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

  • Elliott H. Lieb


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.


Asymptotic Expansion Invariant Measure Singular Perturbation Harnack Inequality Distributional Equation 
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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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