A short LHFT note showing that the hydrogenic Coulomb problem, rewritten in the logarithmic structural coordinate u = ln (r/r0), yields the exact bound-state spectrum En ∝ -1/n² and the principal shell radius rn = (a0/Z) n². The text gives the microscopic bridge to un = 2 ln (n) and k (n) = 2 ln (n) / λ, interprets Bohr’s radius law as the projected principal-shell image of the atomic Coulomb sector in LHFT coordinates, and includes a first LHFT atomic Hamiltonian with possible log-harmonic correction terms. It presents a microscopic weak-form closure of the principal shell hierarchy, but does not claim a full operator-theoretic closure of the complete atomic LHFT sector.
CHRISTIAN BAGANZ (Thu,) studied this question.