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Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics


The nonlinear Hartree equation describes the macroscopic dynamics of initially factorized N-boson states, in the limit of large N. In this paper we provide estimates on the rate of convergence of the microscopic quantum mechanical evolution towards the limiting Hartree dynamics. More precisely, we prove bounds on the difference between the one-particle density associated with the solution of the N-body Schrödinger equation and the orthogonal projection onto the solution of the Hartree equation.

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  1. Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: Univ. Texas Math. Physics Archive,, No. 05-211, 2005

  2. Bardos C., Golse F., Mauser N.: Weak coupling limit of the N-particle Schrödinger equation. Meth. Appl. Anal. 7, 275–293 (2000)

    MATH  MathSciNet  Google Scholar 

  3. 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(2), 265–283 (2006)

    MATH  Article  Google Scholar 

  4. Elgart A., Schlein B.: Mean field dynamics of Boson stars. Commun. Pure Appl. Math. 60(4), 500–545 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  5. 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)

    Article  ADS  MathSciNet  Google Scholar 

  6. Erdős, L., Schlein, B., Yau, H.-T.: Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate. To appear in Ann. of Math., 2006

  7. 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)

    MathSciNet  Google Scholar 

  8. 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)

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

    Article  ADS  MathSciNet  Google Scholar 

  10. Krasikov I.: Inequalities for Laguerre polynomials. East J. Approx. 11, 257–268 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Spohn H.: Kinetic equations from Hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  12. Szegö, G.: Orthogonal Polynomials. Colloq. pub. AMS. V. 23, New York: Amer. Math. Soc., 1959

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Correspondence to Benjamin Schlein.

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Communicated by H.-T. Yau

Partially supported by NSF grant DMS-0702270.

On leave from the University of Cambridge. Supported by a Sofja Kovalevskaja Award of the Humboldt Foundation. Current address: University of Cambridge, Centre for Mathematical Sciences, DPMMS, Wilberforce Rd, Cambridge CB3 0WB, UK.

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Rodnianski, I., Schlein, B. Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Commun. Math. Phys. 291, 31–61 (2009).

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  • Coherent State
  • Quantum Fluctuation
  • Field Dynamics
  • Singular Potential
  • Canonical Commutation Relation