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Archive for Rational Mechanics and Analysis

, Volume 220, Issue 3, pp 1119–1158 | Cite as

The Gross–Pitaevskii Hierarchy on General Rectangular Tori

  • Sebastian HerrEmail author
  • Vedran Sohinger
Article

Abstract

In this work, we study the Gross–Pitaevskii hierarchy on general—rational and irrational—rectangular tori of dimensions two and three. This is a system of infinitely many linear partial differential equations which arises in the rigorous derivation of the nonlinear Schrödinger equation. We prove a conditional uniqueness result for the hierarchy. In two dimensions, this result allows us to obtain a rigorous derivation of the defocusing cubic nonlinear Schrödinger equation from the dynamics of many-body quantum systems. On irrational tori, this question was posed as an open problem in the previous work of Kirkpatrick, Schlein, and Staffilani.

Keywords

Density Matrice Strichartz Estimate Rigorous Derivation Pitaevskii Equation Hartree Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ammari Z., Nier F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. H. Poincaré 9, 1503–1574 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Anderson M.H., Ensher J.R., Matthews M.R., Wieman C.E., Cornell E.A.: Observations of Bose–Einstein condensation in a dilute atomic vapor. Science 269, 198–201 (1995)ADSCrossRefGoogle Scholar
  4. 4.
    Bardos C., Erdős L., Golse F., Mauser N., Yau H.-T.: Derivation of the Schrödinger–Poisson equation from the quantum N-body problem. C.R. Math. Acad. Sci. Paris 334(6), 515–520 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bardos C., Golse F., Mauser N.: Weak coupling limit of the N-particle Schrödinger equation. Methods Appl. Anal. 7, 275–293 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beckner, W.: Convolution estimates and the Gross–Pitaevskii hierarchy (2011) (preprint). arXiv:1111.3857
  7. 7.
    Beckner W.: Multilinear embedding estimates for the fractional Laplacian. Math. Res. Lett. 19(1), 175–189 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bose S.N.: Plancks Gesetz und Lichtquantenhypothese. Zeitschrift für Physik 26, 178 (1924)ADSCrossRefzbMATHGoogle Scholar
  9. 9.
    Bourgain J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrödinger equations. Geom. Funct. Anal. 3, 107–156 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bourgain, J.: On Strichartz’s inequalities and the nonlinear Schrödinger equation on irrational tori. Mathematical Aspects of Nonlinear Dispersive Equations. Annals of Mathematics Studies, Vol. 163. Princeton University Press, Princeton, pp. 1–20, 2007Google Scholar
  11. 11.
    Bourgain J., Demeter C.: The proof of the l 2 decoupling conjecture. Ann. Math. (2) 182(1), 351–389 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bourgain, J., Demeter, C., l p decouplings for hypersurfaces with nonzero Gaussian curvature (2014) (preprint). arXiv:1407.0291
  13. 13.
    Bourgain, J., Demeter, C.: Decouplings for curves and hypersurfaces with nonzero Gaussian curvature (2014) (preprint). arXiv:1409.1634
  14. 14.
    Burq N., Gérard P., Tzvetkov N.: Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces. Invent. Math. 159(1), 187–223 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Catoire F., Wang W.-M.: Bounds on Sobolev norms for the defocusing nonlinear Schrödinger equation on general flat tori. Commun. Pure Appl. Anal. 9(2), 483–491 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen T., Hainzl C., Pavlović N., Seiringer R.: Unconditional uniqueness for the cubic Gross–Pitaevskii hierarchy via quantum de Finetti. Commun. Pure Appl. Math. 68(10), 1845–1884 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Chen T., Hainzl C., Pavlović N., Seiringer R.: On the well-posedness and scattering for the Gross–Pitaevskii hierarchy via quantum de Finetti. Lett. Math. Phys. 104(7), 871–891 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen T., Pavlović N.: On the Cauchy problem for focusing and defocusing Gross–Pitaevskii hiearchies. Discret. Contin. Dyn. Syst. 27(2), 715–739 (2010)CrossRefzbMATHGoogle Scholar
  19. 19.
    Chen T., Pavlović N.: Recent results on the Cauchy problem for focusing and defocusing Gross–Pitaevskii hierarchies. Math. Model. Nat. Phenom. 5(4), 54–72 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen T., Pavlović N.: The quintic NLS as the mean field limit of a Boson gas with three-body interactions. J. Funct. Anal. 260(4), 959–997 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen T., Pavlović N.: A new proof of existence of solutions for focusing and defocusing Gross–Pitaevskii hierarchies. Proc. Amer. Math. Soc. 141(1), 279–293 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chen T., Pavlović N.: Higher order energy conservation and global well-posedness for Gross–Pitaevskii hierarchies. Commun. Partial Differ. Equ. 39(9), 1597–1634 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chen T., Pavlović N.: Derivation of the cubic NLS and Gross–Pitaevskii hierarchy from many-body dynamics in d =  2, 3 based on spacetime norms. Ann. Henri Poincaré 15(3), 543–588 (2014)ADSCrossRefzbMATHGoogle Scholar
  24. 24.
    Chen T., Pavlović N., Tzirakis N.: Energy conservation and blowup of solutions for focusing and defocusing Gross–Pitaevskii hierarchies. Ann. Inst. H. Poincaré Anal. Non Linéaire. 27(5), 1271–1290 (2010)ADSCrossRefzbMATHGoogle Scholar
  25. 25.
    Chen, T., Pavlović, N., Tzirakis, N., Multilinear Morawetz identities for the Gross–Pitaevskii hierarchy. Recent Advances in Harmonic Analysis and Partial Differential Equations. Contemporary Mathematics, Vol. 581. American Mathematical Society, Providence, pp. 39–62, 2012Google Scholar
  26. 26.
    Chen T., Taliaferro K.: Positive semidefiniteness and global well-posedness of solutions to the Gross–Pitaevskii hierarchy. Commun. Partial Differ. Equ. 39(9), 1658–1693 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Chen, X.: Collapsing estimates and the rigorous derivation of the 2d cubic nonlinear Schrödinger equation with anisotropic switchable quadratic traps. J. Math. Pures Appl. (9) 98(4), 450–478 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Chen, X., Holmer, J.: On the Klainerman–Machedon conjecture of the quantum BBGKY hierarchy with self-interaction. J. Eur. Math. Soc. (JEMS) (2013) (to appear, preprint). arXiv:1303.5385
  29. 29.
    Chen, X., Holmer, J.: Correlation structures, many-body scattering processes and the derivation of the Gross–Pitaevskii hierarchy (2014) (preprint). arXiv:1409.1425
  30. 30.
    Chen X., Smith P.: On the unconditional uniqueness of solutions to the infinite radial Chern–Simons–Schrödinger hierarchy. Anal. PDE 7(7), 1683–1712 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    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(22), 3969–3973 (1995)ADSCrossRefGoogle Scholar
  32. 32.
    Demirbas, S.: Local well-posedness for 2-D Schrödinger equation on irrational tori and bounds on Sobolev norms (2013) (preprint). arXiv:1307.0051
  33. 33.
    Einstein, A.: Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften 1: 3. (1925)Google Scholar
  34. 34.
    Elgart A., Erdős L., Schlein B., Yau H.-T.: Gross–Pitaevskii equation as the mean field limit of weakly coupled bosons. Arch. Ration. Mech. Anal. 179, 265–283 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Erdős L., Schlein B., Yau H.-T.: Derivation of the Gross–Pitaevskii hierarchy for the dynamics of Bose–Einstein condensate. Commun. Pure Appl. Math. 59(12), 1659–1741 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Erdős L., Schlein B., Yau H.-T.: Derivation of the cubic non-linear Schrödinger equation from quantum dynamics of many-body systems. Invent. Math. 167(3), 515–614 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation. Phys. Rev. Lett. 98(4), 040404 (2007)ADSCrossRefGoogle Scholar
  38. 38.
    Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross–Pitaevskii equation with a large interaction potential. J. Amer. Math. Soc. 22(4), 1099–1156 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Ann. Math. (2) 172(1), 291–370 (2010)Google Scholar
  40. 40.
    Erdős L., Yau H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Fröhlich J., Knowles A., Schwarz S.: On the mean-field limit of bosons with Coulomb two-body interaction. Commun. Math. Phys. 288(3), 1023–1059 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Fröhlich J., Tsai T.-P., Yau H.-T.: On the point-particle (Newtonian) limit of the non-linear Hartree equation. Commun. Math. Phys. 225(2), 223–274 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ginibre J., Velo G.: The classical field limit of scattering theory for nonrelativistic many-boson systems I. Commun. Math. Phys. 66(1), 37–76 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Ginibre J., Velo G.: The classical field limit of scattering theory for nonrelativistic many-boson systems II. Commun. Math. Phys. 68(1), 45–68 (1979)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Gressman P., Sohinger V., Staffilani G.: On the uniqueness of solutions to the 3D periodic Gross–Pitaevskii hierarchy. J. Funct. Anal. 266(7), 4705–4764 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Gross E.: Structure of a quantized vortex in boson systems. Nuovo Cimento 20, 454–466 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Guo Z., Oh T., Wang Y.: Strichartz estimates for Schrödinger equations on irrational tori. Proc. Lond. Math. Soc. 109(4), 975–1013 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Herr S.: The quintic nonlinear Schrödinger equation on three-dimensional Zoll manifolds. Amer. J. Math. 135(5), 1271–1290 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Hardy, G.H., Wright, E.M., An Introduction to the Theory of Numbers, 3rd edn. Clarendon Press, Oxford, 1954Google Scholar
  50. 50.
    Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)ADSMathSciNetCrossRefGoogle Scholar
  51. 51.
    Herr S., Tataru D., Tzvetkov N.: Global well-posedness of the energy critical nonlinear Schrödinger equation with small initial data in \({{H^{1}}({\mathbb{T}^{3}})}\). Duke Math. J. 159(2), 329–349 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Hong Y., Taliaferro K., Xie Z.: Unconditional uniqueness of the cubic Gross–Pitaevskii hierarchy with low regularity. SIAM J. Math. Anal. 47(5), 3314–3341 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Hong, Y., Taliaferro, K., Xie, Z.: Uniqueness of solutions to the 3D quintic Gross–Pitaevskii hierarchy (2014) (preprint). arXiv:1410.6961
  54. 54.
    Killip, R., Vişan, M.: Scale invariant Strichartz estimates on tori and applications (2014) (preprint). arXiv:1409.3603
  55. 55.
    Kirkpatrick K., Schlein B., Staffilani G.: Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems: the periodic case. Amer. J. Math. 133(1), 91–130 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Klainerman S., Machedon M.: On the uniqueness of solutions to the Gross–Pitaevskii hierarchy. Commun. Math. Phys. 279(1), 169–185 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Knowles A., Limiting dynamics in large quantum systems, Ph.D. Thesis, ETH, June 2009Google Scholar
  58. 58.
    Knowles A., Pickl P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–139 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Lewin M., Nam P.T., Rougerie N.: Derivation of Hartree’s theory for generic mean-field Bose systems. Adv. Math. 254, 570–621 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Lewin M., Nam P.T., Rougerie N.: Derivation of nonlinear Gibbs measures from many-body quantum mechanics. J. Éc. Polytech. Math. 2, 65–115 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    Lewin M., Sabin J.: The Hartree equation for infinitely many particles. I. Well-posedness theory. Commun. Math. Phys. 334(1), 117–170 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Lewin M., Sabin J.: The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D. Anal. PDE 7(6), 1339–1363 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Lieb, E., Seiringer, R., Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409-1-4 (2002)Google Scholar
  64. 64.
    Lieb, E., Seiringer, R., Solovej, J.P.: The Quantum-Mechanical Many-Body Problems: The Bose Gas, Perspectives in Analysis. Mathematical Physics Studies, Vol. 27. Springer, Berlin, 97–183, 2005Google Scholar
  65. 65.
    Lieb, E., Seiringer, R., Solovej, J.P., Yngvason, J.: The Mathematics of the Bose Gas and its Condensation. Oberwolfach Seminars, Vol. 34. Birkhauser, Basel, 2005Google Scholar
  66. 66.
    Lieb E., Seiringer R., Yngvason J.: Bosons in a trap: a rigorous derivation of the Gross–Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2000)ADSCrossRefGoogle Scholar
  67. 67.
    Lieb E., Seiringer R., Yngvason J.: A rigorous derivation of the Gross–Pitaevskii energy functional for a two-dimensional Bose gas. Dedicated to Joel L. Lebowitz. Commun. Math. Phys. 224(1), 17–31 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Pitaevskii L.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13, 451–454 (1961)MathSciNetGoogle Scholar
  69. 69.
    Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  70. 70.
    Schlein, B.: Derivation of effective evolution equations from microscopic quantum dynamics. Chapter in Evolution Equations. Clay Mathematics Proceedings, Vol. 17 (Eds. D. Ellwood, I. Rodnianski, G. Staffilani, J. Wunsch), 2013Google Scholar
  71. 71.
    Sohinger, V.: Local existence of solutions to Randomized Gross–Pitaevskii hierarchies. Trans. Am. Math Soc. (2014) (to appear, preprint). arXiv.1401.0326. Available electronically at http://dx.doi.org/10.1016/j.anihpc.2014.09.005
  72. 72.
    Sohinger, V.: A rigorous derivation of the defocusing cubic nonlinear Schrödinger equation on \({{\mathbb{T}^{3}}}\) from the dynamics of many-body quantum systems. Ann. Inst. H. Poincaré (C) Analyse Non Linéaire (2014) (to appear, preprint). arXiv.1405.3003. Available electronically at http://dx.doi.org/10.1016/j.anihpc.2014.09.005
  73. 73.
    Sohinger V., Staffilani G.: Randomization and the Gross–Pitaevskii hierarchy. Arch. Rat. Mech. Anal. 218(1), 417–485 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980)ADSMathSciNetCrossRefGoogle Scholar
  75. 75.
    Strunk N.: Strichartz estimates for Schrödinger equations on irrational tori in two and three dimensions. J. Evol. Equ. 14(4-5), 829–839 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany
  2. 2.Departement MathematikEidgenössische Technische Hochschule ZürichZurichSwitzerland

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