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Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.

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Abstract

Inspired by the works of Rodnianski and Schlein [31] and Wu [34,35], we derive a new nonlinear Schrödinger equation that describes a second-order correction to the usual tensor product (mean-field) approximation for the Hamiltonian evolution of a many-particle system in Bose-Einstein condensation. We show that our new equation, if it has solutions with appropriate smoothness and decay properties, implies a new Fock space estimate. We also show that for an interaction potential \({v(x)= \epsilon \chi(x) |x|^{-1}}\), where \({\epsilon}\) is sufficiently small and \({\chi \in C_0^{\infty}}\) even, our program can be easily implemented locally in time. We leave global in time issues, more singular potentials and sophisticated estimates for a subsequent part (Part II) of this paper.

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References

  1. 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 

  2. Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations I, II. Geom. Funct. Anal. 3, 107–156 (1993) and 202–262 (1993)

    Google Scholar 

  3. Constantin P., Saut S.: Local smoothing properties of dispersive equations. JAMS 1, 431–439 (1988)

    MathSciNet  Google Scholar 

  4. 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 

  5. Dyson F.J.: Ground-state energy of a hard sphere gas. Phys. Rev. 106, 20–26 (1957)

    Article  MATH  ADS  Google Scholar 

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

    Article  MATH  Google Scholar 

  7. Erdős 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)

    MathSciNet  Google Scholar 

  8. 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, 1659–1741 (2006)

    Article  Google Scholar 

  9. 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, 515–614 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  10. Erdős L., Schlein B., Yau H.-T.: Rigorous derivation of the Gross-Pitaevskii equation. Phys. Rev. Lett. 98, 040404 (2007)

    Article  ADS  Google Scholar 

  11. Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Annals Math

  12. Folland, G.B.: Harmonic Analysis in Phase Space. Annals of Math Studies, Vol. 122 Princeton Univerity Press, Princeton, NJ, 1989

  13. Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems, I and II. Commun. Math. Phys. 66, 37–76 (1979) and 68, 45–68 (1979)

  14. Grillakis M.G., Margetis D.: A priori estimates for many-body Hamiltonian evolution of interacting Boson system. J. Hyperb. Diff. Eqs. 5, 857–883 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gross E.P.: Structure of a quantized vortex in boson systems. Nuovo Cim. 20, 454–477 (1961)

    Article  MATH  Google Scholar 

  16. Gross E.P.: Hydrodynamics of a superfluid condensate. J. Math. Phys. 4, 195–207 (1963)

    Article  ADS  Google Scholar 

  17. Hepp K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)

    Article  MathSciNet  ADS  Google Scholar 

  18. Kenig C., Ponce G., Vega L.: The Cauchy problem for the K-dV equation in Sobolev spaces with negative indices. Duke Math. J. 71, 1–21 (1994)

    Article  MathSciNet  Google Scholar 

  19. Klainerman S., Machedon M.: Space-time estimates for null forms and the local existence theorem. Commun. Pure Appl. Math. 46, 1221–1268 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Klainerman S., Machedon M.: Smoothing estimates for null forms and applications. Duke Math. J. 81, 99–103 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lee T.D., Huang K., Yang C.N.: Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties. Phys. Rev. 106, 1135–1145 (1957)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Lieb E.H., Seiringer R.: Derivation of the Gross-Pitaevskii Equation for rotating Bose gases. Commun. Math. Phys. 264, 505–537 (2006)

    Article  MathSciNet  ADS  Google Scholar 

  23. Lieb E.H., Seiringer R., Solovej J.P., Yngvanson J.: The Mathematics of the Bose Gas and its Condensation. Birkhaüser Verlag, Basel (2005)

    MATH  Google Scholar 

  24. Lieb E.H., Seiringer R., Yngvanson J.: Bosons in a trap: a rigorous derivation of the Gross-Pitaevskii energy functional. Phys. Rev. A 61, 043602 (2006)

    Article  ADS  Google Scholar 

  25. Lieb E.H., Seiringer R., Yngvason J.: A rigorous derivation of the Gross-Pitaevskii energy functional for a two-dimensional Bose gas. Commun. Math. Phys. 224, 17–31 (2001)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  26. Margetis, D.: Studies in Classical Electromagnetic Radiation and Bose-Einstein Condensation. Ph.D. thesis, Harvard University, 1999

  27. Margetis, D.: Solvable model for pair excitation in trapped Boson gas at zero temperature. J. Phys. A: Math. Theor. 41, 235004 (2008); Corrigendum. J. Phys. A: Math. Theor. 41, 459801 (2008)

    Google Scholar 

  28. Pitaevskii L.P.: Vortex lines in an imperfect Bose gas. Soviet Phys. JETP 13, 451–454 (1961)

    MathSciNet  Google Scholar 

  29. Pitaevskii L.P., Stringari S.: Bose-Einstein Condensation. Oxford University Press, Oxford (2003)

    MATH  Google Scholar 

  30. Riesz F., Nagy B.: Functional analysis. Frederick Ungar Publishing, New York (1955)

    Google Scholar 

  31. Rodnianski I., Schlein B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(2), 31–61 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  32. Sjölin P.: Regularity of solutions to the Schrödinger equation. Duke Math. J. 55, 699–715 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  33. Vega L.: Schrödinger equations: Pointwise convergence to the initial data. Proc. AMS 102, 874–878 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Wu T.T.: Some nonequilibrium properties of a Bose system of hard spheres at extremely low temperatures. J. Math. Phys. 2, 105–123 (1961)

    Article  MATH  ADS  Google Scholar 

  35. Wu T.T.: Bose-Einstein condensation in an external potential at zero temperature: General theory. Phys. Rev. A 58, 1465–1474 (1998)

    Article  ADS  Google Scholar 

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Correspondence to Matei Machedon.

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Communicated by H.-T. Yau

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Grillakis, M.G., Machedon, M. & Margetis, D. Second-Order Corrections to Mean Field Evolution of Weakly Interacting Bosons. I.. Commun. Math. Phys. 294, 273–301 (2010). https://doi.org/10.1007/s00220-009-0933-y

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  • DOI: https://doi.org/10.1007/s00220-009-0933-y

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