×

zbMATH — the first resource for mathematics

On some periodic Hartree-type models for crystals. (English) Zbl 1005.81101
Summary: We continue here our study of the thermodynamic limit for various models of quantum chemistry. More specifically, we study the Hartree and the restricted Hartree model. For the restricted Hartree model, we prove the existence of the thermodynamic limit for the energy per unit volume. We also define a periodic problem associated to the Hartree model, and we prove that it is well-posed.

MSC:
81V70 Many-body theory; quantum Hall effect
82D25 Statistical mechanics of crystals
PDF BibTeX XML Cite
Full Text: DOI Numdam EuDML
References:
[1] Amerio, L.; Prouse, G., Almost periodic functions and functional equations, (1971), Van Nostrand Reinhold · Zbl 0215.15701
[2] Ashcroft, N.W.; Mermin, N.D., Solid-state physics, (1976), Saunders College Publishing · Zbl 1118.82001
[3] ()
[4] Balian, R., From microphysics to macrophysics; methods and applications of statistical physics, I & II, (1991), Springer-Verlag · Zbl 1131.82300
[5] Benguria, R.; Brézis, H.; Lieb, E.H., The thomas – fermi – von Weizsäcker theory of atoms and molecules, Comm. math. phys., 79, 167-180, (1981) · Zbl 0478.49035
[6] Bohr, H., Almost periodic functions, (1947), Chelsea
[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, 179-186, (1993) · 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, 377-404, (1993) · Zbl 0828.35013
[9] Callaway, J., Quantum theory of the solid state, (1974), Academic Press
[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, 357-364, (1996) · Zbl 0849.35114
[11] Catto, I.; Le Bris, C.; Lions, P.-L., Mathematical theory of thermodynamic limits: thomas – fermi type models, (1998), Oxford University Press · 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, 259-266, (1998) · 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 · Zbl 0994.35115
[14] Dreizler, R.M.; Gross, E.K.U., Density functional theory, (1990), Springer-Verlag
[15] Eastham, M.S.P., The spectral theory of periodic differential equations, (1973), Scottish Acad. Press Edinburgh · Zbl 0287.34016
[16] Ekeland, I., Nonconvex minimization problems, Bull. amer. math. soc., 1, 3, 443-474, (1979) · Zbl 0441.49011
[17] Fefferman, C., The thermodynamic limit for a crystal, Comm. math. phys., 98, 289-311, (1985) · Zbl 0603.35079
[18] Gregg, J.N., The existence of the thermodynamic limit in Coulomb-like systems, Comm. math. phys., 123, 255-276, (1989) · 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, 89-132, (1928) · 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, 341-354, (1992) · Zbl 0767.35006
[21] Jeanjean, L., Solutions in the spectral gap for a nonlinear equation of Schrödinger type, J. differential equations, 112, 53-80, (1994) · Zbl 0804.35033
[22] Jeanjean, L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear analysis TMA, 28, 10, 1633-1659, (1997) · Zbl 0877.35091
[23] Kittel, C., Introduction to solid-state physics, (1986), Wiley
[24] Lebowitz, J.L.; Lieb, E.H., Existence of thermodynamics for real matter with Coulomb forces, Phys. rev. lett., 22, 13, 631-634, (1969)
[25] Lieb, E.H.; Lebowitz, J.L., The constitution of matter: existence of thermodynamics for systems composed of electrons and nuclei, Adv. math., 9, 316-398, (1972) · Zbl 1049.82501
[26] Lieb, E.H.; Lebowitz, J.L., Lectures on the thermodynamic limit for Coulomb systems, (), 136-161
[27] Lieb, E.H., The stability of matter, Rev. mod. phys., 48, 553-569, (1976)
[28] Lieb, E.H., Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in appl. math., 57, 93-105, (1977) · Zbl 0369.35022
[29] Lieb, E.H., Thomas – fermi and related theories of atoms and molecules, Rev. mod. phys., 53, 4, 603-641, (1981) · Zbl 1114.81336
[30] Lieb, E.H., The stability of matter: from atoms to stars, Bull. amer. math. soc., 22, 1, 1-49, (1990) · Zbl 0698.35135
[31] Lieb, E.H.; Narnhofer, H., The thermodynamic limit for jellium, J. stat. phys., 12, 291-310, (1975) · Zbl 0973.82500
[32] Lieb, E.H.; Simon, B., The thomas – fermi theory of atoms, molecules and solids, Adv. math., 23, 22-116, (1977) · Zbl 0938.81568
[33] Lieb, E.H.; Simon, B., The hartree – fock theory for Coulomb systems, Comm. math. phys., 53, 185-194, (1977)
[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, 109-145, (1984), and 223-283 · Zbl 0541.49009
[35] Lions, P.-L., Solutions of hartree – fock equations for Coulomb systems, Comm. math. phys., 109, 33-97, (1987) · Zbl 0618.35111
[36] Lopes, O., A constrained minimization problem with integrals on the entire space, Bol. soc. bras. mat., nova ser., 25, 1, 77-92, (1994) · Zbl 0805.49005
[37] Lopes, O., Sufficient conditions for minima of some translation invariant functionals, Differential integral equations, 10, 2, 231-244, (1997) · Zbl 0891.49001
[38] Lopes, O., Variational problems defined by integrals on the entire space and periodic coefficients, Comm. appl. nonlinear anal., 5, 2, 87-120, (1998) · Zbl 1108.49300
[39] Madelung, O., Introduction to solid-state theory, solid state sciences, vol. 2, (1981), Springer
[40] Pisani, C., Quantum mechanical ab initio calculation of the properties of crystalline materials, Lecture notes in chemistry, 67, (1996), Springer
[41] Parr, R.G.; Yang, W., Density-functional theory of atoms and molecules, (1989), Oxford University Press Oxford
[42] Quinn, Ch.M., An introduction to the quantum theory of solids, (1973), Clarendon Press Oxford
[43] Reed, M.; Simon, B., Methods of modern mathematical physics, IV, (1978), Academic Press New York
[44] Ruelle, D., Statistical mechanics: rigorous results, (1969), Benjamin New York, and Advanced Books Classics, Addison-Wesley, 1989 · Zbl 0177.57301
[45] Senechal, M., Quasicrystals and geometry, (1995), Cambridge University Press · Zbl 0828.52007
[46] Simon, B., Schrödinger semi-groups, Bull. amer. math. soc., 7, 3, 447-526, (1982) · Zbl 0524.35002
[47] Slater, J.C., Quantum theory of molecules and solids, (1963), Mac Graw Hill · Zbl 0115.23803
[48] Slater, J.C., Symmetry and energy bands in crystals, (1972), Dover · Zbl 0164.57001
[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, (1994), International Press
[50] Stuart, Ch., Bifurcation into spectral gaps, Bulletin of the Belgian mathematical society, (1995)
[51] Tolman, R.C., The principles of statistical mechanics, (1962), Oxford University Press · JFM 64.0886.07
[52] Wilcox, C., Theory of Bloch waves, J. analyse math., 33, 146-167, (1978) · Zbl 0408.35067
[53] Ziman, J., Principles of the theory of solids, (1972), Cambridge University Press · Zbl 0121.44801
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.