Skip to main content
SpringerLink
Log in

On the leading correction of the Thomas-Fermi model: Lower bound

  • Published:
Inventiones mathematicae Aims and scope

Summary

We prove that the quantum mechanical ground state energy of an atom with nuclear chargeZ can be bounded from below by the sum of the Thomas-Fermi energy of the problem plusq/8Z 2 plus terms of ordero(Z 2).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Antosiewicz, H.A.: Bessel functions of fractional order. In: Abramowitz, M., Stegun, A. (eds.). Handbook of mathematical functions with formulas, graphs, and mathematical tables. Chap. 10. New York: Dover 1965, pp. 435–478

    Google Scholar 

  2. Birkhoff, G., Rota, G.-C.: Ordinary differential equations. New York: John Wiley, 3 ed., 1978

    Google Scholar 

  3. Erdélyi, A.: Asymptotic expansions. New York: Dover, 1 ed., 1956

    Google Scholar 

  4. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. Vol. 2. New York: McGraw-Hill, 1 ed., 1953

    Google Scholar 

  5. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher transcendental functions. Vol. 1. New York: McGraw-Hill, 1 ed., 1953

    Google Scholar 

  6. Hughes, W.: An atomic energy lower bound that gives Scott's correction. PhD thesis, Princeton, Department of Mathematics, 1986. See also: Adv. Math., (to appear)

    Google Scholar 

  7. Langer, R.E.: On the connection formulas and the solutions of the wave equation. Phys. Rev.51, 669–676 (1937)

    Google Scholar 

  8. Lieb, E.H.: A lower bound for Coulomb energies. Phys. Lett.70A, 444–446 (1979)

    Google Scholar 

  9. Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22–116 (1977)

    Google Scholar 

  10. Lieb, E.H., Thirring, W.E.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Lieb, E.H., Simon, B., Wightman, A.S. (eds.). Studies in Mathematical Physics: Essays in Honor of Valentine Bargmann. Princeton: Princeton University Press 1976

    Google Scholar 

  11. Siedentop, H., Weikard, R.: On the leading energy correction for the statistical model of the atom: interacting case. Commun. Math. Phys.112, 471–490 (1987)

    Google Scholar 

  12. Siedentop, H., Weikard, R.: Upper bound on the ground state energy of atoms that proves Scott's conjecture. Phys. Lett. A120, 341–342 (1987)

    Google Scholar 

  13. Siedentop, H.K.H., Weikard, R.: On the leading energy correction for the statistical model of the atom: non-interacting case. Abh. Braunschweig. Wiss. Ges. (Germany)38, 145–158 (1986)

    Google Scholar 

  14. Slater, J.C.: A simplification of the Hartree-Fock method. Phys. Rev.81, 385–390 (1951)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Carolo-Wilhelmina, Institut für Mathematische Physik, Mendelssohnstraße 3, D-3300, Braunschweig, FRG

    Heinz Siedentop, Rudi Weikard & A. M. Klaus Müller

Authors
  1. Heinz Siedentop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rudi Weikard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. M. Klaus Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Siedentop, H., Weikard, R. & Müller, A.M.K. On the leading correction of the Thomas-Fermi model: Lower bound. Invent Math 97, 159–193 (1989). https://doi.org/10.1007/BF01850659

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01850659

Keywords

Search

Navigation