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It is well known that $N$-electron atoms undergoes unbinding for a criticalcharge 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 deriveupper 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 bestable for $Z=Z_c$ without any change in the ground state energy. We show thatthe eigenvector decays faster as $\exp\left(-C\sum\sqrt{|x|_{k}}\right)$ wherewe sum K largest values of $|x_j|$, $j\in\{1,\ldots,N\}$. Our method do notrequire Born-Oppenheimer approximation.

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