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 $$\widehat{H}_{\hbar }{:}{=}-\frac{\hbar ^2}{2}\Delta _{\mathbb {R}^3}-\frac{1}{|x|}$$ H ^ ħ : = - ħ 2 2 Δ R 3 - 1 | x | . In particular, any Radon probability measure on the fixed negative energy hypersurface $$\Sigma _E$$ Σ 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 $$\widehat{H}_{\hbar }$$ H ^ ħ with eigenvalue E as $$\hbar \rightarrow 0$$ ħ → 0 . The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of $$\widehat{H}_{\hbar }$$ H ^ ħ and $$-\Delta _{\mathbb {S}^3}$$ - Δ S 3 . We first prove that for any Kepler orbit $$\gamma $$ γ on $$\Sigma _E$$ Σ E , there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on $$\gamma $$ γ as $$\hbar \rightarrow 0$$ ħ → 0 , and we finish using a density argument in the weak-* topology.