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Séminaire Laurent Schwartz — EDP et applications

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Mathieu Lewin
Gaz de bosons dans le régime de champ moyen : les théories de Hartree et Bogoliubov
Séminaire Laurent Schwartz — EDP et applications (2012-2013), Exp. No. 3, 22 p., doi: 10.5802/slsedp.33
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Résumé - Abstract

Nous étudions le spectre du Hamiltonien d’un gaz de bosons, à la limite d’un grand nombre $N$ de particules et dans le régime de champ moyen (l’interaction est multipliée par $1/N$). Le premier terme du développement est donné par le modèle non linéaire de Hartree, alors que le second terme est donné par la théorie de Bogoliubov.

Bibliography

[1] Z. Ammari and F. Nier, Mean field limit for bosons and infinite dimensional phase-space analysis, Annales Henri Poincaré, 9 (2008), pp. 1503–1574. http://dx.doi.org/10.1007/s00023-008-0393-5.  MR 2465733 |  Zbl 1171.81014
[2] Z. Ammari and F. Nier, Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl., 95 (2011), pp. 585–626.  MR 2802894 |  Zbl 1251.81062
[3] V. Bach, Ionization energies of bosonic Coulomb systems, Lett. Math. Phys., 21 (1991), pp. 139–149.  MR 1093525 |  Zbl 0725.47049
[4] V. Bach, R. Lewis, E. H. Lieb, and H. Siedentop, On the number of bound states of a bosonic $N$-particle Coulomb system, Math. Z., 214 (1993), pp. 441–459.  MR 1245205 |  Zbl 0852.47036
[5] B. Baumgartner, On Thomas-Fermi-von Weizsäcker and Hartree energies as functions of the degree of ionisation, J. Phys. A, 17 (1984), pp. 1593–1601.  MR 750572 |  Zbl 0541.49020
[6] G. Ben Arous, K. Kirkpatrick, and B. Schlein, A Central Limit Theorem in Many-Body Quantum Dynamics, arXiv :1111.6999, (2011).
[7] R. Benguria and E. H. Lieb, Proof of the Stability of Highly Negative Ions in the Absence of the Pauli Principle, Physical Review Letters, 50 (1983), pp. 1771–1774.
[8] F. Berezin, The method of second quantization, Pure and applied physics. A series of monographs and textbooks, Academic Press, 1966.  MR 208930 |  Zbl 0151.44001
[9] N. N. Bogoliubov, About the theory of superfluidity, Izv. Akad. Nauk SSSR, 11 (1947), p. 77.  MR 22177
[10] F. Calogero, Solution of the one-dimensional $N$-body problems with quadratic and/or inversely quadratic pair potentials, J. Mathematical Phys., 12 (1971), pp. 419–436.  MR 280103 |  Zbl 1002.70558
[11] F. Calogero and C. Marchioro, Lower bounds to the ground-state energy of systems containing identical particles, J. Mathematical Phys., 10 (1969), pp. 562–569.  MR 339719
[12] H. D. Cornean, J. Derezinski, and P. Zin, On the infimum of the energy-momentum spectrum of a homogeneous bose gas, J. Math. Phys., 50 (2009), p. 062103.  MR 2541168 |  Zbl 1216.82006
[13] B. De Finetti, Funzione caratteristica di un fenomeno aleatorio. Atti della R. Accademia Nazionale dei Lincei, 1931. Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali.  JFM 57.0610.01
[14] P. Diaconis and D. Freedman, Finite exchangeable sequences, Ann. Probab., 8 (1980), pp. 745–764.  MR 577313 |  Zbl 0434.60034
[15] L. Erdös, B. Schlein, and H.-T. Yau, Ground-state energy of a low-density Bose gas : A second-order upper bound, Phys. Rev. A, 78 (2008), p. 053627.
[16] J. Ginibre and G. Velo, The classical field limit of scattering theory for nonrelativistic many-boson systems. II, Commun. Math. Phys., 68 (1979), pp. 45–68.  MR 539736 |  Zbl 0443.35068
[17] M. Girardeau, Relationship between systems of impenetrable bosons and fermions in one dimension, J. Mathematical Phys., 1 (1960), pp. 516–523.  MR 128913 |  Zbl 0098.21704
[18] A. Giuliani and R. Seiringer, The ground state energy of the weakly interacting Bose gas at high density, J. Stat. Phys., 135 (2009), pp. 915–934.  MR 2548599 |  Zbl 1172.82006
[19] P. Grech and R. Seiringer, The excitation spectrum for weakly interacting bosons in a trap, arXiv :1205.5259, (2012).  MR 2824481
[20] M. G. Grillakis, M. Machedon, and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. I, Commun. Math. Phys., 294 (2010), pp. 273–301.  MR 2575484 |  Zbl 1208.82030
[21] M. G. Grillakis, M. Machedon, and D. Margetis, Second-order corrections to mean field evolution of weakly interacting bosons. II, Adv. Math., 228 (2011), pp. 1788–1815.  MR 2824569
[22] K. Hepp, The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., 35 (1974), pp. 265–277.  MR 332046
[23] E. Hewitt and L. J. Savage, Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., 80 (1955), pp. 470–501.  MR 76206 |  Zbl 0066.29604
[24] R. L. Hudson and G. R. Moody, Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33 (1975/76), pp. 343–351.  MR 397421 |  Zbl 0304.60001
[25] L. Landau, Theory of the Superfluidity of Helium II, Phys. Rev., 60 (1941), pp. 356–358.  Zbl 0027.18505
[26] M. Lewin, Geometric methods for nonlinear many-body quantum systems, J. Funct. Anal., 260 (2011), pp. 3535–3595.  MR 2781970 |  Zbl 1216.81180
[27] M. Lewin, P. T. Nam, and N. Rougerie, Derivation of Hartree’s theory for generic mean-field Bose gases, preprint arXiv :1303.0981, (2013).
[28] M. Lewin, P. T. Nam, S. Serfaty, and J. P. Solovej, Bogoliubov spectrum of interacting Bose gases, preprint arXiv :1211.2778, (2012).
[29] E. H. Lieb, Exact analysis of an interacting Bose gas. II. The excitation spectrum, Phys. Rev. (2), 130 (1963), pp. 1616–1624.  MR 156631 |  Zbl 0138.23002
[30] E. H. Lieb, A lower bound for Coulomb energies, Phys. Lett. A, 70 (1979), pp. 444–446.  MR 588128
[31] E. H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. (2), 130 (1963), pp. 1605–1616.  MR 156630 |  Zbl 0138.23001
[32] E. H. Lieb and S. Oxford, Improved lower bound on the indirect Coulomb energy, Int. J. Quantum Chem., 19 (1980), pp. 427–439.
[33] E. H. Lieb and R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge Univ. Press, 2010.  MR 2583992 |  Zbl 1179.81004
[34] E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason, The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, Birkhäuser, 2005.  MR 2143817 |  Zbl 1104.82012
[35] E. H. Lieb and J. P. Solovej, Ground state energy of the one-component charged Bose gas, Commun. Math. Phys., 217 (2001), pp. 127–163.  MR 1815028 |  Zbl 1042.82004
[36] E. H. Lieb and J. P. Solovej, Ground state energy of the two-component charged Bose gas., Commun. Math. Phys., 252 (2004), pp. 485–534.  MR 2104887 |  Zbl 1124.82303
[37] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 109–149. Numdam |  MR 778970 |  Zbl 0704.49004
[38] P.-L. Lions, Mean-field games and applications. Lectures at the Collège de France, unpublished, Nov 2007.  Zbl 1205.91027
[39] P. T. Nam, Contributions to the rigorous study of the structure of atoms, PhD thesis, University of Copenhagen, 2011.
[40] O. Penrose and L. Onsager, Bose-Einstein Condensation and Liquid Helium, Phys. Rev., 104 (1956), pp. 576–584.  Zbl 0071.44701
[41] D. Petz, G. A. Raggio, and A. Verbeure, Asymptotics of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys., 121 (1989), pp. 271–282.  MR 985399 |  Zbl 0682.46054
[42] G. A. Raggio and R. F. Werner, Quantum statistical mechanics of general mean field systems, Helv. Phys. Acta, 62 (1989), pp. 980–1003.  MR 1034151 |  Zbl 0938.82501
[43] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975.  MR 493420 |  Zbl 0242.46001
[44] E. Sandier and S. Serfaty, 2D Coulomb Gases and the Renormalized Energy, arXiv :1201.3503, (2012).
[45] E. Sandier and S. Serfaty, From the Ginzburg-Landau model to vortex lattice problems, Commun. Math. Phys., 313 (2012), pp. 635–743.  MR 2945619 |  Zbl 1252.35034
[46] R. Seiringer, The excitation spectrum for weakly interacting bosons, Commun. Math. Phys., 306 (2011), pp. 565–578.  MR 2824481 |  Zbl 1226.82039
[47] J. P. Solovej, Asymptotics for bosonic atoms, Lett. Math. Phys., 20 (1990), pp. 165–172.  MR 1065245 |  Zbl 0712.35075
[48] J. P. Solovej, Upper bounds to the ground state energies of the one- and two-component charged Bose gases, Commun. Math. Phys., 266 (2006), pp. 797–818.  MR 2238912 |  Zbl 1126.82006
[49] J. P. Solovej, Many body quantum mechanics. LMU, 2007. Lecture notes.
[50] E. Størmer, Symmetric states of infinite tensor products of $C^{\ast } $-algebras, J. Functional Analysis, 3 (1969), pp. 48–68.  MR 241992 |  Zbl 0167.43403
[51] B. Sutherland, Quantum Many-Body Problem in One Dimension : Ground State, J. Mathematical Phys., 12 (1971), pp. 246–250.
[52] B. Sutherland, Quantum Many-Body Problem in One Dimension : Thermodynamics, J. Mathematical Phys., 12 (1971), pp. 251–256.
[53] A. Sütő, Thermodynamic limit and proof of condensation for trapped bosons, J. Statist. Phys., 112 (2003), pp. 375–396.  MR 1991602 |  Zbl 1035.82007
[54] R. F. Werner, Large deviations and mean-field quantum systems, in Quantum probability & related topics, QP-PQ, VII, World Sci. Publ., River Edge, NJ, 1992, pp. 349–381.  MR 1186674 |  Zbl 0788.60126
[55] T. T. Wu, Bose-Einstein condensation in an external potential at zero temperature : General theory, Phys. Rev. A, 58 (1998), pp. 1465–1474.
[56] H.-T. Yau and J. Yin, The second order upper bound for the ground energy of a Bose gas, J. Stat. Phys., 136 (2009), pp. 453–503.  MR 2529681 |  Zbl 1200.82002
[57] J. Yngvason, The interacting Bose gas : A continuing challenge, Phys. Particles Nuclei, 41 (2010), pp. 880–884.
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