  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.
