Analysis of the Thomas-Fermi-von Weizsäcker Equation for an Infinite Atom Without Electron Repulsion

  • Elliott H. Lieb

Abstract

The equation
$$ \left\{ { - \Delta + {{\left| {\psi (x)} \right|}^{2p - 2}} - {{\left| x \right|}^{ - 1}}} \right\}\psi (x) = 0$$
in three dimensions is investigated. Uniqueness and other properties of the positive solution are proved for 3/2<p<2. There are two physical interpretations of this equation for p = 5/3 : (i) As the TFW equation for an infinite atom without electron repulsion ; (ii) The positive solution, xp, suitably scaled, is asymptotically equal to the solution of the TFW equation for an atom or molecule with electron repulsion in the regime where the nuclear charges are large and x is close to one of the nuclei.

Keywords

Kato 

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References

  1. 1.
    Adams, R.A.: Sobolev spaces. New York: Academic Press 1975MATHGoogle Scholar
  2. 2.
    Benguria, R. : The von Weizsäcker and exchange corrections in Thomas-Fermi theory. Ph. D. thesis, Princeton University 1979 (unpublished)Google Scholar
  3. 3.
    Benguria, R., Brezis, H., Lieb, E.H. : The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)MathSciNetADSCrossRefMATHGoogle Scholar
  4. 4.
    Fermi, E. : Un metodo statistico per la determinazione di alcune priorieta dell’atome. Rend. Accad. Naz. Lincei 6, 602–607 (1927)Google Scholar
  5. 5.
    Gilbarg, D., Trudinger, N. : Elliptic partial differential equations of second order. Berlin, Heidelberg. Berlin, Heidelberg, New York : Springer 1977CrossRefGoogle Scholar
  6. 6.
    Kato, T. : On the eigenfunctions of many particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–171 (1957)CrossRefMATHGoogle Scholar
  7. 7.
    Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math. 13, 135–148 (1973)CrossRefGoogle Scholar
  8. 8.
    Liberman, D., Lieb, E.H. : Numerical calculation of the Thomas-Fermi-von Weizsäcker function for an infinite atom without electron repulsion, Los Alamos National Laboratory report (in preparation)Google Scholar
  9. 9.
    Lieb, E.H. : Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Lieb, E.H., Simon, B. : The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math. 23, 22–116(1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Morrey, C.B., Jr.: Multiple integrals in the calculus of variations, Berlin, Heidelberg New York: Springer 1966MATHGoogle Scholar
  12. 12.
    Stampacchia, G.: Equations elliptiques du second ordre à coefficients discontinus. Montréal: Presses de l’Univ. 1965Google Scholar
  13. 13.
    Thomas, L.H.: The calculation of atomic fields. Proc. Camb. Phil. Soc. 23, 542–548 (1927)ADSCrossRefMATHGoogle Scholar
  14. 14.
    von Weizsäcker, C.F.: Zur Theorie der Kernmassen. Z. Phys. 96, 431–458 (1935)ADSCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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