Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential
Authors:
László Erdos, Benjamin Schlein and Horng-Tzer Yau
Journal:
J. Amer. Math. Soc. 22 (2009), 1099-1156
MSC (2000):
Primary 82C10, 35Q55
DOI:
https://doi.org/10.1090/S0894-0347-09-00635-3
Published electronically:
May 6, 2009
MathSciNet review:
2525781
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Consider a system of bosons in three dimensions interacting via a repulsive short range pair potential
, where
denotes the positions of the particles. Let
denote the Hamiltonian of the system and let
be the solution to the Schrödinger equation. Suppose that the initial data
satisfies the energy condition







- 1. Riccardo Adami, Claude Bardos, François Golse, and Alessandro Teta, Towards a rigorous derivation of the cubic NLSE in dimension one, Asymptot. Anal. 40 (2004), no. 2, 93–108. MR 2104130
- 2. Riccardo Adami, François Golse, and Alessandro Teta, Rigorous derivation of the cubic NLS in dimension one, J. Stat. Phys. 127 (2007), no. 6, 1193–1220. MR 2331036, https://doi.org/10.1007/s10955-006-9271-z
- 3. Anderson, M.H.; Ensher, J.R.; Matthews, M.R.; Wieman, C.E.; Cornell, E.A.: Observation of Bose-Einstein condensation in a dilute atomic vapor. Science (269), 198 (1995).
- 4. Claude Bardos, François Golse, and Norbert J. Mauser, Weak coupling limit of the 𝑁-particle Schrödinger equation, Methods Appl. Anal. 7 (2000), no. 2, 275–293. Cathleen Morawetz: a great mathematician. MR 1869286, https://doi.org/10.4310/MAA.2000.v7.n2.a2
- 5. J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999), no. 1, 145–171. MR 1626257, https://doi.org/10.1090/S0894-0347-99-00283-0
- 6. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ℝ³, Ann. of Math. (2) 167 (2008), no. 3, 767–865. MR 2415387, https://doi.org/10.4007/annals.2008.167.767
- 7. Davis, K.B.; Mewes, M.-O.; Andrews, M.R.; van Druten, N.J.; Durfee, D.S.; Kurn, D.M.; Ketterle, W.: Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett. (75), 3969 (1995).
- 8. Alexander Elgart, László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal. 179 (2006), no. 2, 265–283. MR 2209131, https://doi.org/10.1007/s00205-005-0388-z
- 9. Alexander Elgart and Benjamin Schlein, Mean field dynamics of boson stars, Comm. Pure Appl. Math. 60 (2007), no. 4, 500–545. MR 2290709, https://doi.org/10.1002/cpa.20134
- 10. László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Comm. Pure Appl. Math. 59 (2006), no. 12, 1659–1741. MR 2257859, https://doi.org/10.1002/cpa.20123
- 11. László Erdős, Benjamin Schlein, and Horng-Tzer Yau, Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems, Invent. Math. 167 (2007), no. 3, 515–614. MR 2276262, https://doi.org/10.1007/s00222-006-0022-1
- 12. Erdős, L.; Schlein, B.; Yau, H.-T.: Derivation of the Gross-Pitaevskii Equation for the Dynamics of Bose-Einstein Condensate. Preprint arXiv:math-ph/0606017. To appear in Ann. of Math.
- 13. László Erdős and Horng-Tzer Yau, Derivation of the nonlinear Schrödinger equation from a many body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), no. 6, 1169–1205. MR 1926667, https://doi.org/10.4310/ATMP.2001.v5.n6.a6
- 14.
J.
Ginibre and G.
Velo, The classical field limit of scattering theory for
nonrelativistic many-boson systems. I, Comm. Math. Phys.
66 (1979), no. 1, 37–76. MR
530915
J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. II, Comm. Math. Phys. 68 (1979), no. 1, 45–68. MR 539736 - 15. Jean Ginibre and Giorgio Velo, On a class of nonlinear Schrödinger equations with nonlocal interaction, Math. Z. 170 (1980), no. 2, 109–136. MR 562582, https://doi.org/10.1007/BF01214768
- 16. J. Ginibre and G. Velo, Scattering theory in the energy space for a class of nonlinear Schrödinger equations, J. Math. Pures Appl. (9) 64 (1985), no. 4, 363–401. MR 839728
- 17. Klaus Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys. 35 (1974), 265–277. MR 332046
- 18. Lieb, E.H.; Seiringer, R.: Proof of Bose-Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88 (2002), 170409-1-4.
- 19. Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, and Jakob Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, vol. 34, Birkhäuser Verlag, Basel, 2005. MR 2143817
- 20. Lieb, E.H.; Seiringer, R.; Yngvason, J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev A 61 (2000), 043602.
- 21. Tosio Kato, On nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 113–129 (English, with French summary). MR 877998
- 22. Sergiu Klainerman and Matei Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Comm. Math. Phys. 279 (2008), no. 1, 169–185. MR 2377632, https://doi.org/10.1007/s00220-008-0426-4
- 23. Michelangeli, A.: Equivalent definitions of asymptotic 100% BEC. Il Nuovo Cimento B 123 (2008), no. 2, 181-192.
- 24. Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
- 25. Rodnianski, I.; Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Preprint arXiv:math-ph/0711.3087. To appear in Comm. Math. Phys.
- 26. Walter Rudin, Functional analysis, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR 0365062
- 27. Herbert Spohn, Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys. 52 (1980), no. 3, 569–615. MR 578142, https://doi.org/10.1103/RevModPhys.52.569
- 28. Walter A. Strauss, Nonlinear scattering theory at low energy, J. Functional Analysis 41 (1981), no. 1, 110–133. MR 614228, https://doi.org/10.1016/0022-1236(81)90063-X
- 29. Kenji Yajima, The 𝑊^{𝑘,𝑝}-continuity of wave operators for Schrödinger operators, J. Math. Soc. Japan 47 (1995), no. 3, 551–581. MR 1331331, https://doi.org/10.2969/jmsj/04730551
- 30. Kenji Yajima, The 𝑊^{𝑘,𝑝}-continuity of wave operators for Schrödinger operators, Proc. Japan Acad. Ser. A Math. Sci. 69 (1993), no. 4, 94–98. MR 1222831
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Additional Information
László Erdos
Affiliation:
Institute of Mathematics, University of Munich, Theresienstrasse 39, D-80333 Munich, Germany
Benjamin Schlein
Affiliation:
DPMMS, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom
Horng-Tzer Yau
Affiliation:
Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
DOI:
https://doi.org/10.1090/S0894-0347-09-00635-3
Received by editor(s):
April 15, 2008
Published electronically:
May 6, 2009
Additional Notes:
The first author was partially supported by SFB/TR12 Project from DFG
The second author was supported by a Kovalevskaja Award from the Humboldt Foundation
The third author was partially supported by NSF grants DMS-0602038, 0757425, and 0804279
Article copyright:
© Copyright 2009
American Mathematical Society