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 $\mathrm{He}^-, \mathrm{Be}^-, \mathrm{N}^-, \mathrm{Ne}^-$, etc., many numerical results give strong evidence of the condition $E(N, Z)= E(N-1, Z)$.