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 $E(N,Z)=-{1/2}Z^2\ln Z+(E^{\TF}(\lambda)+{1/2}c^{\rm H})Z^2+o(Z^2)$when$Z\to \infty$and$N/Z\to \lambda$,where$E^{\TF}(\lambda)$isgivenbyaThomas-Fermitypevariationalproblemand$c^{\rm H}\approx -2.2339$ is an explicit constant. We also show that the radius of a two-dimensional neutral atom is unbounded when $Z\to \mathrm{\infty }$, which is contrary to the expected behavior of three-dimensional atoms.