Quantum Systems at The Brink: Properties of Atomic Bound States at The Ionization Threshold

TL;DR

This paper rigorously proves that atomic systems with long-range repulsion can have stable ground states at the ionization threshold, with detailed bounds on their decay properties, confirming longstanding quantum chemistry predictions.

Contribution

It provides the first rigorous analysis of atomic bound states exactly at the ionization threshold, including decay bounds and sharpness for helium-like systems.

Findings

Ground states exist at the ionization threshold due to long-range repulsion.

Decay of ground states is slower than WKB predictions, following an exponential of sqrt{|x|}.

Bounds are sharp for helium-like systems, confirming quantum chemists' predictions.

Abstract

We give a rigorous argument that long--range repulsion stabilizes quantum systems; ground states of such quantum systems exist even when the ground state energy is precisely at the ionization threshold. For atomic systems at the critical nuclear charge, our bounds show that the ground state falls off like $\exp(-c\sqrt{|x|})$ for large $|x|$. This is much slower than what the WKB method predicts for bound states with energies strictly below the ionization threshold. For helium type systems at critical nuclear charge, we show that our upper bounds are sharp. This rigorously confirms predictions by quantum chemists.