Advertisement

Journal of Statistical Physics

, Volume 176, Issue 2, pp 358–381 | Cite as

On the Time Dependence of the Rate of Convergence Towards Hartree Dynamics for Interacting Bosons

  • Jinyeop LeeEmail author
Article
  • 48 Downloads

Abstract

We consider interacting N-Bosons in three dimensions. It is known that the difference between the many-body Schrödinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order 1 / N. We investigate the time dependence of the difference. To have sub-exponential bound, we use the results of time decay estimate for small initial data. We also refine time dependent bound for singular potential using Strichartz estimate. We consider the interaction potential V(x) of type \(\lambda \exp (-\mu |x|)|x|^{-\gamma }\) for \(\lambda \in \mathbb {R}\), \(\mu \ge 0\), and \(0<\gamma <3/2\), which covers the Coulomb and Yukawa interaction.

Keywords

Many body quantum dynamics Hartree equation Rate of convergence Mean field limit 

Notes

Acknowledgements

The author is grateful to numerous helpful discussions and suggestions from Ji Oon Lee. The author also would like to thank the anonymous referee for carefully reading the manuscript and providing helpful comments. This research is supported in part by KIA Motors Scholarship.

References

  1. 1.
    Cazenave, T.: Semilinear Schrödinger Equations. Courant Lecture Notes in Mathematics, vol. 10. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (2003)Google Scholar
  2. 2.
    Chen, L., Lee, J.O.: Rate of convergence in nonlinear Hartree dynamics with factorized initial data. J. Math. Phys. 52(5), 052108, 25 (2011)Google Scholar
  3. 3.
    Chen, L., Lee, J.O., Schlein, B.: Rate of convergence towards Hartree dynamics. J. Stat. Phys. 144(4), 872–903 (2011)ADSMathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, L., Lee, J.O., Lee, J.: Rate of convergence toward Hartree dynamics with singular interaction potential. J. Math. Phys. 59(3), 031902, 20 (2018)Google Scholar
  5. 5.
    Chen, X.: Second order corrections to mean field evolution for weakly interacting bosons in the case of three-body interactions. Arch. Ration. Mech. Anal. 203, 455–497 (2012)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Cho, Y., Ozawa, T.: On the semirelativistic hartree-type equation. SIAM J. Math. Anal. 38(4), 1060–1074 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chong, J.J.W.: Dynamics of large boson systems with attractive interaction and a derivation of the cubic focusing nls in \(\mathbb{R}^3\). arXiv:1608.01615
  8. 8.
    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)MathSciNetzbMATHGoogle Scholar
  9. 9.
    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)ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    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)ADSMathSciNetzbMATHGoogle Scholar
  11. 11.
    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)ADSMathSciNetzbMATHGoogle Scholar
  12. 12.
    Grillakis, M.G., Machedon, M., Margetis, D.: Second-order corrections to mean field evolution of weakly interacting Bosons. II. Adv. Math. 228, 1788–1815 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Grillakis, M., Machedon, M.: Pair excitations and the mean field approximation of interacting Bosons. I. Commun. Math. Phys. 324(2), 601–636 (2013)ADSMathSciNetzbMATHGoogle Scholar
  14. 14.
    Grillakis, M., Machedon, M.: Pair excitations and the mean field approximation of interacting Bosons. II. Commun. Partial Differ. Equ. 42(1), 24–67 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Hayashi, N., Naumkin, P.I.: Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations. Am. J. Math. 120(2), 369–389 (1998)zbMATHGoogle Scholar
  16. 16.
    Hayashi, N., Naumkin, P.I.: Scattering theory and large time asymptotics of solutions to the Hartree type equations with a long range potential. Hokkaido Math. J. 30(1), 137–161 (2001)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Hayashi, N., Ozawa, T.: Time decay of solutions to the Cauchy problem for time-dependent Schrödinger–Hartree equations. Commun. Math. Phys. 110(3), 467–478 (1987)ADSzbMATHGoogle Scholar
  18. 18.
    Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)ADSMathSciNetGoogle Scholar
  19. 19.
    Hott, M.: Convergence rate towards the fractional hartree-equation with singular potentials in higher Sobolev norms. arXiv:1805.01807
  20. 20.
    Knowles, A., Pickl, P.: Mean-field dynamics: singular potentials and rate of convergence. Commun. Math. Phys. 298(1), 101–138 (2010)ADSMathSciNetzbMATHGoogle Scholar
  21. 21.
    Lee, J.O.: Rate of convergence towards semi-relativistic Hartree dynamics. Ann. Henri Poincaré 14(2), 313–346 (2013)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Lewin, M., Nam, P.T., Schlein, B.: Fluctuations around Hartree states in the mean-field regime. Am. J. Math. 137(6), 1613–1650 (2015)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Nakanishi, K.: Modified wave operators for the Hartree equation with data, image and convergence in the same space. II. Ann. Henri Poincaré 3(3), 503–535 (2002)ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Nakanishi, K.: Modified wave operators for the hartree equation with data, image and convergence in the same space. Commun. Pure Appl. Anal. 1(2), 237–252 (2002)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Rodnianski, I., Schlein, B.: Quantum fluctuations and rate of convergence towards mean field dynamics. Commun. Math. Phys. 291(1), 31–61 (2009)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Spohn, H.: Kinetic equations from Hamiltonian dynamics: Markovian limits. Rev. Mod. Phys. 52(3), 569–615 (1980)ADSMathSciNetGoogle Scholar
  27. 27.
    Tao, T.: Nonlinear Dispersive Equations. CBMS Regional Conference Series in Mathematics, vol. 106. American Mathematical Society, Providence, RI (2006)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesKorea Advanced Institute of Science and TechnologyDaejeonRepublic of Korea

Personalised recommendations