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 $Z>0$ and $N$ quantum electrons of charge -1 is $E(N,Z)=-{1/2}Z^2\ln Z+(E^{\TF}(λ)+{1/2}c^{\rm H})Z^2+o(Z^2)$ when $Z\to \infty$ and $N/Z\to \lambda$, where $E^{\TF}(λ)$ is given by a Thomas-Fermi type variational problem and $c^{\mathrm{H}}\approx -2.2339$ 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.

Revised version to appear in Ann. Henri Poincaré

References in corpus (1)

• Geometric methods for nonlinear many-body quantum systems