Stability of Coulomb Systems with Magnetic Fields

I. The One-Electron Atom
  • Jürg Fröhlich
  • Elliott H. Lieb
  • Michael Loss


The ground state energy of an atom in the presence of an external magnetic field B (with the electron spin-field interaction included) can be arbitrarily negative when B is arbitrarily large. We inquire whether stability can be restored by adding the self energy of the field, ∫ B 2. For a hydrogenic like atom we prove that there is a critical nuclear charge, z c , such that the atom is stable for z < z c , and unstable for z > z c .


Magnetic Field Vector Field External Magnetic Field Ground State Energy Sobolev Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields: III. Atoms in homogeneous magnetic field. Commun. Math. Phys. 79, 529–572 (1981)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Remark 3 in Brezis, H., Lieb, E.H.: Minimum action solution of some vector field equations. Commun. Math. Phys. 96, 97–113 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Daubechies, I., Lieb, E.H.: One-electron relativistic molecules with Coulomb interaction. Commun. Math. Phys. 90, 497–510 (1983)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Kato, T.: Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148 (1972)Google Scholar
  5. 5.
    Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74, 441–448 (1983)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Lieb, E.H., Loss, M.: Stability of Coulomb systems with magnetic fields: II. The many-electron atom and the one-electron molecule. Commun. Math. Phys. 104, 271–282 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    Lieb, E.H., Thirring, W.: Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In: Studies in mathematical physics, essays in honor of Valentine Bargmann. Lieb, E.H., Simon, B., Wightman, A.S. (eds.). Princeton, NJ: Princeton University Press 1976Google Scholar
  8. 8.
    Loss, M., Yau, H.T.: Stability of Coulomb systems with magnetic fields: III. Zero energy bound states of the Pauli operator. Commun. Math. Phys. 104, 283–290 (1986)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Michel, F.C.: Theory of pulsar magnetospheres. Rev. Mod. Phys. 54, 1–66 (1982)ADSCrossRefGoogle Scholar
  10. 10.
    Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton, NJ: Princeton University Press 1970Google Scholar
  11. 11.
    Straumann, N.: General relativity and relativistic astrophysics. Berlin, Heidelberg, New York, Tokyo: Springer 1984CrossRefGoogle Scholar
  12. 12.
    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev. A 16, 1782–1785 (1977)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    Avron, J., Herbst, I., Simon, B.: Schrödinger operators with magnetic fields: I. General Interactions. Duke Math. J. 45, 847–883 (1978)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Jürg Fröhlich
    • 1
  • Elliott H. Lieb
    • 2
  • Michael Loss
    • 2
  1. 1.Theoretical PhysicsETH-HönggerbergZürichSwitzerland
  2. 2.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

Personalised recommendations