Annales Henri Poincaré

, Volume 13, Issue 2, pp 333–362 | Cite as

Asymptotics for Two-Dimensional Atoms

  • Phan Thanh Nam
  • Fabian Portmann
  • Jan Philip Solovej


We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z > 0 and N quantum electrons of charge −1 is \({E(N,Z)=-\frac{1}{2}Z^2{\rm ln} Z+(E^{\rm TF}(\lambda)+\frac{1}{2}c^{\rm H})Z^2+o(Z^2)}\) when Z → ∞ and N/Z → λ, where ETF(λ) is given by a Thomas–Fermi type variational problem and cH ≈ −2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z → ∞, which is contrary to the expected behavior of three-dimensional atoms.


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Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Phan Thanh Nam
    • 1
  • Fabian Portmann
    • 2
  • Jan Philip Solovej
    • 1
  1. 1.Department of Mathematical SciencesUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of MathematicsRoyal Institute of TechnologyStockholmSweden

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