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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