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

TL;DR

This paper investigates the existence and decay rates of bound states at the critical nuclear charge in multi-electron atoms, providing bounds and decay estimates without relying on the Born-Oppenheimer approximation.

Contribution

It derives upper bounds for bound states at the critical charge and characterizes their decay rates, extending understanding of atomic stability at thresholds.

Findings

Eigenstates decay faster at the critical charge, with decay rate involving exponential of sum of square roots of electron positions.

Provides bounds on the eigenvector decay without using Born-Oppenheimer approximation.

Shows conditions under which atoms remain stable at the critical charge.

Abstract

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.