Global structure and geodesics for Koenigs superintegrable systems

TL;DR

This paper analyzes the global structure and geodesic flows of Koenigs superintegrable systems, demonstrating their quantum superintegrability and spectrum in specific cases, on non-compact manifolds.

Contribution

It derives the global structure of four types of Koenigs metrics and shows quantum superintegrability preservation, including spectrum computation for closed geodesic cases.

Findings

Global structure of Koenigs metrics established

Quantum superintegrability preserved under Carter quantization

Discrete spectrum computed for cases with closed geodesics

Abstract

Starting from the framework defined by Matveev and Shevchishin we derive the local and the global structure for the four types of super-integrable Koenigs metrics. These dynamical systems are always defined on non-compact manifolds, namely $\,{\mb R}^2\,$ and $\,{\mb H}^2$. The study of their geodesic flows is made easier using their linear and quadratic integrals. Using Carter (or minimal) quantization we show that the formal superintegrability is preserved at the quantum level and in two cases, for which all of the geodesics are closed, it is even possible to compute the discrete spectrum of the quantum hamiltonian.