Morse Bridge between Planar Kepler and Hyperbolic Landau Dynamics

Abstract

We show that two paradigmatic systems, the planar Kepler--Coulomb problem and the Landau problem on the hyperbolic plane $H^2$, are connected by a common one-dimensional mediator: the Morse Hamiltonian. On the Kepler side, a Liouville transformation and coupling-constant metamorphosis turn the radial dynamics into the Morse problem, with the Kepler polar angle becoming the Morse evolution parameter. On the Landau side, horocyclic reduction of the hyperbolic magnetic dynamics gives the same Morse Hamiltonian, with a quantum half-density correction. Consequently, the radial Kepler problem and the fixed-horocyclic-momentum sectors of the hyperbolic Landau problem are mapped to one Morse spectral problem, relating their bound spectra, continuum thresholds, resonances and scattering data. 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, and turns the magnetic $SL(2,\mathbb R)$ symmetry of the Landau problem into the spectrum-generating algebraic structure of the Morse system.