Lie-algebraic interpretation of the maximal superintegrability and exact solvability of the Coulomb-Rosochatius potential in n dimensions

TL;DR

This paper uses the potential group method to analyze the n-dimensional Coulomb-Rosochatius potential, revealing its superintegrability and exact solvability through algebraic structures and explicit calculations of bound and scattering states.

Contribution

It provides a Lie-algebraic framework explaining the superintegrability and solvability of the Coulomb-Rosochatius system in multiple dimensions, linking constants of motion to Casimir operators.

Findings

Explicit bound and scattering states derived.

S-matrix elements computed via intertwining operators.

Superintegrability explained through potential group structure.

Abstract

The potential group method is applied to the n-dimensional Coulomb-Rosochatius potential, whose bound states and scattering states are worked out in detail. As far as scattering is concerned, the S-matrix elements are computed by the method of intertwining operators and an integral representation is obtained for the scattering amplitude. It is shown that the maximal superintegrability of the system is due to the underlying potential group and that the 2n-1 constants of motion are related to Casimir operators of subgroups.