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


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 |.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Coulomb Interaction 
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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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