Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stäckel transform, coupling constant metamorphosis and algebraization of quasi-exactly solvable systems

Siyu Li, Ian Marquette, Yao-Zhong Zhang

TL;DR

This work develops a general framework that uses the St{"a}ckel transform and coupling constant metamorphosis (CCM) to convert non-Lie-algebraic quasi-exactly solvable (QES) systems into St{"a}ckel equivalents that admit an $sl(2)$ algebraization. Once transformed, the $sl(2)$ structure yields closed-form polynomials and a Jacobi matrix that determine the first few eigenvalues and eigenfunctions; CCM then maps these results back to the original models, providing energies, wavefunctions, and parameter constraints. The authors apply the method to a broad set of physically relevant problems, including a 2D hydrogen atom in a magnetic field, Hooke-type two-electron systems, electrons on a sphere, inverse quartic and sextic potentials, and quantum Newtonian cosmology, deriving explicit $n$-dependent spectra and analytic solutions. The approach extends the reach of Lie-algebraic QES methods and suggests pathways to higher-rank algebras, expanding the toolbox for analytic treatment of complex quantum systems.

Abstract

We generalize the notions of the Stäckel 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 Stäckel 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.

Stäckel transform, coupling constant metamorphosis and algebraization of quasi-exactly solvable systems

TL;DR

This work develops a general framework that uses the St{"a}ckel transform and coupling constant metamorphosis (CCM) to convert non-Lie-algebraic quasi-exactly solvable (QES) systems into St{"a}ckel equivalents that admit an algebraization. Once transformed, the structure yields closed-form polynomials and a Jacobi matrix that determine the first few eigenvalues and eigenfunctions; CCM then maps these results back to the original models, providing energies, wavefunctions, and parameter constraints. The authors apply the method to a broad set of physically relevant problems, including a 2D hydrogen atom in a magnetic field, Hooke-type two-electron systems, electrons on a sphere, inverse quartic and sextic potentials, and quantum Newtonian cosmology, deriving explicit -dependent spectra and analytic solutions. The approach extends the reach of Lie-algebraic QES methods and suggests pathways to higher-rank algebras, expanding the toolbox for analytic treatment of complex quantum systems.

Abstract

We generalize the notions of the Stäckel 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 algebraizations can only be achieved via coupling constant metamorphosis after appropriate Stäckel 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.

Paper Structure

This paper contains 11 sections, 1 theorem, 122 equations.

Table of Contents

  1. Introduction
  2. Stäckel transform and algebraization of QES systems
  3. 2D hydrogen atom in a uniform magnetic field
  4. Hooke-type models of two charged particles
  5. Two electrons in an external oscillator potential
  6. Two planar charged particles in uniform magnetic field
  7. Two Coulombically repelling electrons on a sphere
  8. Inverse quartic power potential
  9. Inverse sextic power potential
  10. Quantum Newtonian cosmology
  11. Conclusion

Key Result

Theorem 1

zhang2016 The differential operator $\mathcal{H}'$ allows for a $sl(2)$ algebraization, i.e. has a hidden $sl(2)$ algebraic structure, if and only if where $n$ is a nonnegative integer.

Theorems & Definitions (1)

  • Theorem 1