Abstract
The equation
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, ψ, 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.
Work partially supported by U.S. National Science Foundation grant PHY-7825390 A02
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Adams, R.A.: Sobolev spaces. New York: Academic Press 1975
Benguria, R.: The von Weizsäcker and exchange corrections in Thomas-Fermi theory. Ph. D. thesis, Princeton University 1979 (unpublished)
Benguria, R., Brezis, H., Lieb, E.H.: The Thomas-Fermi-von Weizsäcker theory of atoms and molecules. Commun. Math. Phys. 79, 167–180 (1981)
Fermi, E.: Un metodo statistico per la determinazione di alcune priorieta dell’atome. Rend. Accad. Naz. Lincei 6, 602–607 (1927)
Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977
Kato, T.: On the eigenfunctions of many particle systems in quantum mechanics. Commun. Pure Appl. Math. 10, 151–171 (1957)
Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math. 13, 135–148 (1973)
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)
Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys. 53, 603–641 (1981)
Lieb, E.H., Simon, B.: The Thomas-Fermi theory of atoms, molecules, and solids. Adv. Math. 23, 22–116 (1977)
Morrey, C.B., Jr.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966
Stampacchia, G.: Equations elliptiques du second ordre à coefficients discontinus. Montréal: Presses de l’Univ. 1965
Thomas, L.H.: The calculation of atomic fields. Proc. Camb. Phil. Soc. 23, 542–548 (1927)
von Weizsäcker, C.F.: Zur Theorie der Kernmassen. Z. Phys. 96, 431–458 (1935)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg New York
About this chapter
Cite this chapter
Lieb, E.H. (2005). Analysis of the Thomas-Fermi-von Weizsäcker Equation for an Infinite Atom Without Electron Repulsion. In: Thirring, W. (eds) The Stability of Matter: From Atoms to Stars. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-27056-6_26
Download citation
DOI: https://doi.org/10.1007/3-540-27056-6_26
Received:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22212-5
Online ISBN: 978-3-540-27056-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)