St\"ackel transform, coupling constant metamorphosis and algebraization of quasi-exactly solvable systems

TL;DR

This paper extends the concepts of Stäckel transform and coupling constant metamorphosis to quasi-exactly solvable systems, revealing that their algebraizations often require these transformations, and applies this to derive solutions for systems like Hooke's atoms.

Contribution

It introduces the first application of coupling constant metamorphosis to quasi-exactly solvable systems, broadening the algebraization methods available for these systems.

Findings

Algebraizations of quasi-exactly solvable systems often require coupling constant metamorphosis.

Derived algebraizations and energies for systems like Hooke's atoms in magnetic fields.

Established a new approach for analyzing quasi-exactly solvable systems using Stäckel transforms.

Abstract

We generalize the notions of the St\"ackel transform and the coupling constant metamorphosis to quasi-exactly solvable systems. We discover that for a variety of one-dimensional and separable multidimensional quasi-exactly solvable systems, their $sl(2)$ algebraizations can only be achieved via coupling constant metamorphosis after appropriate St\"ackel transformations. This discovery has interesting applications, allowing us to derive algebraizations and energies for a wide class of quasi-exactly solvable systems, such as Hooke's atoms in magnetic fields and Newtonian cosmology. The approach of coupling constant metamorphosis was successfully applied previously in the context of exactly solvable, integrable and superintegrable systems. To our knowledge, the present work is the first to apply the idea and approach in the context of quasi-exactly solvable systems.