Asymptotics of the Ground State Energy in the Relativistic Settings

  • Victor IvriiEmail author


The purpose of this paper is to derive sharp asymptotics of the ground state energy for the heavy atoms and molecules in the relativistic settings, and, in particular, to derive relativistic Scott correction term and also Dirac, Schwinger and relativistic correction terms. Also we will prove that Thomas-Fermi density approximates the actual density of the ground state, which opens the way to estimate the excessive negative and positive charges and the ionization energy.

Key words and phrases

Relativistic Schrödinger operator Heavy atoms and Molecules Thomas-Fermi theory Scott correction term Microlocal Analysis Sharp Spectral Asymptotics 

2010 Mathematics Subject Classification:

35P20 81Q10 


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  1. [Bach]
    V. Bach. Error bound for the Hartree-Fock energy of atoms and molecules. Commun. Math. Phys. 147:527–548 (1992).MathSciNetCrossRefGoogle Scholar
  2. [Dau]
    I. Daubechies. An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. 90(4):511–520 (1983).MathSciNetCrossRefGoogle Scholar
  3. [EFS1]
    L. Erdös, S. Fournais, J. P. Solovej. Scott correction for large atoms and molecules in a self-generated magnetic field. Commun. Math. Physics, 312(3):847–882 (2012).MathSciNetCrossRefGoogle Scholar
  4. [EFS2]
    L. Erdös, S. Fournais, J. P. Solovej. Relativistic Scott correction in self-generated magnetic fields. Journal of Mathematical Physics 53, 095202 (2012), 27pp.MathSciNetCrossRefGoogle Scholar
  5. [FLS]
    R. L. Frank, E. H. Lieb, R. Seiringer. Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Amer. Math. Soc. 21(4), 925–950 (2008).CrossRefGoogle Scholar
  6. [FSW]
    R. L. Frank, H. Siedentop, S. Warzel. The ground state energy of heavy atoms: relativistic lowering of the leading energy correction. Comm. Math. Phys. 278(2):549–566 (2008).MathSciNetCrossRefGoogle Scholar
  7. [GS]
    G. M. Graf, J. P Solovej. A correlation estimate with applications to quantum systems with Coulomb interactions Rev. Math. Phys., 6(5a):977–997 (1994). Reprinted in The state of matter a volume dedicated to E. H. Lieb, Advanced series in mathematical physics, 20, M. Aizenman and H. Araki (Eds.), 142–166, World Scientific (1994).Google Scholar
  8. [Herb]
    I. W. Herbst. Spectral Theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\), Commun. Math. Phys. 53(3):285–294 (1977).Google Scholar
  9. [Ivr1]
    V. Ivrii. Microlocal Analysis, Sharp Spectral Asymptotics and Applications.Google Scholar
  10. [Ivr2]
    V. Ivrii. Asymptotics of the ground state energy in the relativistic settings and with self-generated magnetic field.Google Scholar
  11. [LT]
    E. H. Lieb, W. E. Thirring, Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities, in Studies in Mathematical Physics (E. H. Lieb, B. Simon, and A. S. Wightman, eds.), Princeton Univ. Press, Princeton, New Jersey, 1976, pp. 269–303.CrossRefGoogle Scholar
  12. [LY]
    E. H. Lieb, H. T. Yau. The Stability and instability of relativistic matter. Commun. Math. Phys. 118(2): 177–213 (1988).Google Scholar
  13. [SSS]
    J. P. Solovej, T. Ø.  Sørensen, W. L. Spitzer. The relativistic Scott correction for atoms and molecules. Comm. Pure Appl. Math., 63:39–118 (2010).MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TorontoTorontoCanada

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