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Solutions of Hartree-Fock equations for Coulomb systems. (English) Zbl 0618.35111
Author’s summary: ”This paper deals with the existence of multiple solutions of Hartree-Fock equations for Coulomb systems and related equations such as the Thomas-Fermi-Dirac-von Weizsäcker equation”.
Reviewer: P.Hillion

MSC:
35Q99 Partial differential equations of mathematical physics and other areas of application
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
46N99 Miscellaneous applications of functional analysis
81V10 Electromagnetic interaction; quantum electrodynamics
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References:
[1] Agmon, S.: Lower bounds for solutions of Schrödinger equations. J. Anal Math. 1-25 (1970) · Zbl 0211.40703
[2] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critica point theory and applications. J. Funct. Anal.14, 349-381 (1973) · Zbl 0273.49063
[3] Bader, P.: Variational method for the Hartree equation of the Helium atom. Proc. R. Soc. Edin.82A, 27-39 (1978) · Zbl 0403.45021
[4] Bahri, A.: Une méthode perturbative en théorie de Morse. Thèse d’Etat, Univ. P. et M. Curie, Paris 1981
[5] Bahri, A., Lions, P.L.: Remarques sur la théorie variationnelle des points critiques et applications. C.R. Acad. Sci. Paris301, 145-147 (1985) · Zbl 0589.58007
[6] Bazley, N., Reeken, M., Zwahlen, B.: Global properties of the minimal branch of a class of nonlinear variational problems. Math. Z.123, 301-309 (1971) · Zbl 0217.45302
[7] Bazley, N., Seydel, R.: Existence and bounds for critical energies of the Hartree operator. Chem. Phys. Lett.24, 128-132 (1974)
[8] Bazley, N., Zwahlen, B.: Estimation of the bifurcation coefficient for nonlinear eigenvalue problems. J. Appl Math. Phys.20, 281-288 (1969) · Zbl 0177.42703
[9] Bazley, N., Zwahlen, B.: A branch of positive solutions of nonlinear eigenvalue problems. Manuscr. Math.2, 365-374 (1970) · Zbl 0192.49501
[10] Benguria, R., Brézis, H., Lieb, E.H.: The Thomas-Fermi-von Weizäcker theory of atoms and molecules. Commun. Math. Phys.79, 167-180 (1981) · Zbl 0478.49035
[11] Berestycki, H., Lions, P.L.: Nonlinear scalar field equations. Arch. Rat. Mech. Anal.82, 313-345 and 347-375 (1983) · Zbl 0533.35029
[12] Berestycki, H., Taubes, C.: In preparation
[13] Bethe, H., Jackiw, R.: Intermediate quantum mechanics. New York: Benjamin 1969
[14] Bongers. A.: Behandlung verallgemeinerter indlineover Eigenwertprobleme und Ljusternik-Schnirelman Theorie. Univ. Mainz 1979 · Zbl 0445.34006
[15] Bourgain, J.: La propriété de Radon-Nikodym. Cours de 3e cycle polypié no 36, Univ. P. et M. Curie, Paris 1979
[16] Brézis, H., Coron, J. M.: Convergence of solutions of H systems or how to blow bubbles. Arch. Rat. Mech. Anal. (to appear) · Zbl 0584.49024
[17] Brézis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. (1983) · Zbl 0526.46037
[18] Brézis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal., Theory Methods Appl.10, 55-64 (1986) · Zbl 0593.35045
[19] Cazenave, T., Lions, P.L.: Orbital stability of standing waves for some nonlinear Schrödinger equations. Commun. Math. Phys.85, 549-561 (1982) · Zbl 0513.35007
[20] Clarke, D.C.: A variant of the Ljusternik-Schnirelman theory. Indiana Univ. Math. J.22, 65-74 (1972) · Zbl 0228.58006
[21] Coffman, C.V.: Ljusternik-Schnirelman theory: Complementary principles and the Morse index. Preprint · Zbl 0661.58008
[22] Ekeland, I.: Nonconvex minimization problems. Bull. Am. Math. Soc.1, 443-474 (1979) · Zbl 0441.49011
[23] Ekeland, I., Lebourg, G.: Generic Frechet-differentiability and perturbed optimization problems in Banach spaces. Trans. Am. Math. Soc.224, 193-216 (1976) · Zbl 0313.46017
[24] Fock, V.: Näherungsmethode zur Lösung des quantenmechanischen Mehrkörperproblems. Z. Phys.61, 126-148 (1930) · JFM 56.1313.08
[25] Gogny, D., Lions, P.L.: Hartree-Fock theory in Nuclear Physics. RAIRO M2AN, 1986 · Zbl 0607.35078
[26] Gustafson, K., Sather, D.: A branching analysis of the Hartree equation. Rend. Mat. Appl.4, 723-734 (1971)
[27] Hartree, D.: The wave mechanics of anatom with a non-coulomb central field. Part I. Theory and methods. Proc. Camb. Phil. Soc.24, 89-312 (1928)
[28] Hartree, D.: The calculation of atomic structures. New York: Wiley 1957 · Zbl 0079.21401
[29] Krasnosel’skii, M.A.: Topological methods in the theory of nonlinear integral equations. New York: Mac Millan 1964
[30] Léon, J.F.: work in preparation
[31] Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53, 603-641 (1981) · Zbl 1114.81336
[32] Lieb, E.H.: Statistical theories of large atoms and molecules. Comment. At. Mol. Phys.11, 147-155 (1982)
[33] Lieb, E.H.: Thomas-Fermi and Hartree-Fock theory. In Proceedings of the International Congress of Mathematicians, Vancouver, Vol. 2, pp. 383-386
[34] Lieb, E.H.: Bound on the maximum negative ionization of atoms and molecules. Phys. Rev. A.29, 3018-3028 (1984)
[35] Lieb, E.H.: Analysis of TFW equation for an infinite atom without electron repulsion. Commun. Math. Phys.85, 15-25 (1982) · Zbl 0514.35074
[36] Lieb, E.H., Liberman, D.A.: Numerical calculation of the TFW function for an infinite atom without electron repulsion. Los Alamos report No. LA-9186-MS (1982)
[37] Lieb, E.H., Simon, B.: The Hartree-Fock theory for Coulomb systems. Commun. Math. Phys.53, 185-194 (1977)
[38] Lieb, E.H., Simon, B.: On solutions to the Hartree-Fock problem for atoms and molecules. J. Chem. Phys.61, 735-736 (1974)
[39] Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22-116 (1977) · Zbl 0938.81568
[40] Lions, P.L.: Sur l’existence d’états excités dans la théorie de Hartree-Fock. C.R. Acad. Sci. Paris294, 377-379 (1982) · Zbl 0488.35072
[41] Lions, P.L.: Some remarks on Hartree equations. Nonlinear Anal., Theory Methods Appl.5, 1245-1256 (1981) · Zbl 0472.35074
[42] Lions, P.L.: Compactness and topological methods for some nonlinear variational problems of Mathematical Physics. In: Nonlinear problems, present and future. Bishop, A., Campbell, D., Nicolaenko, B. (eds.) Amsterdam: North-Holland 1982 · Zbl 0496.35079
[43] Lions, P.L.: The concentration-compactness principle in the Calculus of Variations. The locally compact case. Ann. Inst. H. Poincaré1, 109-145 and 223-283 (1984). See also C.R. Acad. Sci. Paris294, 261-264 (1982) · Zbl 0541.49009
[44] Lions, P.L.: On the concentration-compactness principle. In: Contributions to nonlinear partial differential equations. London: Pitman 1983
[45] Lions, P.L.: The concentration-compactness principle in the Calculus of Variations. The limit case. Riv. Mat. Iberoam.1, 145-201 and 45-121 (1985) · Zbl 0704.49005
[46] Lions, P.L.: Symmetries and the concentration-compactness method. In: Nonlinear variational problems. London: Pitman 1987 · Zbl 0829.49010
[47] Ljusternik, L.A., Schnirelman, L.G.: Topological methods in the calculus of variations. Paris: Herman 1934
[48] Palais, R.S.: Ljusternik-Schnirelman theory on Banach manifolds. Topology5, 115-132 (1966) · Zbl 0143.35203
[49] Palais, R.S.: Critical point theory and the minimax principle. Proc. Symp. Pure Math.15, Providence, R.I.: A.M.S. 185-212 (1970) · Zbl 0212.28902
[50] Rabinowitz, P.H.: Variational methods for nonlinear eigenvalue problems, In Eigenvalues of nonlinear problems, C.I.M.E., Rome, Ediz. Cremonese (1974) · Zbl 0278.35040
[51] Reeken, M.: General theorem on bifurcation and its application to the Hartree equation of the Helium atom. J. Math. Phys.11, 2505-2512 (1970)
[52] Sacks, J., Uhlenbeck, K.: The existence of minimal 2-spheres. Ann. Math.113, 1-24 (1981) · Zbl 0462.58014
[53] Schaeffer, H.F. III: The electronic structure of atoms and molecules. Reading, MA: Addison-Wesley 1972
[54] Slater, J.C.: A note on Hartree’s method. Phys. Rev.35, 210-211 (1930)
[55] Slater, J.C.: Quantum theory of atomic structure, Vol. I. New York: McGraw-Hill 1960 · Zbl 0094.44702
[56] Stegall, C.: Optimization of functions on certain subsets of Banach spaces. Math. Anal.236, 171-176 (1978) · Zbl 0379.49008
[57] Struwe, M.: A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. (1985) · Zbl 0535.35025
[58] Stuart, C.: Existence theory for the Hartree equation. Arch. Rat. Mech. Anal.51, 60-69 (1973) · Zbl 0287.34032
[59] Stuart, C.: An example in nonlinear functional analysis: the Hartree equation. J. Math. Anal. Appl.49, 725-733 (1975) · Zbl 0311.47032
[60] Taubes, C.: The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on ?3. I. Commun. Math. Phys.86, p. 257-298 (1982) · Zbl 0514.58016
[61] Taubes, C.: The existence of a non-minimal solution to the SU (2) Yang-Mills-Higgs equations on ?3. II. Commun. Math. Phys.86, 299-320 (1982) · Zbl 0514.58017
[62] Uhlenbeck, K.: Removable singularities in Yang-Mills fields. Commun. Math. Phys.83, 11-29 (1982) · Zbl 0491.58032
[63] Uhlenbeck, K.: Connections withL p bounds on curvature. Commun. Math. Phys.83, 31-42 (1982) · Zbl 0499.58019
[64] Viterbo, C.: Indice de Morse des points critiques obtenus par minimax (In preparation) · Zbl 0695.58007
[65] Wolkowisky, J.H.: Existence of solutions of the Hartree equations forN electrons. An application of the Schauder-Tychonoff theorem. Indiana Univ. Math. J.22, 551-568 (1972) · Zbl 0237.34006
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