Mathematical Physics of Quantum Mechanics (eBook)

Preface 6
Contents 8
List of Contributors 18
Introduction 23
Part I Quantum Dynamics and Spectral Theory 25
Solving the Ten Martini Problem 27
1 Introduction 27
2 Analytic Extension 30
3 The Liouvillian Side 31
4 The Diophantine Side 32
5 A Localization Result 34
References 36
Swimming Lessons for Microbots 39
Landau-Zener Formulae from Adiabatic Transition Histories 41
1 Introduction 41
2 Exponentially Small Transitions 44
3 The Hamiltonian in the Super-Adiabatic Representation 47
4 The Scattering Regime 49
References 53
Scattering Theory of Dynamic Electrical Transport 55
1 From an Internal Response 55
to a Quantum Pump E.ect 55
2 Quantum Coherent Pumping: A Simple Picture 58
3 Beyond the Frozen Scatterer Approximation: 61
Instantaneous Currents 61
References 66
The Landauer-Büttiker Formula and Resonant Quantum Transport 67
1 The Landauer-Büttiker Formula 67
2 Resonant Transport in a Quantum Dot 69
3 A Numerical Example 70
References 75
Point Interaction Polygons: An Isoperimetric Problem 76
1 Introduction 76
2 The Local Result in Geometric Terms 77
3 Proof of Theorem 1 79
4 About the Global Maximizer 82
5 Some Extensions 83
Acknowledgments 85
References 85
Limit Cycles in Quantum Mechanics 87
1 Introduction 87
2 Definition of the Model 89
3 Renormalization Group 91
4 Limit Cycle 93
5 Marginal and Irrelevant Operators 95
6 Tuning to a Cycle 96
7 Generic Properties of Limit Cycles 97
8 Conclusion 98
Acknowledgments 98
References 98
Cantor Spectrum for Quasi-Periodic Schrödinger Operators 101
1 The Almost Mathieu Operator & the Ten Martini Problem
2 Extension to Real Analytic Potentials 109
3 Cantor Spectrum for Speci.c Models 110
References 112
Part II Quantum Field Theory and Statistical Mechanics 115
Adiabatic Theorems and Reversible Isothermal Processes 117
1 Introduction 117
2 A General Adiabatic Theorem 119
3 The Isothermal Theorem 121
4 (Reversible) Isothermal Processes 123
References 126
Quantum Massless Field in 1+1 Dimensions 129
1 Introduction 129
2 Fields 130
3 Poincar ´ e Covariance 133
4 Changing the Compensating Functions 134
5 Hilbert Space 135
6 Fields in Position Representation 137
7 The SL(2, R) × SL(2, R) Covariance 138
8 Normal Ordering 139
9 Classical Fields 140
10 Algebraic Approach 142
11 Vertex Operators 144
12 Fermions 145
13 Supersymmetry 147
Acknowledgement 148
References 148
Stability of Multi-Phase Equilibria 151
1 Stability of a Single-Phase Equilibrium 151
2 Stability of Multi-Phase Equilibria 159
3 Quantum Tweezers 160
References 170
Ordering of Energy Levels in Heisenberg Models and Applications 171
1 Introduction 171
2 Proof of the Main Result 174
3 The Temperley-Lieb Basis. Proof of Proposition 1 180
4 Extensions 185
5 Applications 187
Acknowledgement 191
References 191
Interacting Fermions in 2 Dimensions 193
1 Introduction 193
2 Fermi Liquids and Salmhofer’s Criterion 193
3 The Models 195
4 A Brief Review of Rigorous Results 196
5 Multiscale Analysis, Angular Sectors 197
6 One and Two Particle Irreducible Expansions 198
References 200
On the Essential Spectrum of the Translation Invariant Nelson Model 201
1 The Model and the Result 201
2 A Complex Function of Two Variables 204
3 The Essential Spectrum 211
A Riemannian Covers 216
References 217
Part III Quantum Kinetics and Bose-Einstein Condensation 219
Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice 221
1 Introduction 221
2 Reflection Positivity 225
3 Proof of BEC for Small . and T 227
4 Absence of BEC and Mott Insulator Phase 232
5 The Non-Interacting Gas 235
6 Conclusion 236
References 236
Long Time Behaviour to the Schrödinger–Poisson–Xa Systems 239
1 Introduction 239
2 On the Derivation of the Slater Approach 242
3 Some Results Concerning Well Posedness and Asymptotic Behaviour 245
4 Long-Time Behaviour 250
5 On the General 252
Case 252
References 253
Towards the Quantum Brownian Motion 255
1 Introduction 255
2 Statement of Main Result 259
3 Sketch of the Proof 264
4 Computation of the Main Term and Its Convergence to a Brownian Motion 277
Acknowledgements 279
References 279
Bose-Einstein Condensation and Superradiance 281
1 Introduction 281
2 Solution of the Model 1 285
3 Model 2 and Matter-Wave Grating 294
4 Conclusion 298
Acknowledgement 299
References 299
Derivation of the Gross-Pitaevskii Hierarchy 301
1 Introduction 301
2 The Main Result 308
3 Sketch of the Proof 312
References 314
Towards a Microscopic Derivation of the Phonon Boltzmann Equation 317
1 Introduction 317
2 Microscopic Model 318
3 Kinetic Limit and Boltzmann Equation 320
4 Feynman Diagrams 322
Acknowledgements 326
References 326
Part IV Disordered Systems and Random Operators 327
On the Quantization of Hall Currents in Presence of Disorder 329
1 The Edge Conductance and General Invariance Principles 329
2 Regularizing the Edge Conductance in Presence of Impurities 332
3 Localization for the Landau Operator with a Half-Plane Random Potential 339
References 343
Equality of the Bulk and Edge Hall Conductances in 2D 347
1 Introduction and Main Result 347
2 Proof of sB = sE 351
References 354
Generic Subsets in Spaces of Measures and Singular Continuous Spectrum 355
1 Introduction 355
2 Generic Subsets in Spaces of Measures 356
3 Singular Continuity of Measures 356
4 Selfadjoint Operators and the Wonderland Theorem 358
5 Operators Associated to Delone Sets 360
Acknowledgment 363
References 363
Low Density Expansion for Lyapunov Exponents 365
1 Introduction 365
2 Model and Preliminaries 366
3 Result on the Lyapunov Exponent 368
4 Proof 369
5 Result on the Density of States 371
References 372
Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles 373
1 Introduction 373
2 Wigner and Band Random Matrices with Heavy Tails of Marginal Distributions 378
3 Real Sample Covariance Matrices with Cauchy Entries 382
4 Conclusion 384
References 385
Part V Semiclassical Analysis and Quantum Chaos 387
Recent Results on Quantum Map Eigenstates 389
1 Introduction 389
2 Perturbed CAT Maps: Classical Dynamics 390
3 Quantum Maps 392
4 What is Known? 393
5 Perturbed Cat Maps 399
Acknowledgments 402
References 402
Level Repulsion and Spectral Type for One-Dimensional Adiabatic Quasi-Periodic Schrödinger Operators 405
1 A Heuristic Description 405
2 Mathematical Results 409
References 423
Low Lying Eigenvalues of Witten Laplacians and Metastability (After Helffer-Klein-Nier and Helffer-Nier) 425
1 Main Goals and Assumptions 425
2 Saddle Points and Labelling 426
3 Rough Semi-Classical Analysis of Witten Laplacians 428
and Applications to Morse Theory 428
4 Main Result in the Case of 429
5 About the Proof in the Case of 430
6 The Main Result in the Case with Boundary 432
7 About the Proof in the Case with Boundary 433
Acknowledgements 436
References 436
The Mathematical Formalism of a Particle in a Magnetic Field 439
1 Introduction 439
2 The Classical Particle in a Magnetic Field 440
3 The Quantum Picture 444
4 The Limit h . 0 448
5 The Schrödinger Representation 449
6 Applications to Spectral Analysis 453
Acknowledgements 454
References 455
Fractal Weyl Law for Open Chaotic Maps 457
1 Introduction 457
2 The Open Baker’s Map and Its Quantization 461
3 A Solvable Toy Model for the Quantum Baker 468
Acknowledgments 471
References 471
Spectral Shift Function for Magnetic Schr¨ odinger Operators 473
1 Introduction 473
2 Auxiliary Results 475
3 Main Results 477
Acknowledgements 485
References 485
Counting String/M Vacua 489
1 Introduction 489
2 Type IIb Flux Compacti.cations of String/M Theory 490
3 Critical Points and Hessians of Holomorphic Sections 492
4 The Critical Point Problem 493
5 Statement of Results 495
6 Comparison to the Physics Literature 496
7 Sketch of Proofs 497
8 Other Formulae for the Critical Point Density 498
9 Black Hole Attractors 502
References 503

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.