Quantum Systems at The Brink. Existence and Decay Rates of Bound States at Thresholds; Atoms

It is well known that $N$-electron atoms undergoes unbinding for a critical charge of the nucleus $Z_c$, i.e. the atom has eigenstates for the case $Z> Z_c$ and it has no bound states for $Z<Z_c$. In the present paper we derive upper bound for the bound state for the case $Z=Z_c$ under the assumption $Z_c<N-K$ where $K$ is the number of electrons to be removed for atom to be stable for $Z=Z_c$ without any change in the ground state energy. We show that the eigenvector decays faster as $\exp\left(-C\sum\sqrt{|x|_{k}}\right)$ where we sum K largest values of $|x_j|$, $j\in\{1,\ldots,N\}$. Our method do not require Born-Oppenheimer approximation.