2023/03/16 by Nicholas Lohr, Lohr, Nicholas
Mathematics · Pharmacology, Toxicology and Pharmaceutics · #Analysis of PDEs (math.AP) #Chemical Reactions and Isotopes #FOS: Mathematics #FOS: Physical sciences #Mathematical Dynamics and Fractals #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2303.09640
openalex publication_date 2023/03/16 · openalex created_date 2023/03/21 · openalex updated_date 2026/07/28
We characterize the set of semiclassical measures corresponding to sequences of eigenfunctions of the attractive Coulomb operator \widehatHℏ:=-(ℏ2)/(2)Δℝ3-(1)/(|x|). In particular, any Radon probability measure on the fixed negative energy hypersurface ΣE 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 \widehatHℏ with eigenvalue E as ℏ → 0. The main tool that we use is the celebrated Fock unitary conjugation map between eigenspaces of \widehatHℏ and -Δ_\mathbbS3. We first prove that for any Kepler orbit γ on ΣE, there is a sequence of eigenfunctions that converge in the sense of semiclassical measures to the delta measure supported on γ as ℏ → 0, and we finish using a density argument in the weak-* topology.