Mathematical physics of quantum mechanics

Lecture Notes in Physics
Editorial Board
R. Beig, Wien, Austria
W. Beiglböck, Heidelberg, Germany
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I. Ojima, Kyoto, Japan
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J. Wess, München, Germany
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Joachim Asch Alain Joye (Eds.)

Mathematical Physics
of Quantum Mechanics
Selected and Refereed Lectures from QMath9

ABC

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Editors
Alain Joye
Institut Fourier
Université Grenoble 1
BP 74
38402 Saint-Martin-d’Hères Cedex
France
E-mail:

Joachim Asch
Université du Sud Toulon Var
Centre de physique théorique
Département de Mathématiques
BP 20132
F-83957 La Garde Cedex
France
E-mail:

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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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 meetings 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 initiated 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 engineering 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, Quantum dynamics, Quantum eld theory, Quantum kinetics, Random Schră
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 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. Effort 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
Erdă
os, Pavel Exner, Svetlana Jitomirskaya, Frederic 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,

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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, Institut Fourier, Universit´e Joseph Fourier.
Toulon
Grenoble
January 2006

Joachim Asch
Alain Joye

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Part I Quantum Dynamics and Spectral Theory
Solving the Ten Martini Problem

A. Avila and S. Jitomirskaya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Rough Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Analytic Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The Liouvillian Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Gaps for Rational Approximants . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Continuity of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 The Diophantine Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Localization and Reducibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 A Localization Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
6
8
9
9
10
10
11
12
12
14

Swimming Lessons for Microbots
Y. Avron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Landau-Zener Formulae from Adiabatic Transition Histories
V. Betz and S. Teufel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Exponentially Small Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The Hamiltonian in the Super-Adiabatic Representation . . . . . . . . . .
4 The Scattering Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

19
19
22
25
27
31

Scattering Theory of Dynamic Electrical Transport
M. Bă
uttiker and M. Moskalets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1 From an Internal Response to a Quantum Pump Effect . . . . . . . . . . 33
2 Quantum Coherent Pumping: A Simple Picture . . . . . . . . . . . . . . . . . 36

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VIII

Contents

3

Beyond the Frozen Scatterer Approximation:
Instantaneous Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
The Landauer-Bă
uttiker Formula
and Resonant Quantum Transport
H.D. Cornean, A. Jensen and V. Moldoveanu . . . . . . . . . . . . . . . . . . . . . .
1 The Landauer-Bă
uttiker Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Resonant Transport in a Quantum Dot . . . . . . . . . . . . . . . . . . . . . . . . .
3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
45
47
48
53

Point Interaction Polygons: An Isoperimetric Problem
P. Exner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Local Result in Geometric Terms . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 About the Global Maximizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Some Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55
55
56
58

61
62
64

Limit Cycles in Quantum Mechanics
S.D. Glazek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Definition of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Renormalization Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Limit Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Marginal and Irrelevant Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Tuning to a Cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Generic Properties of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65
65
67
69
71
73
74
75
76
76

Cantor Spectrum for Quasi-Periodic Schră
odinger Operators
J. Puig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 The Almost Mathieu Operator & the Ten Martini Problem . . . . . . .
1.1 The ids and the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Sketch of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Reducibility of Quasi-Periodic Cocycles . . . . . . . . . . . . . . . . . . . .
1.4 End of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Extension to Real Analytic Potentials . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Cantor Spectrum for Specific Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79
79
80
83
84
86
87
88
90

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Contents

IX

Part II Quantum Field Theory and Statistical Mechanics

Adiabatic Theorems and Reversible Isothermal Processes
W.K. Abou-Salem and J. Fră
ohlich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 A General “Adiabatic Theorem” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The “Isothermal Theorem” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 (Reversible) Isothermal Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95
95
97
99
101
104

Quantum Massless Field in 1+1 Dimensions
J. Derezi´
nski and K.A. Meissner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Poincar´e Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Changing the Compensating Functions . . . . . . . . . . . . . . . . . . . . . . . . .
5 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Fields in Position Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 The SL(2, R) × SL(2, R) Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Normal Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 Classical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10 Algebraic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Vertex Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

12 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107
107
108
111
112
113
115
116
117
118
120
122
123
125
126

Stability of Multi-Phase Equilibria
M. Merkli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Stability of a Single-Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Free Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Spontaneous Symmetry Breaking
and Multi-Phase Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Return to Equilibrium in Absence of a Condensate . . . . . . . . . .
1.4 Return to Equilibrium in Presence of a Condensate . . . . . . . . .
1.5 Spectral Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Stability of Multi-Phase Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Quantum Tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Non-Interacting System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Interacting System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Stability of the Quantum Tweezers, Main Results . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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129
129
129
133
135
135
136
137
138
141
146
147
148


X

Contents

Ordering of Energy Levels in Heisenberg Models
and Applications
B. Nachtergaele and S. Starr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The Temperley-Lieb Basis. Proof of Proposition 1 . . . . . . . . . . . . . . .
3.1 The Basis for Spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Basis for Higher Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 The Spin 1/2 SUq (2)-symmetric XXZ Chain . . . . . . . . . . . . . . .
4.2 Higher Order Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Diagonalization at Low Energy . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The Ground States of Fixed Magnetization
for the XXZ Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Aldous’ Conjecture
for the Symmetric Simple Exclusion Process . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149
149
152
158
158
160
163
163
165
165
165
166
167
169

Interacting Fermions in 2 Dimensions
V. Rivasseau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Fermi Liquids and Salmhofer’s Criterion . . . . . . . . . . . . . . . . . . . . . . . .
3 The Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 A Brief Review of Rigorous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Multiscale Analysis, Angular Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 One and Two Particle Irreducible Expansions . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

171
171
171
173
174
175
176
178

On the Essential Spectrum of the Translation Invariant
Nelson Model
J. Schach-Møller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 The Model and the Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 A Complex Function of Two Variables . . . . . . . . . . . . . . . . . . . . . . . . .
3 The Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A Riemannian Covers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179

179
182
189
194
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

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2 Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Proof of BEC for Small λ and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Absence of BEC and Mott Insulator Phase . . . . . . . . . . . . . . . . . . . . .
5 The Non-Interacting Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Long Time Behaviour to the Schră
odingerPoissonX
Systems
S
O. Bokanowski, J.L. L´
opez, O.

anchez and J. Soler . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 On the Derivation of the Slater Approach . . . . . . . . . . . . . . . . . . . . . .
3 Some Results Concerning Well Posedness
and Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Existence and Uniqueness of Physically Admissible Solutions .
3.2 Minimum of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Optimal Kinetic Energy Bounds . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Long-Time Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 On the General X α Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Towards the Quantum Brownian Motion
L. Erd˝
os, M. Salmhofer and H.-T. Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Statement of Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Sketch of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Renormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Expansion and the Stopping Rules . . . . . . . . . . . . . . . . . . . .
3.3 The L2 Norm of the Non-Repetitive Wavefunction . . . . . . . . . .
3.4 Sketch of the Proof of the Main Technical Theorem . . . . . . . . .
3.5 Point Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Computation of the Main Term
and Its Convergence to a Brownian Motion . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Bose-Einstein Condensation and Superradiance
J.V. Pul´e, A.F. Verbeure and V.A. Zagrebnov . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Solution of the Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 The Effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.2 The Pressure for Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Model 2 and Matter-Wave Grating . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XII

Contents

Derivation of the Gross-Pitaevskii Hierarchy
B. Schlein . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Sketch of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

279
279

286
290
292

Towards a Microscopic Derivation of the Phonon Boltzmann
Equation
H. Spohn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Microscopic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Kinetic Limit and Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . .
4 Feynman Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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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 . . . . . . . . . . . . . . . . . . . . . . .
1 The Edge Conductance
and General Invariance Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Regularizing the Edge Conductance
in Presence of Impurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A Time Averaged Regularization
for a Dynamically Localized System . . . . . . . . . . . . . . . . . . . . . . .

2.3 Regularization Under a Stronger
form of Dynamical Localization . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Localization for the Landau Operator
with a Half-Plane Random Potential . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 A Large Magnetic Field Regime . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A Large Disorder Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Equality of the Bulk and Edge Hall Conductances in 2D
A. Elgart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction and Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Proof of σB = σE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Convergence and Trace Class Properties . . . . . . . . . . . . . . . . . . .
2.3 Edge – Bulk Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 σE = σB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XIII

Generic Subsets in Spaces of Measures
and Singular Continuous Spectrum
D. Lenz and P. Stollmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Generic Subsets in Spaces of Measures . . . . . . . . . . . . . . . . . . . . . . . . .
3 Singular Continuity of Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Selfadjoint Operators and the Wonderland Theorem . . . . . . . . . . . . .
5 Operators Associated to Delone Sets . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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333
334
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341

Low Density Expansion for Lyapunov Exponents
H. Schulz-Baldes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Model and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Result on the Lyapunov Exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Result on the Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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344
346
347
349
350

Poisson Statistics for the Largest Eigenvalues
in Random Matrix Ensembles
A. Soshnikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Wigner Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Band Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Sample Covariance Random Matrices . . . . . . . . . . . . . . . . . . . . . .
1.4 Universality in Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . .
2 Wigner and Band Random Matrices
with Heavy Tails of Marginal Distributions . . . . . . . . . . . . . . . . . . . . .
3 Real Sample Covariance Matrices with Cauchy Entries . . . . . . . . . . .
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Part V Semiclassical Analysis and Quantum Chaos
Recent Results on Quantum Map Eigenstates
S. De Bi`evre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Perturbed CAT Maps: Classical Dynamics . . . . . . . . . . . . . . . . . . . . . .
3 Quantum Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 What is Known? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Perturbed Cat Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XIV

Contents

Level Repulsion and Spectral Type
for One-Dimensional Adiabatic Quasi-Periodic Schră
odinger
Operators
A. Fedotov and F. Klopp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 A Heuristic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Mathematical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 The Periodic Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 A “Geometric” Assumption
on the Energy Region Under Study . . . . . . . . . . . . . . . . . . . . . . .
2.3 The Definitions of the Phase Integrals
and the Tunelling Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Ergodic Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 A Coarse Description of the Location of the Spectrum in J . .
2.6 A Precise Description of the Spectrum . . . . . . . . . . . . . . . . . . . . .
2.7 The Model Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 When τ is of Order 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.9 Numerical Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Low Lying Eigenvalues of Witten Laplacians and
Metastability (After Helffer-Klein-Nier and Helffer-Nier)
B. Helffer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 Main Goals and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Saddle Points and Labelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 Rough Semi-Classical Analysis of Witten Laplacians
and Applications to Morse Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Witten Laplacians on p-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Morse Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Main Result in the Case of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 About the Proof in the Case of Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Witten Complex, Reduced Witten Complex . . . . . . . . . . . . . . . .
5.3 Singular Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 The Main Result in the Case with Boundary . . . . . . . . . . . . . . . . . . . .
7 About the Proof in the Case with Boundary . . . . . . . . . . . . . . . . . . . .
7.1 Define the Witten Complex and the Associate Laplacian . . . . .
7.2 Rough Localization of the Spectrum of this Laplacian
on 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Construction of WKB Solutions Attached
to the Critical Points of Index 1 . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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XV

The Mathematical Formalism of a Particle in a Magnetic Field
M. M˘
antoiu and R. Purice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Classical Particle in a Magnetic Field . . . . . . . . . . . . . . . . . . . . . .
2.1 Two Hamiltonian Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Magnetic Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3 The Quantum Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Magnetic Moyal Product . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The Magnetic Moyal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 The Twisted Crossed Product . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Abstract Affiliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 The Limit → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 The Schrăodinger Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Representations of the Twisted Crossed Product . . . . . . . . . . . .
5.2 Pseudodifferential Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 A New Justification: Functional Calculus . . . . . . . . . . . . . . . . . .
5.4 Concrete Affiliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Applications to Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 The Essential Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 A Non-Propagation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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417
418
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Fractal Weyl Law for Open Chaotic Maps
S. Nonnenmacher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Generalities on Resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Trapped Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Fractal Weyl Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Open Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 The Open Baker’s Map and Its Quantization . . . . . . . . . . . . . . . . . . .
2.1 Classical Closed Baker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Opening the Classical Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Quantum Baker’s Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Resonances of the Open Baker’s Map . . . . . . . . . . . . . . . . . . . . . .
3 A Solvable Toy Model for the Quantum Baker . . . . . . . . . . . . . . . . . .
3.1 Description of the Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Interpretation of CN as a Walsh-Quantized Baker . . . . . . . . . . .

3.3 Resonances of CN =3k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

435
435
435
436
437
438
439
439
440
441
442
446
446
446
447
449

Spectral Shift Function for Magnetic Schră
odinger Operators
G. Raikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Notations and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

451
451
453

453

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XVI

Contents

2.2 A. Pushnitski’s Representation of the SSF . . . . . . . . . . . . . . . . .
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Singularities of the SSF at the Landau Levels . . . . . . . . . . . . . .
3.2 Strong Magnetic Field Asymptotics of the SSF . . . . . . . . . . . . .
3.3 High Energy Asymptotics of the SSF . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

453
455
455
460
462
463

Counting String/M Vacua
S. Zelditch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Type IIb Flux Compactifications of String/M Theory . . . . . . . . . . . .
3 Critical Points and Hessians of Holomorphic Sections . . . . . . . . . . . .
4 The Critical Point Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Statement of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Comparison to the Physics Literature . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Sketch of Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Other Formulae for the Critical Point Density . . . . . . . . . . . . . . . . . .
9 Black Hole Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467
467
468
470
471
473
474
475
476
480
481

3

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List of Contributors

W.K. Abou-Salem
Institute for Theoretical
Physics, ETH-Hă
onggerberg

CH-8093 Ză
urich, Switzerland


de Mathematiques et Laboratoire
Painlev´e, 59655 Villeneuve d’Ascq
Cedex, France


M. Aizenman
Departments of Mathematics
and Physics
Jadwin Hall Princeton University
P. O. Box 708, Princeton
New Jersey, 08544
USA

O. Bokanowski
Laboratoire Jacques Louis Lions
(Paris VI)
UFR Math´ematique B. P. 7012
& Universit´e Paris VII
Paris, Paris Cedex 05
France


A. Avila
CNRS UMR 7599
Laboratoire de Probabilit´es
et Mod`eles al´eatoires

Universit´e Pierre
et Marie Curie–Boite
courrier 188, 75252–Paris Cedex 05
France

V. Betz
Institute for Biomathematics
and Biometry GSF
Forschungszentrum Postfach
1129 D-85758
Oberschleißheim
Germany

S. De Bi`
evre
Universit´e des Sciences et
Technologies de Lille, UFR

M. Bă
uttiker
Departement de Physique Theorique
Universite de Gen`eve
CH-1211 Gen`eve 4, Switzerland

J.-M. Combes
D´epartement de Math´ematiques
Universit´e de Toulon-Var
BP 132, 83957 La Garde C´edex
France

H.D. Cornean
Department of Mathematical
Sciences
Aalborg University
Fredrik Bajers Vej 7G
9220 Aalborg, Denmark


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XVIII List of Contributors

J. Derezi´
nski
Department of Mathematics
and Methods in Physics
Warsaw University Hoza 74
00-682 Warsaw
Poland


J. Fră
ohlich
Institute for Theoretical
Physics
ETH-Hăonggerberg
CH-8093 Ză
urich
Switzerland

A. Elgart
Department of Mathematics
Stanford University
Stanford, CA 94305-2125
USA


F. Germinet
D´epartement de Math´ematiques
Universit´e de Cergy-Pontoise
Site de Saint-Martin
2 avenue Adolphe Chauvin
95302 Cergy-Pontoise C´edex
France


L. Erd˝
os
Institute of Mathematics
University of Munich
Theresienstr.
39 D-80333 Munich
Germany

de
P. Exner
Department of Theoretical Physics
Nuclear Physics Institute

Academy of Sciences
ˇ z near Prague
25068 Reˇ
Czechia
and
Doppler Institute
Czech Technical University
Bˇrehov´
a 7, 11519
Prague, Czechia

A. Fedotov
Department of Mathematical
Physics
St Petersburg State University 1
Ulianovskaja 198904
St Petersburg-Petrodvorets, Russia
mailto:


S.D. Glazek
Institute of Theoretical Physics
Warsaw University
00-681 Hoza 69
Poland

B. Helffer
D´epartement de Math´ematiques
Bˆat. 425, Universit´e Paris
Sud UMR CNRS 8628, F91405

Orsay Cedex
France

P.D. Hislop
Department of Mathematics
University of Kentucky
Lexington KY 40506-0027
USA

A. Jensen
Department of Mathematical
Sciences
Aalborg University
Fredrik Bajers Vej 7G
9220 Aalborg, Denmark


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List of Contributors

S. Jitomirskaya
University of California
Irvine Department of Mathematics
243 Multipurpose Science
& Technology Building
Irvine, CA 92697-3875
USA

F. Klopp
LAGA, Institut Galil´ee
U.M.R 7539 C.N.R.S
Universit´e Paris-Nord
Avenue J.-B. Clement
F-93430 Villetaneuse, France

D. Lenz
Fakultă
at fă
ur Mathematik
Technische Universităat
09107 Chemnitz, Germany
d.lenz@@mathematik.tuchemnitz.de
E.H. Lieb
Departments of Mathematics
and Physics
Jadwin Hall Princeton University
P. O. Box 708, Princeton
New Jersey, 08544
USA
J.L. L´
opez
Departamento de Matem´atica
Aplicada Facultad de Ciencias
Universidad de Granada
Campus Fuentenueva
18071 Granada
Spain

M. M˘
antoiu
“Simion Stoilow” Institute of
Mathematics, Romanian Academy
21, Calea Grivitei Street
010702-Bucharest, Sector 1
Romania


XIX

K.A. Meissner
Institute of Theoretical Physics
Warsaw University
Ho˙za 69 00-681 Warsaw
Poland

M. Merkli
Department of Mathematics and
Statistics, McGill University
805 Sherbrooke W., Montreal
QC Canada, H3A 2K6
and
Centre des Recherches
Math´ematiques, Universit´e de
Montr´eal, Succursale
centre-ville Montr´eal
QC Canada, H3C 3J7

V. Moldoveanu

National Institute of Materials
Physics, P.O. Box MG-7
Magurele, Romania

J.S. Møller
Aarhus University Department of
Mathematical Sciences
8000 Aarhus C, Denmark

M. Moskalets
Department of Metal
and Semiconductor Physics
National Technical University
“Kharkiv Polytechnic Institute”
61002 Kharkiv, Ukraine

B. Nachtergaele
Department of Mathematics
University of California
Davis One Shields Avenue
Davis, CA 95616-8366, USA


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XX

List of Contributors

S. Nonnenmacher
Service de Physique Th´eorique
CEA/DSM/PhT, Unit´e de recherche
associ´ee au CNRS, CEA/Saclay
91191 Gif-sur-Yvette
France

J. Puig
Departament de Matem`atica
Aplicada I
Universitat Polit`ecnica
de Catalunya. Av. Diagonal
647, 08028 Barcelona, Spain

M. Salmhofer
Theoretical Physics
University of
Leipzig Augustusplatz 10
D-04109 Leipzig
Germany
and
Max–Planck Institute for
Mathematics, Inselstr. 22
D-04103 Leipzig
Germany


J.V. Pul´
e
Department of Mathematical

Physics
University College
Dublin, Belfield, Dublin 4
Ireland


´ S´
O.
anchez
Departamento de Matem´atica
Aplicada, Facultad de Ciencias
Universidad de Granada
Campus Fuentenueva
18071 Granada
Spain


R. Purice
“Simion Stoilow” Institute of
Mathematics, Romanian Academy
21, Calea Grivitei Street
010702-Bucharest, Sector 1
Romania


B. Schlein
Department of Mathematics
Stanford University
Stanford, CA 94305, USA

H. Schulz-Baldes
Mathematisches Institut
Friedrich-Alexander-Universită
at
Erlangen-Nă
urnberg Bismarckstr. 1
D-91054, Erlangen


G. Raikov
Facultad de Ciencias
Universidad de Chile
Las Palmeras 3425
Santiago, Chile

V. Rivasseau
Laboratoire de Physique Th´eorique
CNRS, UMR 8627
Universit´e de Paris-Sud
91405 Orsay
France


R. Seiringer
Departments of Mathematics
and Physics, Jadwin Hall
Princeton University
P. O. Box 708, Princeton
New Jersey, 08544
USA

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List of Contributors

J. Soler
Departamento de Matem´atica
Aplicada, Facultad de Ciencias
Universidad de Granada
Campus Fuentenueva
18071 Granada
Spain

J.P. Solovej
Department of Mathematics
University of Copenhagen
Universitetsparken 5
DK-2100 Copenhagen, Denmark
and
Institut fă
ur Theoretische Physik
Universităat Wien
Boltzmanngasse 5
A-1090 Vienna, Austria

A. Soshnikov
University of California at Davis
Department of Mathematics
Davis, CA 95616, USA

H. Spohn
Zentrum Mathematik and
Physik Department
TU Mă
unchen D-85747 Garching
Boltzmannstr 3, Germany


XXI

S. Teufel
Mathematisches Institut
Universităat Tă
ubingen
Auf der Morgenstelle 10
72076 Tă
ubingen
Germany

A.F. Verbeure
Instituut voor Theoretische
Fysika, Katholieke Universiteit
Leuven Celestijnenlaan
200D, 3001 Leuven, Belgium

ac.be
H.-T. Yau
Department of Mathematics
Stanford University, CA-94305, USA

J. Yngvason
Institut fă
ur Theoretische Physik
Universităat Wien
Boltzmanngasse 5
A-1090 Vienna, Austria
and
Erwin Schră
odinger Institute for
Mathematical Physics
Boltzmanngasse 9
A-1090 Vienna, Austria

S. Starr
Department of Mathematics
University of California
Los Angeles, Box 951555
Los Angeles, CA 90095-1555
USA


V.A. Zagrebnov
Universit´e de la
M´editerran´ee and Centre de
Physique Theorique, Luminy-Case
907, 13288 Marseille, Cedex 09
France

P. Stollmann
Fakultă
at fă
ur Mathematik
Technische Universităat
09107 Chemnitz, Germany


S. Zelditch
Department of Mathematics
Johns Hopkins University
Baltimore MD 21218
USA


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Introduction

QMath9 gave a particular importance to summarize the state of the art of
the field in a perspective to transmit knowledge to younger scientists. The
main contributors to the field were gathered in order to communicate results,
open questions and motivate new research by the confrontation of different
view points. The edition of this book follows this spirit; the main effort of
the authors and editors is to help finding an access to the variety of themes
and the sometimes very sophisticated literature in Mathematical Physics.
The contributions of this book are organized in five topical groups: Quantum Dynamics and Spectral Theory, Quantum Field Theory and Statistical
Mechanics, Quantum Kinetics and Bose-Einstein Condensation, Disordered
Systems and Random Operators, Semiclassical Analysis and Quantum Chaos.

This splitting is admittedly somewhat arbitrary, since there are overlaps between the topics and the frontiers between the chosen groups may be quite
fuzzy. Moreover, there are close connections between the tools and techniques
used in the analysis of quite different physical phenomena. An introduction
to each theme is given. This plan is intended as a readers guide rather than
as an attempt to put contributions into well defined categories.

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Solving the Ten Martini Problem
Artur Avila1 and Svetlana Jitomirskaya2
1

2

CNRS UMR 7599, Laboratoire de Probabilit´es et Mod`eles al´eatoires
Universit´e Pierre et Marie Curie–Boite courrier 188
75252–Paris Cedex 05, France

University of California, Irvine, California
The research of S.J. was partially supported by NSF under DMS-0300974


Abstract. We discuss the recent proof of Cantor spectrum for the almost Mathieu
operator for all conjectured values of the parameters.

1 Introduction
The almost Mathieu operator (a.k.a. the Harper operator or the Hofstadter
model) is a Schră
odinger operator on 2 (Z),

(H,, u)n = un+1 + un−1 + 2λ cos 2π(θ + nα)un ,
where λ, α, θ ∈ R are parameters (the coupling, the frequency and the phase).
This model first appeared in the work of Peierls [21]. It arises in physics
literature as related, in two different ways, to a two-dimensional electron
subject to a perpendicular magnetic field [15, 23]. It plays a central role in
the Thouless et al theory of the integer quantum Hall effect [27]. The value
of λ of most interest from the physics point of view is λ = 1. It is called the
critical value as it separates two different behaviors as far as the nature of
the spectrum is concerned.
If α = pq is rational, it is well known that the spectrum consists of the
union of q intervals possibly touching at endpoints. In the case of irrational
α the spectrum (which then does not depend on θ) has been conjectured for
a long time to be a Cantor set for all λ = 0 [7]. To prove this conjecture has
been dubbed the Ten Martini problem by Barry Simon, after an offer of Kac
in 1981, see Problem 4 in [25].
In 1984 Bellissard and Simon [8] proved the conjecture for generic pairs of
(λ, α). In 1987 Sinai [26] proved Cantor spectrum for a.e. α in the perturbative regime: for λ = λ(α) sufficiently large or small. In 1989 Heler-Sjăostrand
proved Cantor spectrum for the critical value = 1 and an explicitly defined generic set of α [16]. Most developments in the 90s were related to the
following observation. For α = pq the spectrum of Hλ,α,θ can have at most
q − 1 gaps. It turns out that all these gaps are open, except for the middle
one for even q [11, 20]. Choi, Eliott, and Yui obtained in fact an exponential
A. Avila and S. Jitomirskaya: Solving the Ten Martini Problem, Lect. Notes Phys. 690, 5–16
(2006)
c Springer-Verlag Berlin Heidelberg 2006
www.springerlink.com

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6

A. Avila and S. Jitomirskaya

lower bound on the size of the individual gaps from which they deduced Cantor spectrum for Liouville (exponentially well approximated by the rationals)
α [11]. In 1994 Last, using certain estimates of Avron, van Mouche and Simon [6], proved zero measure Cantor spectrum for a.e. α (for an explicit set
that intersects with but does not contain the set in [16]) and λ = 1 [18]. Just
extending this result to the case of all (rather than a.e.) α was considered a
big challenge (see Problem 5 in [25]).
A major breakthrough came recently with an influx of ideas coming from
dynamical systems. Puig, using Aubry duality [1] and localization for θ = 0
and λ > 1 [13], proved Cantor spectrum for Diophantine α and any noncritical
λ [22]. At about the same time, Avila and Krikorian proved zero measure
Cantor spectrum for λ = 1 and α satisfying a certain Diophantine condition,
therefore extending the result of Last to all irrational α [3]. The solution of
the Ten Martini problem as originally stated was finally given in [2]:
Main Theorem [2]. The spectrum of the almost Mathieu operator is a Cantor set for all irrational α and for all λ = 0.
Here we present the broad lines of the argument of [2]. For a much more
detailed account of the history as well as of the physics background and
related developments see a recent review [19].
While the ten martini problem was solved, a stronger version of it, dubbed
by B. Simon the Dry Ten Martini problem is still open. The problem is to
prove that all the gaps prescribed by the gap labelling theorem are open. This
fact would be quite meaningful for the QHE related applications [4]. Dry ten
martini was only established for Liouville α [2, 11] and for Diophantine α in
the perturbative regime [22], using a theorem of Eliasson [12].
1.1 Rough Strategy
The history of the Ten Martini problem we described shows the existence of
a number of different approaches, applicable on different parameter ranges.
Denote by Σλ,α the union over θ ∈ R of the spectrum of Hλ,α,θ (recall
that the spectrum is actually θ-independent if α ∈ R \ Q). Due to the obvious

symmetry Σλ,α = −Σ−λ,α , we may assume that λ > 0. Aubry duality gives
a much more interesting symmetry, which implies that Σλ,α = λΣλ−1 ,α . The
critical coupling λ = 1 separates two very distinct regimes. The transition
at λ = 1 can be clearly seen by consideration of the Lyapunov exponent
L(E) = Lλ,α (E), for which we have the following statement.
Theorem 1. [9] Let λ > 0, α ∈ R \ Q. For every E ∈ Σλ,α , Lλ,α (E) =
max{ln λ, 0}.
With respect to the frequency α, one can broadly distinguish two approaches, applicable depending on whether α is well approximated by rationals or not (the Liouville and the Diophantine cases):

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