Classical and quantum three-dimensional integrable systems with axial symmetry

TL;DR

This paper classifies and analyzes three-dimensional classical and quantum integrable systems with axial symmetry, deriving explicit potentials, separation of variables, and spheroidal wave equations for quantization.

Contribution

It provides the explicit form of potentials for integrable systems with axial symmetry and derives their separation in oblate spheroidal coordinates, including quantization leading to spheroidal wave equations.

Findings

Explicit potentials compatible with integrability are derived.

Classical equations are separable in oblate spheroidal coordinates.

Quantization yields spheroidal wave equations.

Abstract

We study the most general form of a three dimensional classical integrable system with axial symmetry and invariant under the axis reflection. We assume that the three constants of motion are the Hamiltonian, $H$, with the standard form of a kinetic part plus a potential dependent on the position only, the $z$-component of the angular momentum, $L$, and a Hamiltonian-like constant, $\widetilde H$, for which the kinetic part is quadratic in the momenta. We find the explicit form of these potentials compatible with complete integrability. The classical equations of motion, written in terms of two arbitrary potential functions, is separated in oblate spheroidal coordinates. The quantization of such systems leads to a set of two differential equations that can be presented in the form of spheroidal wave equations.