Morse bridge between planar Kepler and hyperbolic Landau dynamics

Abstract

A common Morse Hamiltonian connects the planar Kepler-Coulomb and hyperbolic Landau systems, thereby linking flat electric dynamics to curved magnetic dynamics. We develop this bridge at both the classical and quantum levels. A radial Liouville transformation combined with coupling-constant metamorphosis converts the separated Kepler problem into the Morse problem, while reduction at fixed horocyclic momentum converts the hyperbolic Landau problem into the same system. In this common normalization, the Coulomb coupling becomes the signed product of the magnetic field and the conserved horocyclic momentum, and the quantum Landau reduction acquires the universal half-density shift 1/4. Going beyond the common reduction, we construct a direct classical orbit map on the regular nonradial sectors, valid for negative, zero and positive Kepler energies. It identifies the Landau height with a scaled inverse Kepler radius, maps ellipses, parabolas and hyperbolas to magnetic circles, horocycles and hypercycles, and sends Hamilton’s hodograph affinely to the corresponding Landau carrier circle. The reduced Kepler symplectic form and the complex structure selected by the Binet evolution further determine a distinguished Poincaré Kähler metric. Its nondegenerate Binet orbits have constant geodesic curvature whose magnitude is the inverse Kepler eccentricity, while the zero-coupling case gives geodesics. The direct orbit map identifies this Binet metric with the Poincaré configuration-space metric of the hyperbolic Landau problem. At the quantum level, composition of the two Liouville maps eliminates the intermediate 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 together with its threshold endpoint. These exact correspondences hold in the stated reduced sectors despite the unitary inequivalence of the complete parent systems.