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Communications in Mathematical Physics

, Volume 90, Issue 4, pp 497–510 | Cite as

One-electron relativistic molecules with Coulomb interaction

  • Ingrid Daubechies
  • Elliott H. Lieb
Article

Abstract

As an approximation to a relativistic one-electron molecule, we study the operator\(H = ( - \Delta + m^2 )^{1/2} - e^2 \sum\limits_{j = 1}^K {Z_j } |x - R_j |^{ - 1}\) withZ j ≧0,e−2=137.04.H is bounded below if and only ife2Z j ≦2/π allj. Assuming this condition, the system is unstable whene2Z j >2/π in the sense thatE0=inf spec(H)→−∞ as the R j →0, allj. We prove that the nuclear Coulomb repulsion more than restores stability; namely\(E_0 + 0.069e^2 \sum\limits_{i< j} {Z_i Z_j } |R_i - R_j |^{ - 1} \geqq 0\). We also show thatE0 is an increasing function of the internuclear distances |R i R j |.

Keywords

Neural Network Statistical Physic Complex System Nonlinear Dynamics Coulomb Interaction 
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.

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References

  1. 1.
    Dyson, F., Lenard, A.: Stability of matter. I. J. Math. Phys.8, 423–434 (1967)Google Scholar
  2. 1a.
    Lenard, A., Dyson, F.: Stability of matter. II. J. Math. Phys.9, 698–711 (1968)Google Scholar
  3. 2.
    Lieb, E., Thirring, W.: A bound for the kinetic energy of fermions which proves the stability of matter. Phys. Rev. Lett.35, 687–689 (1975); Errata: Phys. Rev. Lett.35, 1116 (1975). See also Lieb, E.: The stability of matter. Rev. Mod. Phys.48, 553–569 (1976)Google Scholar
  4. 3.
    Dyson, F.: Ground-state energy of a finite system of charged particles. J. Math. Phys.8, 1538–1545 (1967)Google Scholar
  5. 3a.
    Lieb, E.: TheN 5/3 law for bosons. Phys. Lett.A70, 71–73 (1979)Google Scholar
  6. 4.
    Weder, R.: Spectral analysis of pseudodifferential operators. J. Funct. Anal.20, 319–337 (1975)Google Scholar
  7. 5.
    Herbst, I.: Spectral theory of the operator (p 2+m 2)1/2Ze 2/r. Commun. Math. Phys.53, 285–294 (1977); Errata: Commun. Math. Phys.55, 316 (1977)Google Scholar
  8. 6.
    Kato, T.: Perturbation theory for linear operators. Berlin, New York: Springer 1966 (2nd edn. 1976)Google Scholar
  9. 7.
    Lieb, E., Simon, B.: Monotonicity of the electronic contribution to the Born-Oppenheimer energy. J. Phys.B11, L537–542 (1978)Google Scholar
  10. 8.
    Lieb, E.: Monotonicity of the molecular electronic energy in the nuclear coordinates. J. Phys.B15, L63-L66 (1982)Google Scholar
  11. 9.
    Daubechies, I.: An uncertainty principle for fermions with generalized kinetic energy. Commun. Math. Phys. (1983)Google Scholar
  12. 10.
    Lieb, E., Simon, B.: The Thomas-Fermi theory of atoms, molecules and solids. Adv. Math.23, 22–116 (1977)Google Scholar
  13. 11.
    Kovalenko, V., Perelmuter, M., Semenov, Ya.: Schrödinger operators withL w1/2(ℝl) potentials. J. Math. Phys.22, 1033–1044 (1981)Google Scholar
  14. 12.
    Brascamp, H., Lieb, E., Luttinger, M.: A general rearrangement inequality for multiple integrals. J. Funct. Anal.17, 227–237 (1974)Google Scholar
  15. 13.
    Lieb, E.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities (submitted)Google Scholar
  16. 14.
    Reed, M., Simon, B.: Methods of modern mathematical physics Vol. IV: Analysis of operators. New York: Academic Press 1978Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Ingrid Daubechies
    • 1
  • Elliott H. Lieb
    • 1
  1. 1.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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