Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
1 Introduction 1
1.1 The Ideal Bose Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Concept of Bose-Einstein Condensation . . . . . . . . . . . . . 4
1.3 Overview and Outline . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 The Dilute Bose Gas in 3D 9
2.1 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 The Dilute Bose Gas in 2D 27
4 Generalized Poincar´e Inequalities 33
5 Bose-Einstein Condensation and Superfluidity for Homogeneous Gases 39
5.1 Bose-Einstein Condensation . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6 Gross-Pitaevskii Equation for Trapped Bosons 47
6.1 Three Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2 Two Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7 Bose-Einstein Condensation and Superfluidity for Dilute Trapped Gases 63
8 One-Dimensional Behavior of Dilute Bose Gases in Traps 71
8.1 Discussion of the Results . . . . . . . . . . . . . . . . . . . . . . . . 75
8.2 The 1D Limit of 3D GP Theory . . . . . . . . . . . . . . . . . . . . 78
8.3 Outline of Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
vi Contents
9 Two-Dimensional Behavior in Disc-Shaped Traps 87
9.1 The 2D Limit of 3D GP Theory . . . . . . . . . . . . . . . . . . . . 92
9.2 Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.3 Scattering Length. . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
9.4 Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10 The Charged Bose Gas, the One- and Two-Component Cases 109
10.1 The One-Component Gas . . . . . . . . . . . . . . . . . . . . . . . 109
10.2 The Two-Component Gas . . . . . . . . . . . . . . . . . . . . . . . 112
10.3 The Bogoliubov Approximation . . . . . . . . . . . . . . . . . . . . 113
10.4 The Rigorous Lower Bounds . . . . . . . . . . . . . . . . . . . . . . 116
10.5 The Rigorous Upper Bounds . . . . . . . . . . . . . . . . . . . . . 124
11 Bose-Einstein Quantum Phase Transition in an Optical Lattice Model 131
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.2 Reflection Positivity . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.3 Proof of BEC for Small λ and T . . . . . . . . . . . . . . . . . . . 137
11.4 Absence of BEC andMott Insulator Phase . . . . . . . . . . . . . 142
11.5 The Non-Interacting Gas . . . . . . . . . . . . . . . . . . . . . . . 147
11.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
A Elements of Bogoliubov Theory 149
B An Exactly Soluble Model 165
C Definition and Properties of Scattering Length 171
D c-Number Substitutions and Gauge Symmetry Breaking 177
Bibliography 187
Index 201
