The Mathematics of the Bose Gas and its Condensation

1. Front Matter

   Pages i-viii

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2. Introduction

   Pages 1-8

3. The Dilute Bose Gas in 3D

   Pages 9-25

4. The Dilute Bose Gas in 2D

   Pages 27-32

5. Generalized Poincaré Inequalities

   Pages 33-37

6. Bose-Einstein Condensation and Superfluidity for Homogeneous Gases

   Pages 39-46

7. Gross-Pitaevskii Equation for Trapped Bosons

   Pages 47-62

8. Bose-Einstein Condensation and Superfluidity for Dilute Trapped Gases

   Pages 63-70

9. One-Dimensional Behavior of Dilute Bose Gases in Traps

   Pages 71-86

10. Two-Dimensional Behavior in Disc-Shaped Traps

    Pages 87-107

11. The Charged Bose Gas, the One- and Two-Component Cases

    Pages 109-129

12. Bose-Einstein Quantum Phase Transition in an Optical Lattice Model

    Pages 131-148

13. Back Matter

    Pages 149-203

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The mathematical study of the Bose gas goes back to the ?rst quarter of the twentieth century, with the invention of quantum mechanics. The name refers to the Indian physicist S.N. Bose who realized in 1924 that the statistics governing photons(essentiallyinventedbyMaxPlanckin1900)isdetermined(usingmodern terminology) by restricting the physical Hilbert space to be the symmetric tensor product of single photon states. Shortly afterwards, Einstein applied this idea to massive particles, such as a gas of atoms, and discovered the phenomenon that we now call Bose-Einstein condensation. At that time this was viewed as a mathematical curiosity with little experimental interest, however. The peculiar properties of liquid Helium (?rst lique?ed by Kammerlingh Onnes in 1908) were eventually viewed as an experimental realization of Bose- Einstein statistics applied to Helium atoms. The unresolved mathematical pr- lem was that the atoms in liquid Helium are far from the kind of non-interacting particles envisaged in Einstein’s theory, and the question that needed to be - solved was whether Bose-Einstein condensation really takes place in a strongly interacting system — or even in a weakly interacting system. That question is still with us, three quarters of a century later! The ?rst systematic and semi-rigorous mathematical treatment of the pr- lem was due to Bogoliubov in 1947, but that theory, while intuitively appealing and undoubtedly correct in many aspects, has major gaps and some ?aws. The 1950’s and 1960’s brought a renewed ?urry of interest in the question, but while theoreticalintuitionbene?tedhugelyfromthisactivitythemathematicalstructure did not signi?cantly improve.

• Bose Gas
• Bose-Einstein Condensation
• Helium-Atom-Streuung
• Mathematical Physics
• PAS
• PED
• Phase Transition

"The presentation provides significant insight into a large part of the current issues of interest in the physics of Bose systems and especially into the "kitchen" of several relevant mathematical techniques. As such, it is highly recommended to both advanced researchers and students preparing to work in this field."

(Mathematical Reviews)

• Departments of Mathematics and Physics, Princeton University, Jadwin Hall, Princeton, USA

  Elliott H. Lieb

• Department of Mathematics, University of Copenhagen, Copenhagen, Denmark

  Jan Philip Solovej

• Department of Physics, Princeton University, Jadwin Hall, Princeton, USA

  Robert Seiringer

• Institut für Theoretische Physik, Universität Wien, Wien, Austria

  Jakob Yngvason

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