Lecture Notes in Physics

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Dr. Christian Caron Springer Heidelberg Physics Editorial Department I Tiergartenstrasse 17 69121 Heidelberg/Germany [email protected] Joachim Asch Alain Joye (Eds.)

Mathematical Physics of Quantum Mechanics

Selected and Refereed Lectures from QMath9

ABC Editors

Joachim Asch Alain Joye Université du Sud Toulon Var Institut Fourier Centre de physique théorique Université Grenoble 1 Département de Mathématiques BP 74 BP 20132 38402 Saint-Martin-d’Hères Cedex F-83957 La Garde Cedex France France E-mail: [email protected] E-mail: [email protected]

J. Asch and A. Joye, Mathematical Physics of Quantum Mechanics, Lect. Notes Phys. 690 (Springer, Berlin Heidelberg 2006), DOI 10.1007/b11573432

Library of Congress Control Number: 2005938945

ISSN 0075-8450 ISBN-10 3-540-31026-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-31026-6 Springer Berlin Heidelberg New York

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Typesetting: by the authors and TechBooks using a Springer LATEX macro package Printed on acid-free paper SPIN: 11573432 54/TechBooks 543210 Preface

The topics presented in this book were discussed at the conference “QMath9” held in Giens, France, September 12th-16th 2004. QMath is a series of meet- ings whose aim is to present the state of the art in the Mathematical Physics of Quantum Systems, both from the point of view of physical models and of the mathematical techniques developed for their study. The series was initi- ated in the early seventies as an attempt to enhance collaboration between mathematical physicists from eastern and western European countries. In the nineties it took a worldwide dimension. At the same time, due to engineer- ing achievements, for example in the mesoscopic realm, there was a renewed interest in basic questions of quantum dynamics. The program of QMath9, which was attended by 170 scientists from 23 countries, consisted of 123 talks grouped by the topics: Nanophysics, Quan- tum dynamics, Quantum field theory, Quantum kinetics, Random Schr¨odinger operators, Semiclassical analysis, Spectral theory. QMath9 was also the frame for the 2004 meeting of the European Research Group on “Mathematics and Quantum Physics” directed by Monique Combescure. For a detailed account of the program, see http://www.cpt.univ.mrs.fr/qmath9. Expanded versions of several selected introductory talks presented at the conference are included in this volume. Their aim is to provide the reader with an easier access to the sometimes technical state of the art in a topic. Other contributions are devoted to a pedagogical exposition of quite recent results at the frontiers of research, parts of which were presented in “QMath9”. In addition, the reader will find in this book new results triggered by discussions which took place at the meeting. Hence, while based on the conference “QMath9”, this book is intended to be a starting point for the reader who wishes to learn about the current research in quantum mathematical physics, with a general perspective. Ef- fort has been made by the authors, editors and referees in order to provide contributions of the highest scientific standards to meet this goal. We are grateful to Yosi Avron, Volker Bach, Stephan De Bi`evre, Laszlo Erd¨os, Pavel Exner, Svetlana Jitomirskaya, Fr´ed´eric Klopp who mediated the scientific sessions of “QMath9”. We should like to thank all persons and institutions who helped to organize the conference locally: Sylvie Aguillon, Jean-Marie Barbaroux, VI Preface

Nils Berglund, Jean-Michel Combes, Elisabeth Elophe, Jean-Michel Ghez, Corinne Roux, Corinne Vera, Universit´e du Sud Toulon–Var and Centre de Physique Th´eorique Marseille. We gratefully acknowledge financial support from: European Science Foundation (SPECT), International Association of Mathematical Physics, Minist`ere de l’Education Nationale et de la Recherche, Centre National de la Recherche Scientifique, R´egion Provence-Alpes-Cˆote d’Azur, Conseil G´en´eral du Var, Centre de Physique Th´eorique, Universit´e du Sud Toulon–Var, Insti- tut Fourier, Universit´e Joseph Fourier.

Toulon Joachim Asch Grenoble Alain Joye January 2006 Contents

Introduction ...... 1

Part I Quantum Dynamics and Spectral Theory

Solving the Ten Martini Problem A. Avila and S. Jitomirskaya ...... 5 1 Introduction ...... 5 1.1 RoughStrategy...... 6 2 AnalyticExtension...... 8 3 The Liouvillian Side ...... 9 3.1 Gaps for Rational Approximants ...... 9 3.2 ContinuityoftheSpectrum...... 10 4 TheDiophantineSide...... 10 4.1 Reducibility ...... 11 4.2 Localization and Reducibility...... 12 5 A Localization Result ...... 12 References ...... 14 Swimming Lessons for Microbots Y. Avron ...... 17 Landau-Zener Formulae from Adiabatic Transition Histories V. Betz and S. Teufel...... 19 1 Introduction ...... 19 2 Exponentially Small Transitions ...... 22 3 The Hamiltonian in the Super-Adiabatic Representation...... 25 4 The Scattering Regime ...... 27 References ...... 31 Scattering Theory of Dynamic Electrical Transport M. B¨uttiker and M. Moskalets ...... 33 1 From an Internal Response to a Quantum Pump Effect ...... 33 2 Quantum Coherent Pumping: A Simple Picture ...... 36 VIII Contents

3 Beyond the Frozen Scatterer Approximation: InstantaneousCurrents ...... 39 References ...... 44 The Landauer-B¨uttiker Formula and Resonant Quantum Transport H.D. Cornean, A. Jensen and V. Moldoveanu ...... 45 1 The Landauer-B¨uttiker Formula ...... 45 2 ResonantTransportinaQuantumDot...... 47 3 ANumericalExample ...... 48 References ...... 53 Point Interaction Polygons: An Isoperimetric Problem P. Exner ...... 55 1 Introduction ...... 55 2 TheLocalResultinGeometricTerms...... 56 3 ProofofTheorem1...... 58 4 AbouttheGlobalMaximizer...... 61 5 SomeExtensions...... 62 References ...... 64 Limit Cycles in Quantum Mechanics S.D. Glazek ...... 65 1 Introduction ...... 65 2 Definition of the Model ...... 67 3 Renormalization Group ...... 69 4 Limit Cycle ...... 71 5 Marginal and Irrelevant Operators ...... 73 6 TuningtoaCycle...... 74 7 Generic Properties of Limit Cycles ...... 75 8 Conclusion ...... 76 References ...... 76 Cantor Spectrum for Quasi-Periodic Schr¨odinger Operators J. Puig ...... 79 1 The Almost Mathieu Operator & the Ten Martini Problem ...... 79 1.1 The ids andtheSpectrum...... 80 1.2 SketchoftheProof ...... 83 1.3 Reducibility of Quasi-Periodic Cocycles ...... 84 1.4 EndofProof...... 86 2 Extension to Real Analytic Potentials ...... 87 3 Cantor Spectrum for Specific Models ...... 88 References ...... 90 Contents IX

Part II Quantum Field Theory and Statistical Mechanics

Adiabatic Theorems and Reversible Isothermal Processes W.K. Abou-Salem and J. Fr¨ohlich ...... 95 1 Introduction ...... 95 2 A General “Adiabatic Theorem” ...... 97 3 The“IsothermalTheorem”...... 99 4 (Reversible)IsothermalProcesses...... 101 References ...... 104 Quantum Massless Field in 1+1 Dimensions J. Derezi´nski and K.A. Meissner ...... 107 1 Introduction ...... 107 2 Fields...... 108 3 Poincar´eCovariance...... 111 4 Changing the Compensating Functions ...... 112 5 HilbertSpace...... 113 6 Fields in Position Representation ...... 115 7TheSL(2, R) × SL(2, R)Covariance...... 116 8 NormalOrdering...... 117 9 ClassicalFields ...... 118 10 AlgebraicApproach ...... 120 11 Vertex Operators ...... 122 12 Fermions ...... 123 13 Supersymmetry...... 125 References ...... 126 Stability of Multi-Phase Equilibria M. Merkli ...... 129 1 Stability of a Single-Phase Equilibrium ...... 129 1.1 TheFreeBoseGas...... 129 1.2 Spontaneous Symmetry Breaking and Multi-Phase Equilibrium ...... 133 1.3 Return to Equilibrium in Absence of a Condensate ...... 135 1.4 Return to Equilibrium in Presence of a Condensate ...... 135 1.5 SpectralApproach...... 136 2 Stability of Multi-Phase Equilibria ...... 137 3 Quantum Tweezers ...... 138 3.1 Non-InteractingSystem...... 141 3.2 InteractingSystem...... 146 3.3 Stability of the Quantum Tweezers, Main Results ...... 147 References ...... 148 X Contents

Ordering of Energy Levels in Heisenberg Models and Applications B. Nachtergaele and S. Starr ...... 149 1 Introduction ...... 149 2 ProofoftheMainResult...... 152 3 The Temperley-Lieb Basis. Proof of Proposition 1 ...... 158 3.1 The Basis for Spin 1/2 ...... 158 3.2 TheBasisforHigherSpin...... 160 4 Extensions ...... 163 4.1 The Spin 1/2 SUq(2)-symmetric XXZ Chain ...... 163 4.2 Higher Order Interactions ...... 165 5 Applications ...... 165 5.1 Diagonalization at Low Energy ...... 165 5.2 The Ground States of Fixed Magnetization fortheXXZChain...... 166 5.3 Aldous’ Conjecture for the Symmetric Simple Exclusion Process...... 167 References ...... 169 Interacting Fermions in 2 Dimensions V. Rivasseau ...... 171 1 Introduction ...... 171 2 Fermi Liquids and Salmhofer’s Criterion ...... 171 3 TheModels...... 173 4 A Brief Review of Rigorous Results ...... 174 5 Multiscale Analysis, Angular Sectors ...... 175 6 One and Two Particle Irreducible Expansions ...... 176 References ...... 178 On the Essential Spectrum of the Translation Invariant Nelson Model J. Schach-Møller ...... 179 1 TheModelandtheResult...... 179 2 A Complex Function of Two Variables ...... 182 3 The Essential Spectrum ...... 189 A RiemannianCovers...... 194 References ...... 195

Part III Quantum Kinetics and Bose-Einstein Condensation

Bose-Einstein Condensation as a Quantum Phase Transition in an Optical Lattice M. Aizenman, E.H. Lieb, R. Seiringer, J.P. Solovej and J. Yngvason ...... 199 1 Introduction ...... 199 Contents XI

2 Reflection Positivity ...... 203 3 Proof of BEC for Small λ and T ...... 205 4 Absence of BEC and Mott Insulator Phase ...... 210 5 TheNon-InteractingGas...... 213 6 Conclusion ...... 214 References ...... 214 Long Time Behaviour to the Schr¨odinger–Poisson–Xα Systems O. Bokanowski, J.L. L´opez, O.´ S´anchez and J. Soler ...... 217 1 Introduction ...... 217 2 On the Derivation of the Slater Approach ...... 220 3 Some Results Concerning Well Posedness and Asymptotic Behaviour ...... 223 3.1 Existence and Uniqueness of Physically Admissible Solutions . 223 3.2 MinimumofEnergy...... 225 3.3 Optimal Kinetic Energy Bounds ...... 227 4 Long-Time Behaviour ...... 228 5 On the General Xα Case...... 230 References ...... 231 Towards the Quantum Brownian Motion L. Erd˝os, M. Salmhofer and H.-T. Yau ...... 233 1 Introduction ...... 233 2 Statement of Main Result ...... 237 3 SketchoftheProof...... 242 3.1 Renormalization ...... 242 3.2 TheExpansionandtheStoppingRules...... 243 3.3 The L2 Norm of the Non-Repetitive Wavefunction ...... 245 3.4 Sketch of the Proof of the Main Technical Theorem ...... 250 3.5 Point Singularities ...... 253 4 Computation of the Main Term and Its Convergence to a Brownian Motion ...... 255 References ...... 257 Bose-Einstein Condensation and Superradiance J.V. Pul´e, A.F. Verbeure and V.A. Zagrebnov ...... 259 1 Introduction ...... 259 2 Solution of the Model 1 ...... 263 2.1 The Effective Hamiltonian ...... 263 2.2 ThePressureforModel1...... 270 3 Model 2 and Matter-Wave Grating ...... 272 4 Conclusion ...... 276 References ...... 277 XII Contents

Derivation of the Gross-Pitaevskii Hierarchy B. Schlein...... 279 1 Introduction ...... 279 2 TheMainResult...... 286 3 SketchoftheProof...... 290 References ...... 292 Towards a Microscopic Derivation of the Phonon Boltzmann Equation H. Spohn ...... 295 1 Introduction ...... 295 2 MicroscopicModel ...... 296 3 Kinetic Limit and Boltzmann Equation ...... 298 4 Feynman Diagrams ...... 300 References ...... 304

Part IV Disordered Systems and Random Operators

On the Quantization of Hall Currents in Presence of Disorder J.-M. Combes, F. Germinet and P.D. Hislop ...... 307 1 The Edge Conductance and General Invariance Principles ...... 307 2 Regularizing the Edge Conductance in Presence of Impurities ...... 310 2.1 Generalities ...... 310 2.2 A Time Averaged Regularization for a Dynamically Localized System ...... 312 2.3 Regularization Under a Stronger form of Dynamical Localization ...... 314 3 Localization for the Landau Operator with a Half-Plane Random Potential ...... 317 3.1 A Large Magnetic Field Regime ...... 317 3.2 A Large Disorder Regime ...... 319 References ...... 321

Equality of the Bulk and Edge Hall Conductances in 2D A. Elgart ...... 325 1 Introduction and Main Result ...... 325 2 Proof of σB = σE ...... 329 2.1 Some Preliminaries ...... 329 2.2 Convergence and Trace Class Properties ...... 330 2.3 Edge – Bulk Interpolation ...... 330 2.4 σE = σB ...... 331 References ...... 332 Contents XIII

Generic Subsets in Spaces of Measures and Singular Continuous Spectrum D. Lenz and P. Stollmann...... 333 1 Introduction ...... 333 2 Generic Subsets in Spaces of Measures ...... 334 3 SingularContinuityofMeasures...... 334 4 Selfadjoint Operators and the Wonderland Theorem ...... 336 5 Operators Associated to Delone Sets ...... 338 References ...... 341 Low Density Expansion for Lyapunov Exponents H. Schulz-Baldes ...... 343 1 Introduction ...... 343 2 Model and Preliminaries ...... 344 3 Result on the Lyapunov Exponent ...... 346 4 Proof...... 347 5 Result on the Density of States ...... 349 References ...... 350 Poisson Statistics for the Largest Eigenvalues in Random Matrix Ensembles A. Soshnikov ...... 351 1 Introduction ...... 351 1.1 WignerRandomMatrices...... 351 1.2 BandRandomMatrices ...... 354 1.3 Sample Covariance Random Matrices...... 355 1.4 Universality in Random Matrices ...... 356 2 Wigner and Band Random Matrices with Heavy Tails of Marginal Distributions ...... 356 3 Real Sample Covariance Matrices with Cauchy Entries ...... 360 4 Conclusion ...... 362 References ...... 363

Part V Semiclassical Analysis and Quantum Chaos

Recent Results on Quantum Map Eigenstates S. De Bi`evre ...... 367 1 Introduction ...... 367 2 Perturbed CAT Maps: Classical Dynamics ...... 368 3 QuantumMaps...... 370 4 WhatisKnown?...... 371 5 PerturbedCatMaps...... 377 References ...... 380 XIV Contents

Level Repulsion and Spectral Type for One-Dimensional Adiabatic Quasi-Periodic Schr¨odinger Operators A. Fedotov and F. Klopp ...... 383 1 A Heuristic Description ...... 383 2 Mathematical Results ...... 387 2.1 The Periodic Operator ...... 388 2.2 A “Geometric” Assumption on the Energy Region Under Study ...... 388 2.3 The Definitions of the Phase Integrals and the Tunelling Coefficients ...... 389 2.4 Ergodic Family ...... 391 2.5 A Coarse Description of the Location of the Spectrum in J . . 392 2.6 A Precise Description of the Spectrum ...... 393 2.7 The Model Equation ...... 399 2.8 When τ isofOrder1...... 400 2.9 Numerical Computations ...... 400 References ...... 401 Low Lying Eigenvalues of Witten Laplacians and Metastability (After Helffer-Klein-Nier and Helffer-Nier) B. Helffer ...... 403 1 Main Goals and Assumptions ...... 403 2 Saddle Points and Labelling ...... 404 3 Rough Semi-Classical Analysis of Witten Laplacians and Applications to Morse Theory ...... 406 3.1 PreviousResults...... 406 3.2 Witten Laplacians on p-Forms...... 407 3.3 Morse Inequalities ...... 407 4 Main Result in the Case of Rn ...... 407 5 About the Proof in the Case of Rn ...... 408 5.1 Preliminaries ...... 408 5.2 Witten Complex, Reduced Witten Complex ...... 409 5.3 SingularValues...... 409 6 The Main Result in the Case with Boundary ...... 410 7 About the Proof in the Case with Boundary ...... 411 7.1 Define the Witten Complex and the Associate Laplacian . . . . . 411 7.2 Rough Localization of the Spectrum of this Laplacian on1-Forms ...... 412 7.3 Construction of WKB Solutions Attached to the Critical Points of Index 1 ...... 412 References ...... 414 Contents XV

The Mathematical Formalism of a Particle in a Magnetic Field M. M˘antoiu and R. Purice ...... 417 1 Introduction ...... 417 2 TheClassicalParticleinaMagneticField...... 418 2.1 Two Hamiltonian Formalisms ...... 418 2.2 Magnetic Translations ...... 421 3 TheQuantumPicture ...... 422 3.1 The Magnetic Moyal Product ...... 422 3.2 TheMagneticMoyalAlgebra...... 423 3.3 The Twisted Crossed Product ...... 424 3.4 Abstract Affiliation ...... 426 4 The Limit → 0 ...... 426 5 The Schr¨odinger Representation ...... 427 5.1 Representations of the Twisted Crossed Product ...... 427 5.2 Pseudodifferential Operators ...... 429 5.3 A New Justification: Functional Calculus ...... 429 5.4 Concrete Affiliation ...... 430 6 Applications to Spectral Analysis ...... 431 6.1 The Essential Spectrum ...... 431 6.2 A Non-Propagation Result ...... 432 References ...... 433 Fractal Weyl Law for Open Chaotic Maps S. Nonnenmacher ...... 435 1 Introduction ...... 435 1.1 Generalities on Resonances ...... 435 1.2 TrappedSets...... 436 1.3 FractalWeylLaw...... 437 1.4 OpenMaps...... 438 2 The Open Baker’s Map and Its Quantization ...... 439 2.1 Classical Closed Baker...... 439 2.2 OpeningtheClassicalMap...... 440 2.3 QuantumBaker’sMap ...... 441 2.4 Resonances of the Open Baker’s Map ...... 442 3 ASolvableToyModelfortheQuantumBaker ...... 446 3.1 Description of the Toy Model...... 446 3.2 Interpretation of CN as a Walsh-Quantized Baker ...... 446 3.3 Resonances of CN=3k ...... 447 References ...... 449 Spectral Shift Function for Magnetic Schr¨odinger Operators G. Raikov ...... 451 1 Introduction ...... 451 2 Auxiliary Results ...... 453 2.1 Notations and Preliminaries ...... 453 XVI Contents

2.2 A. Pushnitski’s Representation of the SSF ...... 453 3 MainResults ...... 455 3.1 Singularities of the SSF at the Landau Levels ...... 455 3.2 Strong Magnetic Field Asymptotics of the SSF ...... 460 3.3 High Energy Asymptotics of the SSF ...... 462 References ...... 463 Counting String/M Vacua S. Zelditch ...... 467 1 Introduction ...... 467 2 Type IIb Flux Compactifications of String/M Theory ...... 468 3 Critical Points and Hessians of Holomorphic Sections ...... 470 4 The Critical Point Problem ...... 471 5 Statement of Results ...... 473 6 Comparison to the Physics Literature ...... 474 7 SketchofProofs...... 475 8 Other Formulae for the Critical Point Density ...... 476 9 BlackHoleAttractors...... 480 References ...... 481 List of Contributors

W.K. Abou-Salem de Math´ematiques et Laboratoire Institute for Theoretical Painlev´e, 59655 Villeneuve d’Ascq Physics, ETH-H¨onggerberg Cedex, France CH-8093 Z¨urich, Switzerland [email protected] [email protected] lille1.fr

M. Aizenman O. Bokanowski Departments of Mathematics Laboratoire Jacques Louis Lions and Physics (Paris VI) Jadwin Hall Princeton University UFR Math´ematique B. P. 7012 P. O. Box 708, Princeton & Universit´e Paris VII New Jersey, 08544 Paris, Paris Cedex 05 USA France [email protected] A. Avila CNRS UMR 7599 M. B¨uttiker Laboratoire de Probabilit´es D´epartement de Physique Th´eorique et Mod`eles al´eatoires Universit´e de Gen`eve Universit´e Pierre CH-1211 Gen`eve 4, Switzerland et Marie Curie–Boite [email protected] courrier 188, 75252–Paris Cedex 05 France J.-M. Combes [email protected] D´epartement de Math´ematiques V. Betz Universit´e de Toulon-Var Institute for Biomathematics BP 132, 83957 La Garde C´edex and Biometry GSF France Forschungszentrum Postfach [email protected] 1129 D-85758 Oberschleißheim H.D. Cornean Germany Department of Mathematical [email protected] Sciences Aalborg University S. De Bi`evre Fredrik Bajers Vej 7G Universit´e des Sciences et 9220 Aalborg, Denmark Technologies de Lille, UFR [email protected] XVIII List of Contributors

J. Derezi´nski J. Fr¨ohlich Department of Mathematics Institute for Theoretical and Methods in Physics Physics Warsaw University Ho˙za 74 ETH-H¨onggerberg 00-682 Warsaw CH-8093 Z¨urich Poland Switzerland [email protected] [email protected]

A. Elgart F. Germinet Department of Mathematics D´epartement de Math´ematiques Stanford University Universit´e de Cergy-Pontoise Stanford, CA 94305-2125 Site de Saint-Martin USA 2 avenue Adolphe Chauvin [email protected] 95302 Cergy-Pontoise C´edex France L. Erd˝os [email protected] Institute of Mathematics University of Munich S.D. Glazek Theresienstr. Institute of Theoretical Physics 39 D-80333 Munich Warsaw University Germany 00-681 Hoza 69 [email protected]. Poland de [email protected] B. Helffer P. Exner D´epartement de Math´ematiques Department of Theoretical Physics Bˆat. 425, Universit´e Paris Nuclear Physics Institute Sud UMR CNRS 8628, F91405 Academy of Sciences ˇ Orsay Cedex 25068 Reˇz near Prague France Czechia [email protected] and Doppler Institute P.D. Hislop Czech Technical University Department of Mathematics Bˇrehov´a 7, 11519 University of Kentucky Prague, Czechia Lexington KY 40506-0027 [email protected] USA [email protected] A. Fedotov Department of Mathematical A. Jensen Physics Department of Mathematical St Petersburg State University 1 Sciences Ulianovskaja 198904 Aalborg University St Petersburg-Petrodvorets, Russia Fredrik Bajers Vej 7G mailto:[email protected] 9220 Aalborg, Denmark [email protected] [email protected] List of Contributors XIX

S. Jitomirskaya K.A. Meissner University of California Institute of Theoretical Physics Irvine Department of Mathematics Warsaw University 243 Multipurpose Science Ho˙za 69 00-681 Warsaw & Technology Building Poland Irvine, CA 92697-3875 [email protected] USA [email protected] M. Merkli Department of Mathematics and F. Klopp Statistics, McGill University LAGA, Institut Galil´ee 805 Sherbrooke W., Montreal U.M.R 7539 C.N.R.S QC Canada, H3A 2K6 Universit´e Paris-Nord and Avenue J.-B. Cl´ement Centre des Recherches F-93430 Villetaneuse, France Math´ematiques, Universit´ede [email protected] Montr´eal, Succursale D. Lenz centre-ville Montr´eal Fakult¨at f¨ur Mathematik QC Canada, H3C 3J7 Technische Universit¨at [email protected] 09107 Chemnitz, Germany d.lenz@@mathematik.tu- V. Moldoveanu chemnitz.de National Institute of Materials Physics, P.O. Box MG-7 E.H. Lieb Magurele, Romania Departments of Mathematics [email protected] and Physics Jadwin Hall Princeton University J.S. Møller P. O. Box 708, Princeton Aarhus University Department of New Jersey, 08544 Mathematical Sciences USA 8000 Aarhus C, Denmark J.L. L´opez [email protected] Departamento de Matem´atica Aplicada Facultad de Ciencias M. Moskalets Universidad de Granada Department of Metal Campus Fuentenueva and Semiconductor Physics 18071 Granada National Technical University Spain “Kharkiv Polytechnic Institute” 61002 Kharkiv, Ukraine [email protected] [email protected] M. M˘antoiu “Simion Stoilow” Institute of B. Nachtergaele Mathematics, Romanian Academy Department of Mathematics 21, Calea Grivitei Street University of California 010702-Bucharest, Sector 1 Davis One Shields Avenue Romania Davis, CA 95616-8366, USA [email protected] [email protected] XX List of Contributors

S. Nonnenmacher M. Salmhofer Service de Physique Th´eorique Theoretical Physics CEA/DSM/PhT, Unit´e de recherche University of associ´ee au CNRS, CEA/Saclay Leipzig Augustusplatz 10 91191 Gif-sur-Yvette D-04109 Leipzig France Germany [email protected] and Max–Planck Institute for J. Puig Mathematics, Inselstr. 22 Departament de Matem`atica D-04103 Leipzig Aplicada I Germany Universitat Polit`ecnica [email protected] de Catalunya. Av. Diagonal leipzig.de 647, 08028 Barcelona, Spain O.´ S´anchez J.V. Pul´e Departamento de Matem´atica Department of Mathematical Physics Aplicada, Facultad de Ciencias University College Universidad de Granada Dublin, Belfield, Dublin 4 Campus Fuentenueva Ireland 18071 Granada [email protected] Spain [email protected] R. Purice “Simion Stoilow” Institute of B. Schlein Mathematics, Romanian Academy Department of Mathematics 21, Calea Grivitei Street Stanford University 010702-Bucharest, Sector 1 Stanford, CA 94305, USA Romania [email protected] [email protected] H. Schulz-Baldes G. Raikov Mathematisches Institut Facultad de Ciencias Friedrich-Alexander-Universit¨at Universidad de Chile Erlangen-N¨urnberg Bismarckstr. 1 Las Palmeras 3425 D-91054, Erlangen Santiago, Chile [email protected] [email protected]

V. Rivasseau R. Seiringer Laboratoire de Physique Th´eorique Departments of Mathematics CNRS, UMR 8627 and Physics, Jadwin Hall Universit´e de Paris-Sud Princeton University 91405 Orsay P. O. Box 708, Princeton France New Jersey, 08544 [email protected] USA List of Contributors XXI

J. Soler S. Teufel Departamento de Matem´atica Mathematisches Institut Aplicada, Facultad de Ciencias Universit¨at T¨ubingen Universidad de Granada Auf der Morgenstelle 10 Campus Fuentenueva 72076 T¨ubingen 18071 Granada Germany Spain [email protected] [email protected] A.F. Verbeure J.P. Solovej Instituut voor Theoretische Department of Mathematics Fysika, Katholieke Universiteit University of Copenhagen Leuven Celestijnenlaan Universitetsparken 5 200D, 3001 Leuven, Belgium DK-2100 Copenhagen, Denmark [email protected]. and ac.be Institut f¨ur Theoretische Physik Universit¨at Wien H.-T. Yau Boltzmanngasse 5 Department of Mathematics A-1090 Vienna, Austria Stanford University, CA-94305, USA [email protected] [email protected] A. Soshnikov J. Yngvason University of California at Davis Institut f¨ur Theoretische Physik Department of Mathematics Universit¨at Wien Davis, CA 95616, USA Boltzmanngasse 5 [email protected] A-1090 Vienna, Austria H. Spohn and Zentrum Mathematik and Erwin Schr¨odinger Institute for Physik Department Mathematical Physics TU M¨unchen D-85747 Garching Boltzmanngasse 9 Boltzmannstr 3, Germany A-1090 Vienna, Austria [email protected] S. Starr V.A. Zagrebnov Department of Mathematics Universit´edela University of California M´editerran´ee and Centre de Los Angeles, Box 951555 Physique Th´eorique, Luminy-Case Los Angeles, CA 90095-1555 907, 13288 Marseille, Cedex 09 USA France [email protected] [email protected] P. Stollmann S. Zelditch Fakult¨at f¨ur Mathematik Department of Mathematics Technische Universit¨at Johns Hopkins University 09107 Chemnitz, Germany Baltimore MD 21218 [email protected] USA chemnitz.de [email protected]