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Relativistic molecules with Coulomb interaction. (English) Zbl 0565.35101
Differential equations, Proc. Conf., Birmingham/Ala. 1983, North-Holland Math. Stud. 92, 143-148 (1984).
[For the entire collection see Zbl 0539.00010.]
As an approximation to a relativistic one-electron molecule, we study the operator $$H=(-\Delta +m^ 2)^{1/2}-\sum^{K}_{j=1}Z_ je^ 2/| x-R_ j|,$$ with $$Z_ j\geq 0$$ for all j. H is bounded below if and only if $$e^ 2Z_ j\leq 2/\pi$$ for all j. Under this condition, we show that 1) the system is stable when the nuclear repulsion is taken into account, i.e. $$E_ 0+\sum^{K}_{j,k=1,j>k}Z_ jZ_ ke^ 2/| R_ j-R_ k| \geq 0$$, where $$E_ 0=\inf spec H$$. 2) The ground state energy $$E_ 0$$ is an increasing function of the internuclear distances $$| R_ j-R_ k|$$.
##### MSC:
 35Q99 Partial differential equations of mathematical physics and other areas of application 35B35 Stability in context of PDEs 81V10 Electromagnetic interaction; quantum electrodynamics