Asymptotics for Two-dimensional Atoms

Abstract: We prove that the ground state energy of an atom confined to two dimensions with an infinitely heavy nucleus of charge $Z&gt;0$ and $N$ quantum electrons of charge -1 is $E(N,Z)=-{1/2}Z^2\ln Z+(E^{{\TF}(\lambda)+{1/2}c^{\rm} H})Z<sup>2+o(Z<sup>2)$ when $Z\to \infty$ and $N/Z\to \lambda$, where $E<sup>{\TF}(\lambda)$ is given by a Thomas-Fermi type variational problem and $c^{\rm H}\approx -2.2339$ is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when $Z\to \infty$, which is contrary to the expected behavior of three-dimensional atoms.