One-electron relativistic molecules with Coulomb interaction

As an approximation to a relativistic one-electron molecule, we study the operator (H = ( - Delta + m^2 )^{1/2} - e^2 sumlimits_{j = 1}^K {Z_j } |x - R_j |^{ - 1}) withZ _j ≧0,e ^?2=137.04.H is bounded below if and only ife ² Z _j ≦2/π allj. Assuming this condition, the system is unstable whene ²∑Z _j >2/π in the sense thatE ₀=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 sumlimits_{i< j} {Z_i Z_j } |R_i - R_j |^{ - 1} geqq 0) . We also show thatE ₀ is an increasing function of the internuclear distances |R _i ?R _j |.