Weak Mittleman's conjecture for the Dirac–Fock ground state energy

Let $e_q$ be the physical ground state energy of the electron-positron Hartree–Fock model and let $E_q$ be the ground state energy of the Dirac–Fock model. For $\alpha$ small and $c$ large, the two energies satisfy

$$e_q\leq E_q\leq e_q+o_{\alpha_c\to 0}(c^{-2}).$$

Weak Mittleman's conjecture. The Dirac–Fock model is an approximation of the electron-positron Hartree–Fock model in the precise sense of the displayed inequality. This weaker statement is introduced because the original conjecture may fail when the relevant eigenvalue is not simple or when the stated spectral-gap condition does not hold. It quantifies the error at order smaller than $c^{-2}$ in the indicated nonrelativistic regime.