On some periodic Hartree-type models for crystals
Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 2, p. 143-190
@article{AIHPC_2002__19_2_143_0,
     author = {Catto, I. and Le Bris, Claude and Lions, Pierre-Louis},
     title = {On some periodic Hartree-type models for crystals},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {19},
     number = {2},
     year = {2002},
     pages = {143-190},
     zbl = {1005.81101},
     mrnumber = {1902742},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2002__19_2_143_0}
}
Catto, I.; Le Bris, C.; Lions, P.-L. On some periodic Hartree-type models for crystals. Annales de l'I.H.P. Analyse non linéaire, Volume 19 (2002) no. 2, pp. 143-190. http://www.numdam.org/item/AIHPC_2002__19_2_143_0/

[1] Amerio L., Prouse G., Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold, 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] Balian R., From Microphysics to Macrophysics; Methods and Applications of Statistical Physics, I & II, Springer-Verlag, 1991. | MR 1129462 | Zbl 05116426

[5] Benguria R., Brézis H., Lieb E.H., The Thomas-Fermi-von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 (1981) 167-180. | MR 612246 | Zbl 0478.49035

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

[7] Buffoni B., Jeanjean L., Stuart C.A., Existence of a non-trivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179-186. | MR 1145940 | Zbl 0789.35052

[8] Buffoni B., Jeanjean L., Minimax characterization of solutions for a semi-linear elliptic equation with lack of compactness, Ann. Inst. Henri Poincaré, Anal. non lin. 10 (4) (1993) 377-404. | Numdam | MR 1246458 | Zbl 0828.35013

[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, C. R. Acad. Sci. Paris, Série I 322 (1996) 357-364. | MR 1378513 | 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. | MR 1673212 | 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, C. R. Acad. Sci. Paris, Série I 327 (1998) 259-266. | MR 1650265 | Zbl 0919.35142

[13] Catto I., Le Bris C., Lions P.-L., On the thermodynamic limit for Hartree-Fock type models, Ann. Inst. Henri Poincaré, to appear. | Numdam | Zbl 0994.35115

[14] Dreizler R.M., Gross E.K.U., Density Functional Theory, Springer-Verlag, 1990. | Zbl 0723.70002

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

[16] Ekeland I., Nonconvex minimization problems, Bull. Amer. Math. Soc. 1 (3) (1979) 443-474. | MR 526967 | Zbl 0441.49011

[17] Fefferman C., The thermodynamic limit for a crystal, Comm. Math. Phys. 98 (1985) 289-311. | MR 788776 | Zbl 0603.35079

[18] Gregg J.N., The existence of the thermodynamic limit in Coulomb-like systems, Comm. Math. Phys. 123 (1989) 255-276. | MR 1002039 | Zbl 0676.60097

[19] Hartree D., The wave-mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods, Proc. Comb. Phil. Soc. 24 (1928) 89-132. | JFM 54.0966.05

[20] Heinz H.-P., Küpper T., Stuart C.A., Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation, J. Differential Equations 100 (1992) 341-354. | MR 1194814 | Zbl 0767.35006

[21] Jeanjean L., Solutions in the spectral gap for a nonlinear equation of Schrödinger type, J. Differential Equations 112 (1994) 53-80. | MR 1287552 | Zbl 0804.35033

[22] Jeanjean L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Analysis TMA 28 (10) (1997) 1633-1659. | MR 1430506 | Zbl 0877.35091

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

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

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

[26] Lieb E.H., Lebowitz J.L., Lectures on the thermodynamic limit for Coulomb systems, in: Lecture Notes in Physics, 20, Springer, 1973, pp. 136-161. | MR 395513

[27] Lieb E.H., The stability of matter, Rev. Mod. Phys. 48 (1976) 553-569. | MR 456083

[28] Lieb E.H., Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math. 57 (1977) 93-105. | MR 471785 | Zbl 0369.35022

[29] Lieb E.H., Thomas-Fermi and related theories of atoms and molecules, Rev. Mod. Phys. 53 (4) (1981) 603-641. | MR 629207 | Zbl 1049.81679

[30] 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

[31] Lieb E.H., Narnhofer H., The thermodynamic limit for Jellium, J. Stat. Phys. 12 (1975) 291-310. | MR 401029 | Zbl 0973.82500

[32] Lieb E.H., Simon B., The Thomas-Fermi theory of atoms, molecules and solids, Adv. Math. 23 (1977) 22-116. | MR 428944 | Zbl 0938.81568

[33] Lieb E.H., Simon B., The Hartree-Fock theory for Coulomb systems, Comm. Math. Phys. 53 (1977) 185-194. | MR 452286

[34] Lions P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, Parts 1 & 2, Ann. Inst. H. Poincaré 1 (1984) 109-145, and 223-283. | Numdam | Numdam | MR 778970 | Zbl 0704.49004

[35] Lions P.-L., Solutions of Hartree-Fock equations for Coulomb systems, Comm. Math. Phys. 109 (1987) 33-97. | MR 879032 | Zbl 0618.35111

[36] Lopes O., A constrained minimization problem with integrals on the entire space, Bol. Soc. Bras. Mat., Nova Ser. 25 (1) (1994) 77-92. | MR 1274763 | Zbl 0805.49005

[37] Lopes O., Sufficient conditions for minima of some translation invariant functionals, Differential Integral Equations 10 (2) (1997) 231-244. | MR 1424809 | Zbl 0891.49001

[38] Lopes O., Variational problems defined by integrals on the entire space and periodic coefficients, Comm. Appl. Nonlinear Anal. 5 (2) (1998) 87-120. | MR 1621231 | Zbl 1108.49300

[39] Madelung O., Introduction to Solid-State Theory, Solid State Sciences, Vol. 2, Springer, 1981. | MR 534325

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

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

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

[43] Reed M., Simon B., Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1978. | MR 751959 | Zbl 0401.47001

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

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

[46] Simon B., Schrödinger semi-groups, Bull. Amer. Math. Soc. 7 (3) (1982) 447-526. | Zbl 0524.35002

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

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

[49] Solovej J.P., An improvement on stability of matter in mean field theory, Proceedings of the Conference on PDEs and Mathematical Physics, Univ. of Alabama, International Press, 1994. | MR 1721316 | Zbl 0929.35131

[50] Stuart Ch., Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995. | MR 1361485 | Zbl 0864.47037

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

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

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