Morse Bridge between Planar Kepler and Hyperbolic Landau Dynamics

Abstract

A common Morse Hamiltonian bridges the planar Kepler--Coulomb and hyperbolic Landau systems, linking flat electric dynamics to curved magnetic dynamics. A radial Liouville transformation with coupling-constant metamorphosis maps the separated Kepler problem to the Morse system, while fixed-horocyclic-momentum reduction maps the hyperbolic Landau problem to the same system. In a common normalization, the Coulomb coupling is the signed product of the magnetic field and conserved horocyclic momentum, while the quantum Landau reduction has the universal half-density shift $1/4$. Beyond this shared reduction, we construct a direct classical orbit map on the regular nonradial sectors for all signs of the Kepler energy. It maps the Landau height to a scaled inverse Kepler radius, Kepler ellipses, parabolas and hyperbolas to magnetic circles, horocycles and hypercycles, and Hamilton's hodograph affinely to the Landau carrier circle. The reduced Kepler symplectic form and the Binet-selected complex structure determine a Poincaré Kähler metric. Its nondegenerate Binet orbits have constant geodesic curvature of magnitude equal to the inverse eccentricity, while zero coupling gives geodesics. The orbit map identifies this metric isometrically with the Poincaré configuration-space metric of the hyperbolic Landau problem. Composing the quantum Liouville maps eliminates the Morse wavefunction and yields an exact weighted intertwining identity between the radial Kepler and horocyclic Landau operator pencils. At distinguished half-integer magnetic fields, the angular-momentum grading of a fixed Kepler shell is reorganized into the finite reduced Landau tower and its threshold endpoint. These exact correspondences hold in the stated reduced sectors despite the unitary inequivalence of the complete parent systems.