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

Fröhlich, Jürg; Lenzmann, Enno

Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2003-2004), Exposé no. 18, 26 p.

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.

[AF88] 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. | MR | Zbl

[AFG + 02] 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. | MR | Zbl

[AFS88] 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. | MR | Zbl

[BP92] 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. | Zbl

[BP95] —, On the stability of solitary waves for nonlinear Schrödinger equations, Amer. Math. Soc. Transl. Ser. 2 164 (1995), 74–98. | MR | Zbl

[BW04] J. Bourgain and W. Wang, Quasi-periodic solutions of nonlinear lattice Schrödinger equations with random potential, 2004, submitted to Invent. Math.

[Caz03] T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes 10, Amer. Math. Soc., Providence, 2003. | MR | Zbl

[CFKS87] 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. | MR | Zbl

[CL82] T. Cazenave and P.-L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549–561. | MR | Zbl

[Cuc02] S. Cuccagna, Asymptotic stability of the ground states of the nonlinear Schrödinger equation, Rend. Istit. Mat. Univ. Trieste 32 (2002), 105–118. | MR | Zbl

[EY01] 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. | MR | Zbl

[FGJS04] 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. | MR | Zbl

[FS75] W. Faris and B. Simon, Degenerate and non-degenerate ground states for Schrödinger operators, Duke Math. J. 42 (1975), 559–567. | MR | Zbl

[FSW86] J. Fröhlich, T. Spencer, and C. Wayne, Localization in disordered nonlinear dynamical systems, J. Stat. Phys. 42 (1986), 247–274. | MR | Zbl

[FTY02] 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. | MR | Zbl

[GMP03] 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. | MR | Zbl

[GNT04] 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. | MR | Zbl

[GSS90] 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. | MR | Zbl

[Gun99] C. Gundlach, Critical phenomena in gravitational collpase, Living Rev. Rel. 2 (1999). | MR | Zbl

[GV80] J. Ginibre and G. Velo, On a class of nonlinear Schrödinger equations, Math. Z. 170 (1980), 109–136. | MR | Zbl

[Hep74] K. Hepp, The classical limit for quantum mechanical correlation funcitons, Comm. Math. Phys. 35 (1974), 265–277. | MR

[HMT03] R. Harrison, I. Moroz, and K.P. Tod, A numerical study of the Schrödinger-Newton equations, Nonlinearity 16 (2003), 101–122. | MR | Zbl

[Kup95] A. Kupiainen, Renormalizing partial differential equations, Constructive Physics (V. Rivasseau, ed.), Spinger, 1995, pp. 83–117. | MR

[Kwo89] M.K. Kwong, Uniqueness of positive solutions of $▵u-u+u^p=0$ in ${ℝ}^n$, Arch. Rat. Mech. Anal. 105 (1989), 243–266. | MR | Zbl

[Len04] E. Lenzmann, Dynamical evolution of semi-relativistic boson stars, 2004, in preparation.

[Lie77] E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Stud. Appl. Math. 57 (1977), 93–105. | Zbl

[Lio80] P.-L. Lions, Some remarks on Hartree equation, Nonlinear Anal. 5 (1980), 1245–1256. | MR | Zbl

[Lio84a] —, 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. | Numdam | MR | Zbl

[Lio84b] —, 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. | Numdam | MR | Zbl

[LL01] E.H. Lieb and M. Loss, Analysis, 2nd ed., Graduate Series in Mathematics 14, Amer. Math. Soc., Providence, 2001. | MR | Zbl

[LSSY03] 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. | MR | Zbl

[LY87] 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. | MR | Zbl

[Per03] G. Perel’man, Asymptotic stability of multi-soliton solutions for nonlinear Schrödinger equations, 2003, preprint. | Zbl

[PW97] 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. | MR | Zbl

[RSS03] I. Rodnianski, W. Schlag, and A. Soffer, Asymptotic stability of $N$-soliton states of NLS, 2003, preprint.

[Sch04] S. Schwarz, Ph. D. thesis, 2004, in preparation.

[Sig93] I.M. Sigal, Nonlinear wave and Schrödinger equations I, instability of time-periodic and quasiperiodic solutions, Comm. Math. Phys. 153 (1993), 297–320. | MR | Zbl

[Spo80] H. Spohn, Kinetic equations from Hamiltonian dynamics, Rev. Mod. Phys. 52 (1980), no. 3, 569–615. | MR

[SW90] A. Soffer and M.I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations, Comm. Math. Phys. 133 (1990), 119–146. | MR | Zbl

[SW92] —, Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data, J. Differential Equations 98 (1992), 376–390. | MR | Zbl

[SW99] —, Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. Math. 136 (1999), 9–74. | MR | Zbl

[SW03] —, Selection of the ground state for nonlinear Schrödinger equations, 2003, preprint, ArXiv:nlin.PS/0308020.

[Tao04] 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. | MR | Zbl

[TY02a] 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. | MR | Zbl

[TY02b] —, Classification of asymptotic profiles for nonlinear Schrödinger equations with small initial data, Adv. Theor. Math. Phys. 6 (2002), 107–139. | MR | Zbl

[TY02c] —, Relaxation of excited states in nonlinear Schrödinger equations, Int. Math. Res. Not. 31 (2002), 1629–1673. | MR | Zbl

[Wed00] R. Weder, Center manifold for nonintegrable nonlinear Schrödinger equations on the line, Comm. Math. Phys. 215 (2000), 343–356. | MR | Zbl

[Wei85] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472–491. | MR | Zbl