The Hartree equation for infinite quantum systems
Journées équations aux dérivées partielles (2014), article no. 8, 18 p.

We review some recent results obtained with Mathieu Lewin [21] concerning the nonlinear Hartree equation for density matrices of infinite trace, describing the time evolution of quantum systems with infinitely many particles. Our main result is the asymptotic stability of a large class of translation-invariant density matrices which are stationary solutions to the Hartree equation. We also mention some related result obtained in collaboration with Rupert Frank [13] about Strichartz estimates for orthonormal systems.

@article{JEDP_2014____A8_0,
     author = {Sabin, Julien},
     title = {The Hartree equation for infinite quantum systems},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.111},
     language = {en},
     url = {http://www.numdam.org/item/JEDP_2014____A8_0}
}
Sabin, Julien. The Hartree equation for infinite quantum systems. Journées équations aux dérivées partielles (2014), article  no. 8, 18 p. doi : 10.5802/jedp.111. http://www.numdam.org/item/JEDP_2014____A8_0/

[1] C. Bardos, L. Erdős, F. Golse, N. Mauser, and H.-T. Yau, Derivation of the Schrödinger-Poisson equation from the quantum N-body problem, C. R. Math. Acad. Sci. Paris, 334 (2002), pp. 515–520. | MR 1890644 | Zbl 1018.81009

[2] C. Bardos, F. Golse, A. Gottlieb, and N. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl. (9), 82 (2003), pp. 665–683. | MR 1996777 | Zbl 1029.82022

[3] N. Benedikter, M. Porta, and B. Schlein, Mean-field evolution of fermionic systems, Comm. Math. Phys., 331 (2014), pp. 1087–1131. | MR 3248060

[4] J. Bennett, N. Bez, S. Gutierrez, and S. Lee, On the Strichartz estimates for the kinetic transport equation, arXiv preprint arXiv:1307.1600, (2013). | MR 3250975

[5] A. Bove, G. Da Prato, and G. Fano, An existence proof for the Hartree-Fock time-dependent problem with bounded two-body interaction, Commun. Math. Phys., 37 (1974), pp. 183–191. | MR 424069 | Zbl 0303.34046

[6] By same, On the Hartree-Fock time-dependent problem, Commun. Math. Phys., 49 (1976), pp. 25–33. | MR 456066

[7] E. Cancès and G. Stoltz, A mathematical formulation of the random phase approximation for crystals, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire, 29 (2012), pp. 887–925. | Numdam | MR 2995100 | Zbl 1273.82073

[8] F. Castella and B. Perthame, Estimations de Strichartz pour les équations de transport cinétique, CR Acad. Sci. Paris Sér. I Math, 322 (1996), pp. 535–540. | MR 1383431 | Zbl 0848.35095

[9] J. Chadam, The time-dependent Hartree-Fock equations with Coulomb two-body interaction, Commun. Math. Phys., 46 (1976), pp. 99–104. | MR 411439 | Zbl 0322.35043

[10] A. Elgart, L. Erdős, B. Schlein, and H.-T. Yau, Nonlinear Hartree equation as the mean field limit of weakly coupled fermions, J. Math. Pures Appl., 83 (2004), pp. 1241–1273. | MR 2092307 | Zbl 1059.81190

[11] R. Frank, M. Lewin, E. Lieb, and R. Seiringer, A positive density analogue of the Lieb-Thirring inequality, Duke Math. J., 162 (2012), pp. 435–495. | MR 3024090 | Zbl 1260.35088

[12] R. L. Frank, M. Lewin, E. H. Lieb, and R. Seiringer, Strichartz inequality for orthonormal functions, J. Eur. Math. Soc., (2013). In press. | MR 3254332

[13] R. L. Frank and J. Sabin, Restriction theorems for orthonormal functions, strichartz inequalities, and uniform sobolev estimates, arXiv preprint arXiv:1404.2817, (2014).

[14] J. Fröhlich and A. Knowles, A microscopic derivation of the time-dependent Hartree-Fock equation with Coulomb two-body interaction, J. Stat. Phys., 145 (2011), pp. 23–50. | MR 2841931 | Zbl 1269.82042

[15] G. Giuliani and G. Vignale, Quantum Theory of the Electron Liquid, Cambridge University Press, 2005.

[16] C. Hainzl, M. Lewin, and C. Sparber, Existence of global-in-time solutions to a generalized Dirac-Fock type evolution equation, Lett. Math. Phys., 72 (2005), pp. 99–113. | MR 2154857 | Zbl 1115.81026

[17] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), pp. 955–980. | MR 1646048 | Zbl 0922.35028

[18] C. E. Kenig, A. Ruiz, and C. D. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math. J., 55 (1987), pp. 329–347. | MR 894584 | Zbl 0644.35012

[19] M. Lewin and J. Sabin, The Hartree equation for infinitely many particles. I. Well-posedness theory, Comm. Math. Phys., (2013). To appear.

[20] M. Lewin and J. Sabin, A family of monotone quantum relative entropies, Lett. Math. Phys., 104 (2014), pp. 691–705. | MR 3200935

[21] M. Lewin and J. Sabin, The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D, Analysis and PDE, 7 (2014), pp. 1339–1363. | MR 3270166

[22] C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), pp. 29–201. | MR 2863910 | Zbl 1239.82017

[23] E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), pp. 482–492. | MR 82586 | Zbl 0072.32402

[24] E. M. Stein, Oscillatory integrals in Fourier analysis, in Beijing lectures in harmonic analysis (Beijing, 1984), vol. 112 of Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 1986, pp. 307–355. | MR 864375 | Zbl 0821.42001

[25] R. Strichartz, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., 44 (1977), pp. 705–714. | MR 512086 | Zbl 0372.35001

[26] K. Yajima, Existence of solutions for Schrödinger evolution equations, Comm. Math. Phys., 110 (1987), pp. 415–426. | MR 891945 | Zbl 0638.35036

[27] S. Zagatti, The Cauchy problem for Hartree-Fock time-dependent equations, Ann. Inst. H. Poincaré Phys. Théor., 56 (1992), pp. 357–374. | Numdam | MR 1175475 | Zbl 0763.35089