On some periodic Hartree-type models for crystals

I. Catto; C. Le Bris; P.-L. Lions

Annales de l'I.H.P. Analyse non linéaire (2002)

  • Volume: 19, Issue: 2, page 143-190
  • ISSN: 0294-1449

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Catto, I., Le Bris, C., and Lions, P.-L.. "On some periodic Hartree-type models for crystals." Annales de l'I.H.P. Analyse non linéaire 19.2 (2002): 143-190. <http://eudml.org/doc/78542>.

@article{Catto2002,
author = {Catto, I., Le Bris, C., Lions, P.-L.},
journal = {Annales de l'I.H.P. Analyse non linéaire},
keywords = {thermodynamic limit; quantum chemistry},
language = {eng},
number = {2},
pages = {143-190},
publisher = {Elsevier},
title = {On some periodic Hartree-type models for crystals},
url = {http://eudml.org/doc/78542},
volume = {19},
year = {2002},
}

TY - JOUR
AU - Catto, I.
AU - Le Bris, C.
AU - Lions, P.-L.
TI - On some periodic Hartree-type models for crystals
JO - Annales de l'I.H.P. Analyse non linéaire
PY - 2002
PB - Elsevier
VL - 19
IS - 2
SP - 143
EP - 190
LA - eng
KW - thermodynamic limit; quantum chemistry
UR - http://eudml.org/doc/78542
ER -

References

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  1. [1] Amerio L., Prouse G., Almost Periodic Functions and Functional Equations, Van Nostrand Reinhold, 1971. Zbl0215.15701MR275061
  2. [2] Ashcroft N.W., Mermin N.D., Solid-state Physics, Saunders College Publishing, 1976. 
  3. [3] Axel F., Gratias D. (Eds.), Beyond Quasicrystals, Centre de Physique Les Houches, Les Editions de Physique, Springer, 1995. Zbl0880.00009MR1420414
  4. [4] Balian R., From Microphysics to Macrophysics; Methods and Applications of Statistical Physics, I & II, Springer-Verlag, 1991. Zbl1188.82001MR1129462
  5. [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. Zbl0478.49035MR612246
  6. [6] Bohr H., Almost Periodic Functions, Chelsea, 1947. MR20163
  7. [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. Zbl0789.35052MR1145940
  8. [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. Zbl0828.35013MR1246458
  9. [9] Callaway J., Quantum Theory of the Solid State, Academic Press, 1974. 
  10. [10] Catto I., Le Bris C., Lions P.-L., Limite thermodynamique pour des modèles de type Thomas–Fermi, C. R. Acad. Sci. Paris, Série I322 (1996) 357-364. Zbl0849.35114MR1378513
  11. [11] Catto I., Le Bris C., Lions P.-L., Mathematical Theory of Thermodynamic Limits: Thomas–Fermi Type Models, Oxford University Press, 1998. Zbl0938.81001MR1673212
  12. [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 I327 (1998) 259-266. Zbl0919.35142MR1650265
  13. [13] Catto I., Le Bris C., Lions P.-L., On the thermodynamic limit for Hartree–Fock type models, Ann. Inst. Henri Poincaré, to appear. Zbl0994.35115
  14. [14] Dreizler R.M., Gross E.K.U., Density Functional Theory, Springer-Verlag, 1990. Zbl0723.70002
  15. [15] Eastham M.S.P., The Spectral Theory of Periodic Differential Equations, Scottish Acad. Press, Edinburgh, 1973. Zbl0287.34016
  16. [16] Ekeland I., Nonconvex minimization problems, Bull. Amer. Math. Soc.1 (3) (1979) 443-474. Zbl0441.49011MR526967
  17. [17] Fefferman C., The thermodynamic limit for a crystal, Comm. Math. Phys.98 (1985) 289-311. Zbl0603.35079MR788776
  18. [18] Gregg J.N., The existence of the thermodynamic limit in Coulomb-like systems, Comm. Math. Phys.123 (1989) 255-276. Zbl0676.60097MR1002039
  19. [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. Zbl54.0966.05JFM54.0966.05
  20. [20] Heinz H.-P., Küpper T., Stuart C.A., Existence and bifurcation of solutions for nonlinear perturbations of the periodic Schrödinger equation, J. Differential Equations100 (1992) 341-354. Zbl0767.35006MR1194814
  21. [21] Jeanjean L., Solutions in the spectral gap for a nonlinear equation of Schrödinger type, J. Differential Equations112 (1994) 53-80. Zbl0804.35033MR1287552
  22. [22] Jeanjean L., Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Analysis TMA28 (10) (1997) 1633-1659. Zbl0877.35091MR1430506
  23. [23] Kittel C., Introduction to Solid-State Physics, Wiley, 1986. Zbl0052.45506
  24. [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. [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. Zbl1049.82501MR339751
  26. [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. MR395513
  27. [27] Lieb E.H., The stability of matter, Rev. Mod. Phys.48 (1976) 553-569. MR456083
  28. [28] Lieb E.H., Existence and uniqueness of the minimizing solution of Choquard's nonlinear equation, Studies in Appl. Math.57 (1977) 93-105. Zbl0369.35022MR471785
  29. [29] Lieb E.H., Thomas–Fermi and related theories of atoms and molecules, Rev. Mod. Phys.53 (4) (1981) 603-641. Zbl1049.81679MR629207
  30. [30] Lieb E.H., The stability of matter: from atoms to stars, Bull. Amer. Math. Soc.22 (1) (1990) 1-49. Zbl0698.35135MR1014510
  31. [31] Lieb E.H., Narnhofer H., The thermodynamic limit for Jellium, J. Stat. Phys.12 (1975) 291-310. Zbl0973.82500MR401029
  32. [32] Lieb E.H., Simon B., The Thomas–Fermi theory of atoms, molecules and solids, Adv. Math.23 (1977) 22-116. Zbl0938.81568MR428944
  33. [33] Lieb E.H., Simon B., The Hartree–Fock theory for Coulomb systems, Comm. Math. Phys.53 (1977) 185-194. MR452286
  34. [34] Lions P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case, Parts 1 & 2, Ann. Inst. H. Poincaré1 (1984) 109-145, and 223–283. Zbl0704.49004MR778970
  35. [35] Lions P.-L., Solutions of Hartree–Fock equations for Coulomb systems, Comm. Math. Phys.109 (1987) 33-97. Zbl0618.35111MR879032
  36. [36] Lopes O., A constrained minimization problem with integrals on the entire space, Bol. Soc. Bras. Mat., Nova Ser.25 (1) (1994) 77-92. Zbl0805.49005MR1274763
  37. [37] Lopes O., Sufficient conditions for minima of some translation invariant functionals, Differential Integral Equations10 (2) (1997) 231-244. Zbl0891.49001MR1424809
  38. [38] Lopes O., Variational problems defined by integrals on the entire space and periodic coefficients, Comm. Appl. Nonlinear Anal.5 (2) (1998) 87-120. Zbl1108.49300MR1621231
  39. [39] Madelung O., Introduction to Solid-State Theory, Solid State Sciences, Vol. 2, Springer, 1981. MR534325
  40. [40] Pisani C., Quantum Mechanical Ab Initio Calculation of the Properties of Crystalline Materials, Lecture Notes in Chemistry, 67, Springer, 1996. 
  41. [41] Parr R.G., Yang W., Density-Functional Theory of Atoms and Molecules, Oxford University Press, Oxford, 1989. 
  42. [42] Quinn Ch.M., An Introduction to the Quantum Theory of Solids, Clarendon Press, Oxford, 1973. 
  43. [43] Reed M., Simon B., Methods of Modern Mathematical Physics, IV, Academic Press, New York, 1978. Zbl0401.47001MR751959
  44. [44] Ruelle D., Statistical Mechanics: Rigorous Results, Benjamin, New York, 1969, and Advanced Books Classics, Addison-Wesley, 1989. Zbl0177.57301MR289084
  45. [45] Senechal M., Quasicrystals and Geometry, Cambridge University Press, 1995. Zbl0828.52007MR1340198
  46. [46] Simon B., Schrödinger semi-groups, Bull. Amer. Math. Soc.7 (3) (1982) 447-526. Zbl0524.35002
  47. [47] Slater J.C., Quantum Theory of Molecules and Solids, Mac Graw Hill, 1963. Zbl0115.23803
  48. [48] Slater J.C., Symmetry and Energy Bands in Crystals, Dover, 1972. 
  49. [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. Zbl0929.35131MR1721316
  50. [50] Stuart Ch., Bifurcation into Spectral Gaps, Bulletin of the Belgian Mathematical Society, 1995. Zbl0864.47037MR1361485
  51. [51] Tolman R.C., The Principles of Statistical Mechanics, Oxford University Press, 1962. Zbl1203.82001JFM64.0886.07
  52. [52] Wilcox C., Theory of Bloch waves, J. Analyse Math.33 (1978) 146-167. Zbl0408.35067MR516045
  53. [53] Ziman J., Principles of the Theory of Solids, Cambridge University Press, 1972. Zbl0121.44801MR345569

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