Skip to main content
SpringerLink
Log in

Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We consider the dynamics of N bosons in 1D. We assume that the pair interaction is attractive and given by \({N^{\beta-1}V(N^{\beta}.) where }\) where \({\int V \leqslant 0}\). We develop new techniques in treating the N-body Hamiltonian so that we overcome the difficulties generated by the attractive interaction and establish new energy estimates. We also prove the optimal 1D collapsing estimate which reduces the regularity requirement in the uniqueness argument by half a derivative. We derive rigorously the 1D focusing cubic NLS with a quadratic trap as the \({N \rightarrow \infty}\) limit of the N-body dynamic and hence justify the mean-field limit and prove the propagation of chaos for the focusing quantum many-body system.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adami R., Golse F., Teta A.: Rigorous derivation of the cubic NLS in dimension one. J. Stat. Phys. 127, 1194–1220 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Ammari Z., Nier F.: Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states. J. Math. Pures. Appl. 95, 585–626 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ammari Z., Nier F.: Mean field propagation of infinite dimensional Wigner measures with a singular two-body interaction potential. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 16, 155–220 (2015)

    MathSciNet  MATH  Google Scholar 

  4. 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, 198–201 (1995)

    Article  ADS  Google Scholar 

  5. Beckner W.: Multilinear embedding—convolution estimates on smooth submanifolds. Proc. Am. Math. Soc. 142, 1217–1228 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Benedikter N., Oliveira G., Schlein B.: Quantitative derivation of the Gross–Pitaevskii equation. Commun. Pure Appl. Math. 68, 1399–1482 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Carles R.: Nonlinear Schrödinger equation with time dependent potential. Commun. Math. Sci. 9, 937–964 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen L., Lee J.O., Schlein B.: Rate of convergence towards hartree dynamics. J. Stat. Phys. 144, 872–903 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. Chen T., Pavlović N.: On the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies. Discrete Contin. Dyn. Syst. 27, 715–739 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chen T., Pavlović N.: The quintic NLS as the mean field limit of a boson gas with three-body interactions. J. Funct. Anal. 260, 959–997 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chen T., Pavlović N.: Derivation of the cubic NLS and Gross–Pitaevskii hierarchy from manybody dynamics in d = 3 based on spacetime norms. Ann. H. Poincaré 15, 543–588 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chen T., Pavlović N., Tzirakis N.: Energy conservation and blowup of solutions for focusing Gross–Pitaevskii hierarchies. Ann. I. H. Poincaré 27, 1271–1290 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  13. Chen T., Pavlović N., Tzirakis N.: Multilinear Morawetz identities for the Gross–Pitaevskii hierarchy. Contemp. Math. 581, 39–62 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Chen X.: Classical proofs of Kato type smoothing estimates for the Schrödinger equation with quadratic potential in \({\mathbb{R}^{n+1}}\) with application. Differ. Integr. Equ. 24, 209–230 (2011)

    MATH  Google Scholar 

  15. Chen X.: Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions. Arch. Rational Mech. Anal. 203, 455–497 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Chen X.: Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps. J. Math. Pures Appl. 98, 450–478 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen X.: On the rigorous derivation of the 3D cubic nonlinear Schrödinger equation with a quadratic trap. Arch. Rational Mech. Anal. 210, 365–408 (2013)

    Article  ADS  MATH  Google Scholar 

  18. Chen X., Holmer J.: On the rigorous derivation of the 2D cubic nonlinear Schrödinger equation from 3D quantum many-body dynamics. Arch. Rational Mech. Anal. 210, 909–954 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Chen, X., Holmer, J.: On the Klainerman–Machedon conjecture of the quantum BBGKY hierarchy with self-interaction. J. Eur. Math. Soc. (2016, to appear). arXiv:1303.5385

  20. Chen, X., Holmer, J.: Correlation structures, many-body scattering processes and the derivation of the Gross–Pitaevskii hierarchy. Int. Math. Res. Notices (2015). doi:10.1093/imrn/rnv228

  21. Cornish S.L., Claussen N.R., Roberts J.L., Cornell E.A., Wieman C.E.: Stable \({^{85}}\) Rb Bose–Einstein condensates with widely turnable interactions. Phys. Rev. Lett. 85, 1795–1798 (2000)

    Article  ADS  Google Scholar 

  22. 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, 3969–3973 (1995)

    Article  ADS  Google Scholar 

  23. Elgart A., Erdös L., Schlein B., Yau H.T.: Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Rational Mech. Anal. 179, 265–283 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  24. Erdös L., Yau H.T.: Derivation of the non-linear Schrödinger equation from a many-body coulomb system. Adv. Theor. Math. Phys. 5, 1169–1205 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  25. Erdös L., Schlein B., Yau H.T.: Derivation of the Gross–Pitaevskii Hierarchy for the dynamics of Bose–Einstein condensate. Commun. Pure Appl. Math. 59, 1659–1741 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Erdös L., Schlein B., Yau H.T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167, 515–614 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. Erdös L., Schlein B., Yau H.T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Am. Math. Soc. 22, 1099–1156 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  28. Erdös L., Schlein B., Yau H.T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. Math. 172, 291–370 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Fröhlich J., Knowles A., Schwarz S.: On the mean-field limit of bosons with coulomb two-body interaction. Commun. Math. Phys. 288, 1023–1059 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Gressman P., Sohinger V., Staffilani G.: On the uniqueness of solutions to the periodic 3D Gross–Pitaevskii hierarchy. J. Funct. Anal. 266, 4705–4764 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Grillakis M.G., Margetis D.: A priori estimates for many-body Hamiltonian evolution of interacting boson system. J. Hyperb. Differ. Equ. 5, 857–883 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  32. Grillakis M.G., Machedon M.: Pair excitations and the mean field approximation of interacting Bosons, I. Commun. Math. Phys. 324, 601–636 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Grillakis M.G., Machedon M., Margetis D.: Second order corrections to mean field evolution for weakly interacting bosons. I. Commun. Math. Phys. 294, 273–301 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Grillakis M.G., Machedon M., Margetis D.: Second order corrections to mean field evolution for weakly interacting bosons. II. Adv. Math. 228, 1788–1815 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hayashi N., Naumkin P.: Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations. Am. J. Math. 120, 369–389 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  36. Ketterle W., van Druten N.J.: Evaporative cooling of trapped atoms. Adv. At. Mol. Opt. Phys. 37, 181–236 (1996)

    Article  ADS  Google Scholar 

  37. Khaykovich L., Schreck F., Ferrari G., Bourdel T., Cubizolles J., Carr L.D., Castin Y., Salomon C.: Formation of a matter-wave bright soliton. Science 296, 1290–1293 (2002)

    Article  ADS  Google Scholar 

  38. Kirkpatrick K., Schlein B., Staffilani G.: Derivation of the two dimensional nonlinear Schrödinger equation from many body quantum dynamics. Am. J. Math. 133, 91–130 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  39. Klainerman S., Machedon M.: On the uniqueness of solutions to the Gross–Pitaevskii hierarchy. Commun. Math. Phys. 279, 169–185 (2008)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  40. Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence. Commum. Math. Phys. 298, 101–138 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. Lieb E.H., Seiringer R., Solovej J.P., Yngvason J.: The Mathematics of the Bose Gas and Its Condensation. Birkhaüser, Basel (2005)

    MATH  Google Scholar 

  42. Medley P., Weld D.M., Miyake H., Pritchard D.E., Ketterle W.: Spin gradient demagnetization cooling of ultracold atoms. Phys. Rev. Lett. 106, 195301 (2011)

    Article  ADS  Google Scholar 

  43. Michelangeli A., Schlein B.: Dynamical collapse of boson stars. Commum. Math. Phys. 311, 645–687 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. Pickl P.: Derivation of the time dependent Gross–Pitaevskii equation without positivity condition on the interaction. J. Stat. Phys. 140, 76–89 (2010)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  45. Pickl P.: A simple derivation of mean field limits for quantum systems. Lett. Math. Phys. 97, 151–164 (2011)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  46. Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291, 31–61 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. Simon, B.: Trace Ideals and Their Applications, 2nd edn, Mathematical Surveys Monogr., Vol. 120. Amer. Math. Soc., Providence, 2005

  48. Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52, 569–615 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  49. Stamper-Kurn D.M., Andrews M.R., Chikkatur A.P., Inouye S., Miesner H.-J., Stenger J., Ketterle W.: Optical confinement of a Bose–Einstein condensate. Phys. Rev. Lett. 80, 2027–2030 (1998)

    Article  ADS  Google Scholar 

  50. Strecker K.E., Partridge G.B., Truscott A.G., Hulet R.G.: Formation and propagation of matter-wave soliton trains. Nature 417, 150–153 (2002)

    Article  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rochester, Hylan Building, Rochester, NY, 14627, USA

    Xuwen Chen

  2. Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI, 02912, USA

    Justin Holmer

Authors
  1. Xuwen Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Justin Holmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuwen Chen.

Additional information

Communicated by F. Otto

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Holmer, J. Focusing Quantum Many-body Dynamics: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation. Arch Rational Mech Anal 221, 631–676 (2016). https://doi.org/10.1007/s00205-016-0970-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-016-0970-6

Keywords

Search

Navigation