We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator \(\widehat{H}_{\hbar }{:}{=}-\frac{\hbar ^2}{2}\Delta _{\mathbb {R}^3}-\frac{1}{|x|}\) . 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 \(\widehat{H}_{\hbar }\) with eigenvalue E as \(\hbar \rightarrow 0\) . The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of \(\widehat{H}_{\hbar }\) 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 \(\hbar \rightarrow 0\) , and we finish using a density argument in the weak-* topology.
