Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate

Abstract

Consider a system of $N$ bosons in three dimensions interacting via a repulsive short range pair potential $N^2V(N(x_i-x_j))$, where $\mathbf{x}=(x_1, \ldots, x_N)$ denotes the positions of the particles. Let $H_N$ denote the Hamiltonian of the system and let $\psi_{N,t}$ be the solution to the Schrödinger equation. Suppose that the initial data $\psi_{N,0}$ satisfies the energy condition \[ \langle \psi_{N,0}, H_N^k \psi_{N,0} \rangle \leq C^k N^k \; \] for $k=1,2,\ldots\; $. We also assume that the $k$-particle density matrices of the initial state are asymptotically factorized as $N\to\infty$. We prove that the $k$-particle density matrices of $\psi_{N,t}$ are also asymptotically factorized and the one particle orbital wave function solves the Gross-Pitaevskii equation, a cubic nonlinear Schrödinger equation with the coupling constant given by the scattering length of the potential $V$. We also prove the same conclusion if the energy condition holds only for $k=1$ but the factorization of $\psi_{N,0}$ is assumed in a stronger sense.

  • [ABGT] R. Adami, C. Bardos, F. Golse, and A. Teta, "Towards a rigorous derivation of the cubic NLSE in dimension one," Asymptot. Anal., vol. 40, iss. 2, pp. 93-108, 2004.
    @article {ABGT, MRKEY = {2104130},
      AUTHOR = {Adami, Riccardo and Bardos, Claude and Golse, Fran{ç}ois and Teta, Alessandro},
      TITLE = {Towards a rigorous derivation of the cubic {NLSE} in dimension one},
      JOURNAL = {Asymptot. Anal.},
      FJOURNAL = {Asymptotic Analysis},
      VOLUME = {40},
      YEAR = {2004},
      NUMBER = {2},
      PAGES = {93--108},
      ISSN = {0921-7134},
      CODEN = {ASANEZ},
      MRCLASS = {35Q55 (82B10)},
      MRNUMBER = {2005g:35263},
      MRREVIEWER = {Arthur H. Vartanian},
      ZBLNUMBER = {1069.35082},
      }
  • [AGT] Go to document R. Adami, F. Golse, and A. Teta, "Rigorous derivation of the cubic NLS in dimension one," J. Stat. Phys., vol. 127, iss. 6, pp. 1193-1220, 2007.
    @article {AGT, MRKEY = {2331036},
      AUTHOR = {Adami, Riccardo and Golse, Fran{ç}ois and Teta, Alessandro},
      TITLE = {Rigorous derivation of the cubic {NLS} in dimension one},
      JOURNAL = {J. Stat. Phys.},
      FJOURNAL = {Journal of Statistical Physics},
      VOLUME = {127},
      YEAR = {2007},
      NUMBER = {6},
      PAGES = {1193--1220},
      ISSN = {0022-4715},
      MRCLASS = {82C10 (35Q55)},
      MRNUMBER = {2008i:82055},
      MRREVIEWER = {Benjamin Schlein},
      DOI = {10.1007/s10955-006-9271-z},
      ZBLNUMBER = {1118.81021},
      }
  • [BGM] C. Bardos, F. Golse, and N. J. Mauser, "Weak coupling limit of the $N$-particle Schrödinger equation," Methods Appl. Anal., vol. 7, iss. 2, pp. 275-293, 2000.
    @article {BGM, MRKEY = {1869286},
      AUTHOR = {Bardos, Claude and Golse, Fran{ç}ois and Mauser, Norbert J.},
      TITLE = {Weak coupling limit of the {$N$}-particle {S}chrödinger equation},
      JOURNAL = {Methods Appl. Anal.},
      FJOURNAL = {Methods and Applications of Analysis},
      VOLUME = {7},
      YEAR = {2000},
      NUMBER = {2},
      PAGES = {275--293},
      ISSN = {1073-2772},
      MRCLASS = {81U10 (35A10 35Q40)},
      MRNUMBER = {2003c:81215},
      ZBLNUMBER={1003.81027},
      }
  • [Dav] E. B. Davies, "The functional calculus," J. London Math. Soc., vol. 52, iss. 1, pp. 166-176, 1995.
    @article {Dav, MRKEY = {1345723},
      AUTHOR = {Davies, E. B.},
      TITLE = {The functional calculus},
      JOURNAL = {J. London Math. Soc.},
      FJOURNAL = {Journal of the London Mathematical Society. Second Series},
      VOLUME = {52},
      YEAR = {1995},
      NUMBER = {1},
      PAGES = {166--176},
      ISSN = {0024-6107},
      CODEN = {JLMSAK},
      MRCLASS = {47A60 (47D03 47F05 47N50)},
      MRNUMBER = {96e:47017},
      MRREVIEWER = {El Maati Ouhabaz},
      ZBLNUMBER = {0858.47012},
      }
  • [Dy] F. J. Dyson, "Ground-state energy of a hard-sphere gas," Phys. Rev., vol. 106, pp. 20-26, 1957.
    @article{Dy,
      author={Dyson, F. J.},
      TITLE={Ground-state energy of a hard-sphere gas},
      JOURNAL={Phys. Rev.},
      VOLUME={106},
      YEAR={1957},
      PAGES={20--26},
      }
  • [EESY] Go to document A. Elgart, L. ErdHos, B. Schlein, and H. Yau, "Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons," Arch. Ration. Mech. Anal., vol. 179, iss. 2, pp. 265-283, 2006.
    @article {EESY, MRKEY = {2209131},
      AUTHOR = {Elgart, Alexander and Erd{ő}s, L{á}szl{ó} and Schlein, Benjamin and Yau, Horng-Tzer},
      TITLE = {Gross-{P}itaevskii equation as the mean field limit of weakly coupled bosons},
      JOURNAL = {Arch. Ration. Mech. Anal.},
      FJOURNAL = {Archive for Rational Mechanics and Analysis},
      VOLUME = {179},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {265--283},
      ISSN = {0003-9527},
      MRCLASS = {81V70 (81Q05)},
      MRNUMBER = {2007b:81310},
      MRREVIEWER = {Barbara Prinari},
      DOI = {10.1007/s00205-005-0388-z},
      ZBLNUMBER={1086.81035},
      }
  • [ES] Go to document A. Elgart and B. Schlein, "Mean field dynamics of boson stars," Comm. Pure Appl. Math., vol. 60, iss. 4, pp. 500-545, 2007.
    @article {ES, MRKEY = {2290709},
      AUTHOR = {Elgart, Alexander and Schlein, Benjamin},
      TITLE = {Mean field dynamics of boson stars},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {60},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {500--545},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {85A05 (82C10)},
      MRNUMBER = {2009a:85001},
      DOI = {10.1002/cpa.20134},
      ZBLNUMBER = {1113.81032},
      }
  • [ESY] Go to document L. ErdHos, B. Schlein, and H. Yau, "Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate," Comm. Pure Appl. Math., vol. 59, iss. 12, pp. 1659-1741, 2006.
    @article {ESY, MRKEY = {2257859},
      AUTHOR = {Erd{ő}s, L{á}szl{ó} and Schlein, Benjamin and Yau, Horng-Tzer},
      TITLE = {Derivation of the {G}ross-{P}itaevskii hierarchy for the dynamics of {B}ose-{E}instein condensate},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {59},
      YEAR = {2006},
      NUMBER = {12},
      PAGES = {1659--1741},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {82C10 (35Q55 47N55)},
      MRNUMBER = {2007k:82070},
      MRREVIEWER = {Barbara Prinari},
      DOI = {10.1002/cpa.20123},
      }
  • [ESY2] Go to document L. ErdHos, B. Schlein, and H. Yau, "Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems," Invent. Math., vol. 167, iss. 3, pp. 515-614, 2007.
    @article {ESY2, MRKEY = {2276262},
      AUTHOR = {Erd{ő}s, L{á}szl{ó} and Schlein, Benjamin and Yau, Horng-Tzer},
      TITLE = {Derivation of the cubic non-linear {S}chrödinger equation from quantum dynamics of many-body systems},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {167},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {515--614},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {81V70 (35Q55 47N55 81T18 82C10)},
      MRNUMBER = {2007m:81258},
      MRREVIEWER = {Barbara Prinari},
      DOI = {10.1007/s00222-006-0022-1},
      ZBLNUMBER={1122.82018},
      }
  • [EY] L. ErdHos and H. Yau, "Derivation of the nonlinear Schrödinger equation from a many body Coulomb system," Adv. Theor. Math. Phys., vol. 5, iss. 6, pp. 1169-1205, 2001.
    @article {EY, MRKEY = {1926667},
      AUTHOR = {Erd{ő}s, L{á}szl{ó} and Yau, Horng-Tzer},
      TITLE = {Derivation of the nonlinear {S}chrödinger equation from a many body {C}oulomb system},
      JOURNAL = {Adv. Theor. Math. Phys.},
      FJOURNAL = {Advances in Theoretical and Mathematical Physics},
      VOLUME = {5},
      YEAR = {2001},
      NUMBER = {6},
      PAGES = {1169--1205},
      ISSN = {1095-0761},
      MRCLASS = {82C10 (35Q55 45F15 81Q05)},
      MRNUMBER = {2004c:82075},
      MRREVIEWER = {H. Hogreve},
      ZBLNUMBER={1014.81063},
      }
  • [GV] Go to document J. Ginibre and G. Velo, "The classical field limit of scattering theory for nonrelativistic many-boson systems. I," Comm. Math. Phys., vol. 66, iss. 1, pp. 37-76, 1979.
    @article {GV, MRKEY = {530915},
      AUTHOR = {Ginibre, J. and Velo, G.},
      TITLE = {The classical field limit of scattering theory for nonrelativistic many-boson systems. {I}},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {66},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {37--76},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {81F20 (35P25)},
      MRNUMBER = {80h:81069},
      MRREVIEWER = {Jean-Claude Guillot},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103904940},
      ZBLNUMBER={0443.35067},
      }
  • [G1] E. P. Gross, "Structure of a quantized vortex in boson systems," Nuovo Cimento, vol. 20, pp. 454-477, 1961.
    @article {G1, MRKEY = {0128907},
      AUTHOR = {Gross, E. P.},
      TITLE = {Structure of a quantized vortex in boson systems},
      JOURNAL = {Nuovo Cimento},
      VOLUME = {20},
      YEAR = {1961},
      PAGES = {454--477},
      MRCLASS = {81.99},
      MRNUMBER = {23 \#B1944},
      MRREVIEWER = {G. Baym},
      ZBLNUMBER = {0100.42403},
      }
  • [G2] E. P. Gross, "Hydrodynamics of a superfluid condensate," J. Math. Phys., vol. 4, pp. 195-207, 1963.
    @article{G2,
      author={Gross, E. P.},
      TITLE={Hydrodynamics of a superfluid condensate},
      JOURNAL={J. Math. Phys.},
      VOLUME={4},
      YEAR={1963},
      PAGES={195--207},
      }
  • [H] Go to document K. Hepp, "The classical limit for quantum mechanical correlation functions," Comm. Math. Phys., vol. 35, pp. 265-277, 1974.
    @article {H, MRKEY = {0332046},
      AUTHOR = {Hepp, Klaus},
      TITLE = {The classical limit for quantum mechanical correlation functions},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {35},
      YEAR = {1974},
      PAGES = {265--277},
      ISSN = {0010-3616},
      MRCLASS = {81.47},
      MRNUMBER = {48 \#10373},
      MRREVIEWER = {H. Wakita},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103859623},
      }
  • [Hu] K. Huang, Statistical Mechanics, Second ed., New York: John Wiley & Sons, 1987.
    @book {Hu, MRKEY = {1042093},
      AUTHOR = {Huang, Kerson},
      TITLE = {Statistical Mechanics},
      EDITION = {Second},
      PUBLISHER = {John Wiley \& Sons},
      ADDRESS = {New York},
      YEAR = {1987},
      PAGES = {xiv+493},
      ISBN = {0-471-81518-7},
      MRCLASS = {82-01 (80-01)},
      MRNUMBER = {91d:82001},
      MRREVIEWER = {David E. Miller},
      ZBLNUMBER = {1041.82500},
      }
  • [LS] E. H. Lieb and R. Seiringer, "Proof of Bose-Einstein condensation for dilute trapped gases," Phys. Rev. Lett., vol. 88, p. 170409, 2002.
    @article{LS,
      author={Lieb, Elliott H. and Seiringer, Robert},
      TITLE={Proof of {B}ose-{E}instein condensation for dilute trapped gases},
      JOURNAL={Phys. Rev. Lett.},
      VOLUME={88},
      YEAR={2002},
      PAGES={170409},
      }
  • [LSSY] E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, The Mathematics of the Bose Gas and its Condensation, Basel: Birkhäuser, 2005.
    @book {LSSY, MRKEY = {2143817},
      AUTHOR = {Lieb, Elliott H. and Seiringer, Robert and Solovej, Jan Philip and Yngvason, Jakob},
      TITLE = {The Mathematics of the {B}ose Gas and its Condensation},
      SERIES = {Oberwolfach Seminars},
      NUMBER = {34},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2005},
      PAGES = {viii+203},
      ISBN = {978-3-7643-7336-8; 3-7643-7336-9},
      MRCLASS = {82-02 (82C10)},
      MRNUMBER = {2006e:82001},
      MRREVIEWER = {Nicolae Angelescu},
      ZBLNUMBER = {1104.82012},
      }
  • [LSY1] E. H. Lieb, R. Seiringer, and J. Yngvason, "Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional," Phys. Rev. A, vol. 61, p. 043602, 2000.
    @article{LSY1,
      author={Lieb, Elliott H. and Seiringer, Robert and Yngvason, Jakob},
      TITLE={Bosons in a trap: a rigorous derivation of the {G}ross-{P}itaevskii energy functional},
      JOURNAL={Phys. Rev. {\rm A}},
      VOLUME={61},
      YEAR={2000},
      PAGES={043602},
      }
  • [LY1] E. H. Lieb and J. Yngvason, "Ground state energy of the low density Bose gas," Phys. Rev. Lett., vol. 80, pp. 2504-2507, 1998.
    @article{LY1,
      author={Lieb, Elliott H. and Yngvason, Jakob},
      TITLE={Ground state energy of the low density {B}ose gas},
      JOURNAL={Phys. Rev. Lett.},
      VOLUME={80},
      YEAR={1998},
      PAGES={2504--2507},
      }
  • [P] L. P. Pitaevskii, "Vortex lines in an imperfect Bose gas," Soviet Phys. JETP, vol. 13, pp. 451-454, 1961.
    @article{P,
      author={Pitaevskii, L. P.},
      TITLE={Vortex lines in an imperfect {B}ose gas},
      JOURNAL={Soviet Phys. {\rm JETP}},
      VOLUME={13},
      YEAR={1961},
      PAGES={451--454},
      }
  • [Ru] W. Rudin, Functional Analysis, New York: McGraw-Hill Book Co., 1973.
    @book {Ru, MRKEY = {0365062},
      AUTHOR = {Rudin, Walter},
      TITLE = {Functional Analysis},
      NOTE = {McGraw-Hill Series in Higher Mathematics},
      PUBLISHER = {McGraw-Hill Book Co.},
      ADDRESS = {New York},
      YEAR = {1973},
      PAGES = {xiii+397},
      MRCLASS = {46-01},
      MRNUMBER = {51 \#1315},
      MRREVIEWER = {F. Smithies},
      ZBLNUMBER = {0253.46001},
      }
  • [Sim] B. Simon, Trace Ideals and their Applications, Second ed., Providence, RI: Amer. Math. Soc., 2005.
    @book {Sim, MRKEY = {2154153},
      AUTHOR = {Simon, Barry},
      TITLE = {Trace Ideals and their Applications},
      SERIES = {Mathematical Surveys Monogr.},
      NUMBER = {120},
      EDITION = {Second},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2005},
      PAGES = {viii+150},
      ISBN = {0-8218-3581-5},
      MRCLASS = {47L20 (47A40 47A55 47B10 47B36 47E05 81Q15 81U99)},
      MRNUMBER = {2006f:47086},
      MRREVIEWER = {Pavel B. Kurasov},
      ZBLNUMBER = {1074.47001},
      }
  • [Sp] Go to document H. Spohn, "Kinetic equations from Hamiltonian dynamics: Markovian limits," Rev. Modern Phys., vol. 52, iss. 3, pp. 569-615, 1980.
    @article {Sp, MRKEY = {578142},
      AUTHOR = {Spohn, Herbert},
      TITLE = {Kinetic equations from {H}amiltonian dynamics: {M}arkovian limits},
      JOURNAL = {Rev. Modern Phys.},
      FJOURNAL = {Reviews of Modern Physics},
      VOLUME = {52},
      YEAR = {1980},
      NUMBER = {3},
      PAGES = {569--615},
      ISSN = {0034-6861},
      CODEN = {RMPHAT},
      MRCLASS = {82A05 (60K35 82A15 82A40)},
      MRNUMBER = {81e:82010},
      DOI = {10.1103/RevModPhys.52.569},
      }

Authors

László Erdős

Institute of Mathematics, University of Munich, Theresienstr. 39, D-80333 Munich, Germany

Benjamin Schlein

DPMMS, Room E1.01, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom

Horng-Tzer Yau

Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, United States