Catto, I; Le Bris, C; Lions, P.-L
On the thermodynamic limit for Hartree-Fock type models
Annales de l'I.H.P. Analyse non linéaire, Tome 18 (2001) no. 6 , p. 687-760
Zbl 0994.35115 | 5 citations dans Numdam
URL stable : http://www.numdam.org/item?id=AIHPC_2001__18_6_687_0

Bibliographie

[1] Amerio L, Prouse G, Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold Company, 1971. MR 275061 | Zbl 0215.15701

[2] Ashcroft N.W, Mermin N.D, Solid-state Physics, Saunders College Publishing, 1976.

[3] Axel F, Gratias D (Eds.), Beyond Quasicrystals, Centre de Physique Les Houches, Les Editions de Physique, Springer, 1995. MR 1420414 | Zbl 0880.00009

[4] Bach V, Lieb E.H, Solovej J.P, Generalized Hartree-Fock theory and the Hubbard model, J. Stat. Phys. 76 (1994) 3-90. Zbl 0839.60095

[5] Bach V, Error bound for the Hartree-Fock energy of atoms and molecules, Comm. Math. Phys. 147 (1992) 527-548. Zbl 0771.46038

[6] Balian R, From Microphysics to Macrophysics; Methods and Applications of Statistical Physics, I, II, Springer-Verlag, 1991. MR 1129462 | Zbl 05116426

[7] Bloch F, Über die Quantenmechanik der Electronen in Kristallgittern, Z. Phys. 52 (1928) 555-560. JFM 54.0990.01

[8] Bohr H, Almost Periodic Functions, Chelsea, 1947. MR 20163

[9] Callaway J, Quantum Theory of the Solid State, Academic Press, 1974.

[10] Catto I, Le Bris C, Lions P.-L, Limite thermodynamique pour des modèles de type Thomas-Fermi [Thermodynamic limit for Thomas-Fermi type models], C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 357-364. Zbl 0849.35114

[11] Catto I, Le Bris C, Lions P.-L, Mathematical Theory of Thermodynamic Limits: Thomas-Fermi Type Models, Oxford University Press, 1998. Zbl 0938.81001

[12] Catto I, Le Bris C, Lions P.-L, Sur la limite thermodynamique pour des modèles de type Hartree et Hartree-Fock [On the thermodynamic limit for Hartree and Hartree-Fock type models], C. R. Acad. Sci. Paris Sér. I Math. 327 (1998) 259-266. Zbl 0919.35142

[13] Catto I., Le Bris C., Lions P.-L., On some periodic Hartree-type models for crystals, submitted. Also available at: http://www.math.utexas.edu/mp_arc/c/99/99-392.ps.gz.

[14] Chen L, Moody R.V, Patera J, Non-crystallographic root systems, in: Quasicrystals and Discrete Geometry (Toronto, ON, 1995), Fields Institute Monogr., 10, American mathematical society, Providence, RI, 1998, pp. 135-178. MR 1636777 | Zbl 0916.20026

[15] Conca C, Vanninathan M, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math. 57 (6) (1997) 1639-1659. MR 1484944 | Zbl 0990.35019

[16] Conca C, Planchard J, Vanninathan M, Fluids and Periodic Structures, Collection RAM, 38, Wiley/Masson, Paris, 1995. MR 1652238 | Zbl 0910.76002

[17] Eastham M.S.P, The Spectral Theory of Periodic Differential Equations, Scottish Acad. Press, Edinburgh-London, 1973. Zbl 0287.34016

[18] Figotin A, Kuchment P, Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model, SIAM J. Appl. Math. 56 (1) (1996) 68-88. MR 1372891 | Zbl 0852.35014

[19] Figotin A, Kuchment P, Band-gap structure of spectra of periodic dielectric and acoustic media. II. Two-dimensional photonic crystals, SIAM J. Appl. Math. 56 (6) (1996) 1561-1620. MR 1417473 | Zbl 0868.35009

[20] Floquet G, Sur les équations différentielles linéaires à coefficients périodiques, Ann. Ecole Norm. Sér. 2 12 (1883) 47-89. Numdam | JFM 15.0279.01 | MR 1508722

[21] Friesecke G, Pair correlations and exchange phenomena in the free electron gas, Comm. Math. Phys. 184 (1997) 143-171. MR 1462503 | Zbl 0874.60094

[22] Karpeshina Y.E, Perturbation theory for the Schrödinger operator with a periodic potential, Lecture Notes in Mathematics, 1663, Springer-Verlag, 1997. MR 1472485 | Zbl 0883.35002

[23] Kittel C, Introduction to Solid State Physics, Wiley, 1986. Zbl 0052.45506

[24] Kuchment P, Floquet Theory for Partial Differential Equations, Operator Theory Advances and Applications, 60, Birkhäuser, Basel, 1993. MR 1232660 | Zbl 0789.35002

[25] Lebowitz J.L, Lieb E.H, Existence of thermodynamics for real matter with Coulomb forces, Phys. Rev. Lett. 22 (13) (1969) 631-634.

[26] Lieb E.H, Lebowitz J.L, The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei, Adv. in Maths. 9 (1972) 316-398. MR 339751 | Zbl 1049.82501

[27] Lieb E.H, Lebowitz J.L, Lectures on the thermodynamic limit for Coulomb systems, in: Springer Lecture Notes in Physics, 20, Springer, 1973, pp. 136-161.

[28] Lieb E.H, The stability of matter: from atoms to stars, Bull. Amer. Math. Soc. 22 (1) (1990) 1-49. MR 1014510 | Zbl 0698.35135

[29] Lieb E.H, Thomas-Fermi and related theories of atoms and molecules, Rev. Modern Phys. 53 (4) (1981) 603-641, Errata: Rev. Modern Phys. 54 (1982) 311. Zbl 1049.81679

[30] Lieb E.H, A Variational principle for many-fermion systems, Phys. Rev. Lett. 46 (1981) 457-459, Errata: Rev. Modern Phys. 47 (1981) 69. MR 601336

[31] Lieb E.H, Oxford S, An improved lower bound on the indirect Coulomb energy, Int. J. Quantum Chem. 19 (1981) 427-439.

[32] Lieb E.H, Simon B, The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977) 22-116. Zbl 0938.81568

[33] Lieb E.H, Simon B, The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977) 185-194.

[34] Lieb E.H, Solovej J.P, Yngvason J, Asymptotics of heavy atoms in high magnetic fields: I. Lowest Landau band regions, Comm. Pure. Appl. Math. 47 (4) (1994) 513-591. MR 1272387 | Zbl 0800.49041

[35] Lieb E.H, Thirring W, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in: Lieb E.H, Simon B, Wightman A (Eds.), Studies in Mathematical Physics, Princeton University Press, 1976, pp. 269-303. Zbl 0342.35044

[36] Lieb E.H, Thirring W, Bounds for the kinetic energy of fermions which prove the stability of matter, Phys. Rev. Lett. 35 (1975) 687-689, Errata: Phys. Rev. Lett. 35 (1975) 1116.

[37] Lions P.-L, Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987) 33-97. Zbl 0618.35111

[38] Lions P.-L, Hartree-Fock and related equations, in: Nonlinear Partial Differential Equations and their Applications, Lect. Collège de France Seminar, Vol. IX, Paris, 1985-86, Pitman Res. Notes Math. Ser., 181, 1988, pp. 304-333. Zbl 0693.35047

[39] Lions P.-L, Paul T, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (3) (1993) 553-618. MR 1251718 | Zbl 0801.35117

[40] Madelung O, Introduction to Solid State theory, Solid State Sciences, 2, Springer-Verlag, Berlin, 1981. MR 534325

[41] Parr R.G, Yang W, Density-Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989.

[42] Pisani C, Quantum Mechanical Ab Initio Calculation of the Properties of Crystalline Materials, Lecture Notes in Chemistry, 67, Springer-Verlag, 1996.

[43] Quinn Ch.M, An introduction to the Quantum Theory of Solids, Clarendon Press, Oxford, 1973.

[44] Reed M, Simon B, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press, New-York-London, 1972. MR 493419 | Zbl 0242.46001

[45] Reed M, Simon B, Methods of Modern Mathematical Physics, IV: Analysis of Operators, Academic Press, New-York, 1978. MR 493421 | Zbl 0401.47001

[46] Ruelle D, Statistical Mechanics: Rigorous Results, Benjamin, New-York, 1969, Advanced Books Classics, Addison-Wesley, 1989. MR 289084 | Zbl 0177.57301

[47] Senechal M, Quasicrystals and Geometry, Cambridge University Press, 1995. MR 1340198 | Zbl 0828.52007

[48] Slater J.C, Quantum Theory of Molecules and Solids, Mac Graw Hill, 1963. Zbl 0115.23803

[49] Slater J.C, Symmetry and Energy Bands in Crystals, Dover, 1972.

[50] Solovej J.P, Universality in the Thomas-Fermi-von Weizsäcker model of atoms and molecules, Comm. Math. Phys. 129 (1990) 561-598. Zbl 0708.35071

[51] Solovej J.P, An improvement on stability of matter in mean field theory, in: Differential Equations and Mathematical Physics, Proceedings of the International Conference, Univ. of Alabama, Birmingham, March 1994, International Press, 1995. MR 1721316 | Zbl 0929.35131

[52] Solovej J.P, Proof of the ionization conjecture in a reduced Hartree-Fock model, Invent. Math. 104 (1991) 291-311. Zbl 0732.35066

[53] Stein E, Singular Integrals Operators and Differentiability of Functions, Princeton University Press, Princeton, 1970. MR 290095 | Zbl 0207.13501

[54] Temam R, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer-Verlag, 1988. MR 953967 | Zbl 0662.35001

[55] Tolman R.C, The Principles of Statistical Mechanics, Oxford University Press, 1962. JFM 64.0886.07

[56] Wilcox C, Theory of Bloch waves, J. Anal. Math. 33 (1978) 146-167. MR 516045 | Zbl 0408.35067

[57] Ziman J, Principles of the Theory of Solids, Cambridge University Press, 1972. MR 345569 | Zbl 0121.44801