Semiclassical measures through Coulomb collisions
We prove that $\mu$ is a semiclassical measure associated to a sequence of eigenfunctions of energy $E<0$ of the attractive Coulomb operator if and only if $\mu$ is a probability measure on the energy (hyper)surface $\Sigma_E$ invariant under the regularized Kepler flow due to Moser. The converse was shown in recent work by the author, and the present article proves the other direction (as well as an independent proof of the converse). We prove the main theorem for a general symbol class allowing certain non-decay at infinity, which implies that semiclassical measure mass entirely reflects off of the origin. In the special case of semiclassical measures of sequences of eigenfunctions of the exact Coulomb operator, this article solves an open problem posed by Keraani. The main tools include the celebrated Moser-Fock map along with a technical operator extension lemma in $\Psi_{\hbar}^0(\mathbb{S}^d)$, which utilizes standard eigenfunction concentration bounds.