2026/07/02 by Mikhail S. Plyushchay
#hep-th
We show that two paradigmatic systems, the planar Kepler--Coulomb problem and the Landau problem on the hyperbolic plane H2, are connected by a common one-dimensional mediator: the Morse Hamiltonian. On the Kepler side, a radial Liouville transformation and genuine coupling-constant metamorphosis produce the Morse spectral problem; classically, the Kepler polar angle becomes proportional to the Morse evolution parameter. On the Landau side, horocyclic reduction at fixed momentum gives the same Morse Hamiltonian, while quantum half-density normalization produces the universal 1/4 spectral shift. Consequently, the Kepler bound-state problem and the attractive fixed-horocyclic-momentum sectors of the hyperbolic Landau problem are encoded in a common Morse spectral equation, which organizes the Kepler shell structure together with the threshold, resonance and reflection data of the Morse and reduced Landau systems. At the quantum level, each bound-state Kepler shell selects a Morse system from a distinguished integer-parameter family: the Morse level number coincides with the Kepler radial quantum number, while the zero-angular-momentum member of the shell is represented by the Morse threshold resonance. We further show that the Landau time evolution has a Kepler-conic form and reduces to the bound, threshold and scattering trajectories of the Morse system. The resulting dictionary connects Kepler conics with magnetic circles, horocycles and hypercycles. Algebraically, the classical magnetic SL(2,\mathbb R) Casimir reduces to the classical Morse Hamiltonian, whereas at the quantum level Darboux shape invariance provides a complementary parameter-shifting spectrum-generating structure.