Centre de diffusion de revues académiques mathématiques

 
 
 
 

Journées équations aux dérivées partielles

Table des matières de ce volume | Article précédent | Article suivant
Julien Sabin
The Hartree equation for infinite quantum systems
Journées équations aux dérivées partielles (2014), Exp. No. 8, 18 p., doi: 10.5802/jedp.111
Article PDF

Résumé - Abstract

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.

Bibliographie

[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.  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
Copyright Cellule MathDoc 2018 | Crédit | Plan du site