Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation

Jürg Fröhlich[1]; Enno Lenzmann[2]

  • [1] Institute for Theoretical Physics, ETH Zürich-Hönggerberg, CH-8093 Zürich, Switzerland
  • [2] Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland

Séminaire Équations aux dérivées partielles (2003-2004)

  • Volume: 2003-2004, page 1-26

Abstract

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We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schrödinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.

How to cite

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Fröhlich, Jürg, and Lenzmann, Enno. "Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation." Séminaire Équations aux dérivées partielles 2003-2004 (2003-2004): 1-26. <http://eudml.org/doc/11084>.

@article{Fröhlich2003-2004,
abstract = {We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schrödinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.},
affiliation = {Institute for Theoretical Physics, ETH Zürich-Hönggerberg, CH-8093 Zürich, Switzerland; Department of Mathematics, ETH Zürich, CH-8092 Zürich, Switzerland},
author = {Fröhlich, Jürg, Lenzmann, Enno},
journal = {Séminaire Équations aux dérivées partielles},
language = {eng},
pages = {1-26},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation},
url = {http://eudml.org/doc/11084},
volume = {2003-2004},
year = {2003-2004},
}

TY - JOUR
AU - Fröhlich, Jürg
AU - Lenzmann, Enno
TI - Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
JO - Séminaire Équations aux dérivées partielles
PY - 2003-2004
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2003-2004
SP - 1
EP - 26
AB - We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schrödinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.
LA - eng
UR - http://eudml.org/doc/11084
ER -

References

top
  1. C. Albanese and J. Fröhlich, Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations I, Comm. Math. Phys. 116 (1988), 475–502. Zbl0696.35185MR937771
  2. W.H. Aschbacher, J. Fröhlich, G.M. Graf, K. Schnee, and M. Troyer, Symmetry breaking regime in the nonlinear Hartree equation, J. Math. Phys. 43 (2002), 3879–3891. Zbl1060.81012MR1915631
  3. C. Albanese, J. Fröhlich, and T. Spencer, Periodic solutions of some infinite-dimensional Hamiltonian systems associated with non-linear partial difference equations II, Comm. Math. Phys. 119 (1988), 677–699. Zbl0699.35242MR973022
  4. V.S. Buslaev and G.S. Perel’man, Scattering for the nonlinear Schrödinger equation: states that are close to a soliton, Algebra i Analiz 4 (1992), 63–102. Zbl0853.35112
  5. —, On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Transl. Ser. 2 164 (1995), 74–98. Zbl0841.35108MR1334139
  6. J. Bourgain and W. Wang, Quasi-periodic solutions of nonlinear lattice Schrödinger equations with random potential, 2004, submitted to Invent. Math. 
  7. T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes 10, Amer. Math. Soc., Providence, 2003. Zbl1055.35003MR2002047
  8. H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon, Schrödinger operators, with applications to quantum mechanics and global geometry, Springer Study Edition, Spinger, Berlin Heidelberg, 1987. Zbl0619.47005MR883643
  9. T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561. Zbl0513.35007MR677997
  10. S. Cuccagna, Asymptotic stability of the ground states of the nonlinear Schrödinger equation, Rend. Istit. Mat. Univ. Trieste 32 (2002), 105–118. Zbl1006.35088MR1893394
  11. L. Erdös and H.-T. Yau, Derivation of the nonlinear Schrödinger equation from a many-body Coulomb system, Adv. Theor. Math. Phys. 5 (2001), 1169–1205. Zbl1014.81063MR1926667
  12. J. Fröhlich, S. Gustafson, B.L.G. Jonsson, and I.M. Sigal, Solitary wave dynamics in an external potential, 2004, to appear in Comm. Math. Phys. Zbl1075.35075MR2094474
  13. W. Faris and B. Simon, Degenerate and non-degenerate ground states for Schrödinger operators, Duke Math. J. 42 (1975), 559–567. Zbl0339.35033MR381571
  14. J. Fröhlich, T. Spencer, and C. Wayne, Localization in disordered nonlinear dynamical systems, J. Stat. Phys. 42 (1986), 247–274. Zbl0629.60105MR833019
  15. J. Fröhlich, T.-P. Tsai, and H.-T. Yau, On the point-particle (Newtonian) limit of the non-linear Hartree equation, Comm. Math. Phys. 225 (2002), 223–274. Zbl1025.81015MR1889225
  16. S. Graffi, A. Martinez, and M. Pulvirenti, Mean-field approximation of quantum systems and classical limit, Math. Models Methods Appl. Sci. 13 (2003), 59–73. Zbl1049.81022MR1956958
  17. S. Gustafson, K. Nakanishi, and T.-P. Tsai, Asymptotic stability and completeness in the energy space for nonlinear Schrödinger equations with small solitary waves, 2004, to appear in Int. Math. Res. Not. Zbl1072.35167MR2101699
  18. M. Grillakis, J. Shatah, and W. Strauss, Stability theory of solitary waves in the presence of symmetry II, J. Funct. Anal. 94 (1990), 308–348. Zbl0711.58013MR1081647
  19. C. Gundlach, Critical phenomena in gravitational collpase, Living Rev. Rel. 2 (1999). Zbl0944.83020MR1728882
  20. J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, Math. Z. 170 (1980), 109–136. Zbl0407.35063MR562582
  21. K. Hepp, The classical limit for quantum mechanical correlation funcitons, Comm. Math. Phys. 35 (1974), 265–277. MR332046
  22. R. Harrison, I. Moroz, and K.P. Tod, A numerical study of the Schrödinger-Newton equations, Nonlinearity 16 (2003), 101–122. Zbl1040.81554MR1950778
  23. A. Kupiainen, Renormalizing partial differential equations, Constructive Physics (V. Rivasseau, ed.), Spinger, 1995, pp. 83–117. Zbl0831.35076MR1356027
  24. M.K. Kwong, Uniqueness of positive solutions of ▵ u - u + u p = 0 in ℝ n , Arch. Rat. Mech. Anal. 105 (1989), 243–266. Zbl0676.35032MR969899
  25. E. Lenzmann, Dynamical evolution of semi-relativistic boson stars, 2004, in preparation. 
  26. E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105. Zbl0369.35022
  27. P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal. 5 (1980), 1245–1256. Zbl0472.35074MR636734
  28. —, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 109–145. Zbl0541.49009MR778970
  29. —, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283. Zbl0704.49004MR778974
  30. E.H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Series in Mathematics 14, Amer. Math. Soc., Providence, 2001. Zbl0966.26002MR1817225
  31. E.H. Lieb, R. Seiringer, J.P. Solovej, and J. Yngvason, The quantum-mechanical many-body problem: The Bose gas, Perspectives in Analysis (KTH Stockholm) (M. Benedicks, P. Jones, and S. Smirnov, eds.), Springer, 2003. Zbl1113.82007MR2215726
  32. E.H. Lieb and H.-T. Yau, The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Comm. Math. Phys. 112 (1987), 147–174. Zbl0641.35065MR904142
  33. G. Perel’man, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, 2003, preprint. Zbl1067.35113
  34. C.-A. Pillet and C.E. Wayne, Invariant manifolds for a class of dispersive, Hamiltonian partial differential equations, J. Differential Equations 141 (1997), 310–326. Zbl0890.35016MR1488355
  35. I. Rodnianski, W. Schlag, and A. Soffer, Asymptotic stability of N -soliton states of NLS, 2003, preprint. 
  36. S. Schwarz, Ph. D. thesis, 2004, in preparation. 
  37. I.M. Sigal, Nonlinear wave and Schrödinger equations I, instability of time-periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), 297–320. Zbl0780.35106MR1218303
  38. H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys. 52 (1980), no. 3, 569–615. MR578142
  39. A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), 119–146. Zbl0721.35082MR1071238
  40. —, Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), 376–390. Zbl0795.35073MR1170476
  41. —, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9–74. Zbl0910.35107MR1681113
  42. —, Selection of the ground state for nonlinear Schrödinger equations, 2003, preprint, ArXiv:nlin.PS/0308020. 
  43. T. Tao, On the asymptotic behavior of large radial data for a focusing non-linear Schrödinger equation, 2004, to appear in J. PDE and Dynam. Systems. Zbl1082.35144MR2091393
  44. T.-P. Tsai and H.-T. Yau, Asymptotic dynamics of nonlinear Schrödinger equations: resonance-dominated and dispersion-dominated solutions, Comm. Pure Appl. Math. 55 (2002), 153–216. Zbl1031.35137MR1865414
  45. —, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys. 6 (2002), 107–139. Zbl1033.81034MR1992875
  46. —, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. 31 (2002), 1629–1673. Zbl1011.35120MR1916427
  47. R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Comm. Math. Phys. 215 (2000), 343–356. Zbl1003.37045MR1799850
  48. M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491. Zbl0583.35028MR783974

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