Absence of ground states for anions

We show that the $N$-electron Hamiltonian $H(N,Z)$ with the total nuclear charge $Z$ has no normalizable ground state if the ground state energy $E(N,Z)$ satisfies $E(N,Z)=E(N-1,Z)$ for $Z=N-1$. For anions ${\mathrm{H}\mathrm{e}}^{-},{\mathrm{B}\mathrm{e}}^{-},{\mathrm{N}}^{-},{\mathrm{N}\mathrm{e}}^{-}$, etc., many numerical results give strong evidence of the condition $E(N,Z)=E(N-1,Z)$.