Infinite families of superintegrable systems separable in subgroup coordinates

TL;DR

The paper introduces a method to embed subgroup separable superintegrable systems into infinite families, preserving superintegrability in 2D Euclidean and pseudo-Euclidean spaces, with explicit classical and quantum solutions.

Contribution

It presents a novel method for generating infinite families of superintegrable systems that are exactly solvable and preserves superintegrability in specific spaces.

Findings

Constructed two infinite families of superintegrable systems in 2D pseudo-Euclidean space.

Derived classical trajectories and quantum eigenfunctions for these systems.

Expressed wave-functions using Laguerre and generalized Bessel polynomials.

Abstract

A method is presented that makes it possible to embed a subgroup separable superintegrable system into an infinite family of systems that are integrable and exactly-solvable. It is shown that in two dimensional Euclidean or pseudo-Euclidean spaces the method also preserves superintegrability. Two infinite families of classical and quantum superintegrable systems are obtained in two-dimensional pseudo-Euclidean space whose classical trajectories and quantum eigenfunctions are investigated. In particular, the wave-functions are expressed in terms of Laguerre and generalized Bessel polynomials.