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Analysis of the Thomas-Fermi-von Weizsäcker equation for an infinite atom without electron repulsion

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Abstract

The equation

$$\left\{ { - \Delta + |\psi (x)|^{2p - 2} - |x|^{ - 1} } \right\}\psi (x) = 0$$

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 forp=5/3: (i) As the TFW equation for an infinite atomwithout electron repulsion; (ii) The positive solution, ψ, suitably scaled, is asymptotically equal to the solution of the TFW equation for an atom or moleculewith electron repulsion in the regime where the nuclear charges are large andx is close to one of the nuclei.

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References

  1. Adams, R.A.: Sobolev spaces. New York: Academic Press 1975

    MATH  Google Scholar 

  2. Benguria, R.: The von Weizsäcker and exchange corrections in Thomas-Fermi theory. Ph. D. thesis, Princeton University 1979 (unpublished)

  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)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Fermi, E.: Un metodo statistico per la determinazione di alcune priorieta dell'atome. Rend. Accad. Naz. Lincei6, 602–607 (1927)

    Google Scholar 

  5. Gilbarg, D., Trudinger, N.: Elliptic partial differential equations of second order. Berlin, Heidelberg, New York: Springer 1977

    Book  MATH  Google Scholar 

  6. Kato, T.: On the eigenfunctions of many particle systems in quantum mechanics. Commun. Pure Appl. Math.10, 151–171 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  7. Kato, T.: Schrödinger operators with singular potentials. Isr. J. Math.13, 135–148 (1973)

    Article  MATH  Google Scholar 

  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)

  9. Lieb, E.H.: Thomas-Fermi and related theories of atoms and molecules. Rev. Mod. Phys.53, 603–641 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Morrey, C.B., Jr.: Multiple integrals in the calculus of variations. Berlin, Heidelberg, New York: Springer 1966

    MATH  Google Scholar 

  12. Stampacchia, G.: Equations elliptiques du second ordre à coefficients discontinus. Montréal: Presses de l'Univ. 1965

    MATH  Google Scholar 

  13. Thomas, L.H.: The calculation of atomic fields. Proc. Camb. Phil. Soc.23, 542–548 (1927)

    Article  ADS  MATH  Google Scholar 

  14. von Weizsäcker, C.F.: Zur Theorie der Kernmassen. Z. Phys.96, 431–458 (1935)

    Article  ADS  MATH  Google Scholar 

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Authors and Affiliations

  1. Departments of Mathematics and Physics, Princeton University, P.O.B. 708, 08544, Princeton, NJ, USA

    Elliott H. Lieb

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  1. Elliott H. Lieb
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Communicated by R. Jost

Work partially supported by U.S. National Science Foundation grant PHY-7825390 A02

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Lieb, E.H. Analysis of the Thomas-Fermi-von Weizsäcker equation for an infinite atom without electron repulsion. Commun.Math. Phys. 85, 15–25 (1982). https://doi.org/10.1007/BF02029130

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