Semiclassical measures of eigenfunctions of the attractive Coulomb operator

Abstract

We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator ${\hat{H}}_{\hslash }:=-\frac{{\hslash }^2}{2}{\Delta }_{{\mathbb{R}}^3}-\frac{1}{|x|}$. In particular, any Radon probability measure on the fixed negative energy hypersurface ${\Sigma }_E$ of the Kepler Hamiltonian $H$ in classical phase space that is invariant under the regularized Kepler flow is the semiclassical measure of a sequence of eigenfunctions of ${\hat{H}}_{\hslash }$ with eigenvalue $E$ as $\hslash \to 0$. The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of ${\hat{H}}_{\hslash }$ and $-{\Delta }_{{\mathbb{S}}^3}$. We first prove that for any Kepler orbit $\gamma$ on ${\Sigma }_E$, there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $\gamma$ as $\hslash \to 0$, and we finish using a density argument in the weak-* topology.

23 pages, final version. To appear in Annales Henri Poincaré