Derivation of the gross-pitaevskii equation for the dynamics of bose-einstein condensate

Annals of Mathematics

Volumn 172, Issue 1, 2010, Pages 291-370

  Erd̃O, László ^a   Schlein, Benjamin ^b   Yau, Horng Tzer ^c  

^a UNIVERSITY OF MUNICH   (Germany)

^b UNIVERSITY OF CAMBRIDGE   (United Kingdom)

^c HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 77956460797     PISSN: 0003486X     EISSN: None     Source Type: Journal    
DOI: 10.4007/annals.2010.172.291     Document Type: Article

References (23)

1. R. ADAMI, C. BARDOS, F. GOLSE, and A. TETA, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal. 40 (2004), 93-108. MR 2005g:35263 Zbl 1069.35082
2. R. ADAMI, F. GOLSE, and A. TETA, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), 1193-1220. MR 2008i:82055 Zbl 1118.81021
3. C. BARDOS, F. GOLSE, and N. J. MAUSER, Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), 275-293. MR 2003c:81215 Zbl 1003.81027
4. E. B. DAVIES, The functional calculus, J. London Math. Soc. 52 (1995), 166-176. MR96e:47017 Zbl 0858.47012
5. F. J. DYSON, Ground-state energy of a hard-sphere gas, Phys. Rev. 106 (1957), 20-26.
6. A. ELGART, L. ERD̃OS, B. SCHLEIN, and H.-T. YAU, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), 265-283. MR 2007b:81310 Zbl 1086.81035
7. A. ELGART and B. SCHLEIN, Mean field dynamics of boson stars, Comm. Pure Appl. Math. [60] (2007), 500-545. MR 2009a:85001 Zbl 1113.81032
8. L. ERD̃OS, B. SCHLEIN, and H.-T. YAU, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math. 59 (2006), 1659-1741. MR 2007k:82070
9. L. ERD̃OS, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), 515-614. MR 2007m:81258 Zbl 1122.82018
10. L. ERD̃O S and H.-T. YAU, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), 1169-1205. MR 2004c:82075 Zbl 1014.81063
11. J. GINIBRE and G. VELO, The classical field limit of scattering theory for nonrelativistic manyboson systems. I, Comm. Math. Phys. 66 (1979), 37-76. MR 80h:81069 Zbl 0443.35067
12. E. P. GROSS, Structure of a quantized vortex in boson systems, Nuovo Cimento 20 (1961), 454-477. MR 23 #B1944 Zbl 0100.42403
13. E. P. GROSS, Hydrodynamics of a superfluid condensate, J. Math. Phys. 4 (1963), 195-207.
14. K. HEPP, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. [35] (1974), 265-277. MR 48 #10373
15. K. HUANG, Statistical Mechanics, second ed., John Wiley & Sons, New York, 1987. MR 91d: [82001] Zbl 1041.82500
16. E. H. LIEB and R. SEIRINGER, Proof of Bose-Einstein condensation for dilute trapped gases, Phys. Rev. Lett. 88 (2002), 170409.
17. E. H. LIEB, R. SEIRINGER, J. P. SOLOVEJ, and J. YNGVASON, The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars 34, Birkhäuser, Basel, 2005. MR 2006e:82001 Zbl 1104.82012
18. E. H. LIEB, R. SEIRINGER, and J. YNGVASON, Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional, Phys. Rev. A 61 (2000), 043602.
19. E. H. LIEB and J. YNGVASON, Ground state energy of the low density Bose gas, Phys. Rev. Lett. 80 (1998), 2504-2507.
20. L. P. PITAEVSKII, Vortex lines in an imperfect Bose gas, Soviet Phys. JETP 13 (1961), 451-
21. L. P. PITAEVSKII, W. RUDIN, Functional Analysis, McGraw-Hill Book Co., New York, 1973, McGraw-Hill Series in Higher Mathematics. MR 51 #1315 Zbl 0253.46001
22. B. SIMON, Trace Ideals and their Applications, second ed., Mathematical Surveys Monogr. [120,] Amer. Math. Soc., Providence, RI, 2005. MR 2006f:47086 Zbl 1074.47001
23. H. SPOHN, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), 569-615. MR 81e:82010