Operator formulation of classical mechanics: Levi-Civita map and equivalence of central forces in two dimensions

We study the operator formulation of classical mechanics by explicitly applying it to two central potentials in two dimensions. After constructing the classical Hamiltonian operators and corresponding Schrödinger-like equations, we solve for the corresponding classical wave functions associated with these two potentials, viz, Kepler and harmonic potentials. While satisfying continuity equations, these classical wave functions are shown to be renormalizable only in a finite region of the 2D plane. We also derive the well-known equivalence between these two models within the operator formulation of classical mechanics. This equivalence is shown by relating the Schrödinger-like equations and corresponding classical wave functions of these two systems, using the Levi-Civita map and a reparametrization of time (Sundman map).