Binding-energy implication for successive atomic ground states

Determine whether the strict binding inequality E_N<E_{N-1} for the N-particle atomic Hamiltonian necessarily implies the stronger successive binding inequality E_{N-1}<E_{N-2} for the corresponding (N−1)- and (N−2)-particle atomic Hamiltonians.

Background

The paper assumes E_N\le E_{N-1}<E_{N-2} in its main quantitative asymptotic results. It notes that the strict inequality E_N<E_{N-1} guarantees that E_N is a discrete eigenvalue by the HVZ theorem, while the stronger inequality involving E_{N-2} is needed to ensure that E_{N-1} is an isolated eigenvalue. The authors state that it is generally believed that the first strict inequality implies the second, but that this implication remains unresolved. They also note that Zhislin's theorem provides the chain of strict inequalities when N<Z+1.