Asymptotics for Two-Dimensional Atoms

We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge Z > 0 and N quantum electrons of charge −1 is when Z → ∞ and N / Z → λ, where E TF (λ) is given by a Thomas–Fermi type variational problem and c H ≈ −2.2339 is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when Z → ∞, which is contrary to the expected behavior of three-dimensional atoms.