
6.4.4;4 Extensions;185
6.4.5;5 Applications;187
6.4.6;Acknowledgement;191
6.4.7;References;191
6.5;Interacting Fermions in 2 Dimensions;193
6.5.1;1 Introduction;193
6.5.2;2 Fermi Liquids and Salmhofer’s Criterion;193
6.5.3;3 The Models;195
6.5.4;4 A Brief Review of Rigorous Results;196
6.5.5;5 Multiscale Analysis, Angular Sectors;197
6.5.6;6 One and Two Particle Irreducible Expansions;198
6.5.7;References;200
6.6;On the Essential Spectrum of the Translation Invariant Nelson Model;201
6.6.1;1 The Model and the Result;201
6.6.2;2 A Complex Function of Two Variables;204
6.6.3;3 The Essential Spectrum;211
6.6.4;A Riemannian Covers;216
6.6.5;References;217
7;Part III Quantum Kinetics and Bose-Einstein Condensation;219
7.1;Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice;221
7.1.1;1 Introduction;221
7.1.2;2 Reflection Positivity;225
7.1.3;3 Proof of BEC for Small . and T;227
7.1.4;4 Absence of BEC and Mott Insulator Phase;232
7.1.5;5 The Non-Interacting Gas;235
7.1.6;6 Conclusion;236
7.1.7;References;236
7.2;Long Time Behaviour to the Schrödinger–Poisson–Xa Systems;239
7.2.1;1 Introduction;239
7.2.2;2 On the Derivation of the Slater Approach;242
7.2.3;3 Some Results Concerning Well Posedness and Asymptotic Behaviour;245
7.2.4;4 Long-Time Behaviour;250
7.2.5;5 On the General;252
7.2.6;Case;252
7.2.7;References;253
7.3;Towards the Quantum Brownian Motion;255
7.3.1;1 Introduction;255
7.3.2;2 Statement of Main Result;259
7.3.3;3 Sketch of the Proof;264
7.3.4;4 Computation of the Main Term and Its Convergence to a Brownian Motion;277
7.3.5;Acknowledgements;279
7.3.6;References;279
7.4;Bose-Einstein Condensation and Superradiance;281
7.4.1;1 Introduction;281
7.4.2;2 Solution of the Model 1;285
7.4.3;3 Model 2 and Matter-Wave Grating;294
7.4.4;4 Conclusion;298
7.4.5;Acknowledgement;299
7.4.6;References;299
7.5;Derivation of the Gross-Pitaevskii Hierarchy;301
7.5.1;1 Introduction;301
7.5.2;2 The Main Result;308
7.5.3;3 Sketch of the Proof;312
7.5.4;References;314
7.6;Towards a Microscopic Derivation of the Phonon Boltzmann Equation;317
7.6.1;1 Introduction;317
7.6.2;2 Microscopic Model;318
7.6.3;3 Kinetic Limit and Boltzmann Equation;320
7.6.4;4 Feynman Diagrams;322
7.6.5;Acknowledgements;326
7.6.6;References;326
8;Part IV Disordered Systems and Random Operators;327
8.1;On the Quantization of Hall Currents in Presence of Disorder;329
8.1.1;1 The Edge Conductance and General Invariance Principles;329
8.1.2;2 Regularizing the Edge Conductance in Presence of Impurities;332
8.1.3;3 Localization for the Landau Operator with a Half-Plane Random Potential;339
8.1.4;References;343
8.2;Equality of the Bulk and Edge Hall Conductances in 2D;347
8.2.1;1 Introduction and Main Result;347
8.2.2;2 Proof of sB = sE;351
8.2.3;References;354
8.3;Generic Subsets in Spaces of Measures and Singular Continuous Spectrum;355
8.3.1;1 Introduction;355
8.3.2;2 Generic Subsets in Spaces of Measures;356
8.3.3;3 Singular Continuity of Measures;356
8.3.4;4 Selfadjoint Operators and the Wonderland Theorem;358
8.3.5;5 Operators Associated to Delone Sets;360
8.3.6;Acknowledgment;363
8.3.7;References;363
8.4;Low Density Expansion for Lyapunov Exponents;365
8.4.1;1 Introduction;365
8.4.2;2 Model and Preliminaries;366
8.4.3;3 Result on the Lyapunov Exponent;368
8.4.4;4 Proof;369
8.4.5;5 Result on the Density of States;371
8.4.6;References;372
8.5;Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles;373
8.5.1;1 Introduction;373
8.5.2;2 Wigner and Band Random Matrices with Heavy Tails of Marginal Distributions;378
8.5.3;3 Real Sample Covariance Matrices with Cauchy Entries;382
8.5.4;4 Conclusion;384
8.5.5;References;385
9;Part V Semiclassical Analysis and Quantum Chaos;387
9.1;Recent Results on Quantum Map Eigenstates;389
9.1.1;1 Introduction;389
9.1.2;2 Perturbed CAT Maps: Classical Dynamics;390
9.1.3;3 Quantum Maps;392
9.1.4;4 What is Known?;393
9.1.5;5 Perturbed Cat Maps;399
9.1.6;Acknowledgments;402
9.1.7;References;402
9.2;Level Repulsion and Spectral Type for One-Dimensional Adiabatic Quasi-Periodic Schrödinger Operators;405
9.2.1;1 A Heuristic Description;405
9.2.2;2 Mathematical Results;409
9.2.3;References;423
9.3;Low Lying Eigenvalues of Witten Laplacians and Metastability (After Helffer-Klein-Nier and Helffer-Nier);425
9.3.1;1 Main Goals and Assumptions;425
9.3.2;2 Saddle Points and Labelling;426
9.3.3;3 Rough Semi-Classical Analysis of Witten Laplacians;428
9.3.4;and Applications to Morse Theory;428
9.3.5;4 Main Result in the Case of;429
9.3.6;5 About the Proof in the Case of;430
9.3.7;6 The Main Result in the Case with Boundary;432
9.3.8;7 About the Proof in the Case with Boundary;433
9.3.9;Acknowledgements;436
9.3.10;References;436
9.4;The Mathematical Formalism of a Particle in a Magnetic Field;439
9.4.1;1 Introduction;439
9.4.2;2 The Classical Particle in a Magnetic Field;440
9.4.3;3 The Quantum Picture;444
9.4.4;4 The Limit h . 0;448
9.4.5;5 The Schrödinger Representation;449
9.4.6;6 Applications to Spectral Analysis;453
9.4.7;Acknowledgements;454
9.4.8;References;455
9.5;Fractal Weyl Law for Open Chaotic Maps;457
9.5.1;1 Introduction;457
9.5.2;2 The Open Baker’s Map and Its Quantization;461
9.5.3;3 A Solvable Toy Model for the Quantum Baker;468
9.5.4;Acknowledgments;471
9.5.5;References;471
9.6;Spectral Shift Function for Magnetic Schr¨ odinger Operators;473
9.6.1;1 Introduction;473
9.6.2;2 Auxiliary Results;475
9.6.3;3 Main Results;477
9.6.4;Acknowledgements;485
9.6.5;References;485
9.7;Counting String/M Vacua;489
9.7.1;1 Introduction;489
9.7.2;2 Type IIb Flux Compacti.cations of String/M Theory;490
9.7.3;3 Critical Points and Hessians of Holomorphic Sections;492
9.7.4;4 The Critical Point Problem;493
9.7.5;5 Statement of Results;495
9.7.6;6 Comparison to the Physics Literature;496
9.7.7;7 Sketch of Proofs;497
9.7.8;8 Other Formulae for the Critical Point Density;498
9.7.9;9 Black Hole Attractors;502
9.7.10;References;503

Leseproben

Part I Quantum Dynamics and Spectral Theory (p. 3-4)

Different aspects of the solution of a long-standing major problem in mathematical physics are reported in the contributions of Avila, Jitomirskaya and Puig. It had been conjectured for about thirty years by physicists and mathematicians that the problem of electrons confined to a plane under the influence of a periodic potential and a perpendicular magnetic field exhibits fractal spectral properties. Experimental evidence of Hofstadters butterflylike energy spectrum was found about five years ago.

Here the mathematical physicists Avila, Jitomirskaya and Puig report on the proof that the spectrum of a related operator is a Cantor set. Their proofs rely much on recent techniques in classical dynamical systems.We mention that the mathematical model still has fascinating unsolved aspects which are important to physics and especially the quantum hall effect for example the question whether the spectral gaps are open. Building Micron-size robots which move much faster than bacteria is one of the visions of small scale physics. Y. Avron gave an introduction on recent results on the problem of designing an optimal micro-swimmer.

These have been obtained using methods from geometry and linear response theory. V. Betz and S. Teufel report on their progress in the old Landau–Zener problem. For a time dependent two state problem which is asymptotically constant, a detailed approximate solution which takes into account the adiabatic transitions is obtained for all times. It describes both the exponential smallness of the transition probability and the time scale over which it takes place. The theory of transport in mesoscopic systems is addressed in two contributions. Büttiker and Moskalets treat quantum pumping. If the system is driven by several internal parameters oscillating slowly, a direct current may result.

It can be calculated to leading order in terms of stationary scattering matrices. To take account the energy exchange with the environment the full time dependent scattering matrix is developed to next order. A mathematical proof of the formula relating conductance and transmittance has been given by H.D. Cornean, A. Jensen, V. Moldoveanu in the case of an adiabatically switched on external potential. The formula is applied numerically to a model.

Geometry meets physics again in the contribution of P. Exner who presents a conjecture about an interesting isoperimetric problem arising from the spectral analysis of a quantum model with point interactions. S. Glazek discusses examples of renormalization group analysis applied to Schrödinger operators and in particular the occurrence of a limit cycle as a critical attractor instead of a fixed point.