The Mathematics of the Bose Gas and Its Condensation

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