Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rational Extension of Quantum Anisotropic Oscillator Potentials with Linear and/or Quadratic Perturbations

Rajesh Kumar Yadav, Rajesh Kumar, Avinash Khare

TL;DR

This work extends the rational extension (RE) framework to quantum anisotropic oscillator (QAO) potentials with linear and/or quadratic perturbations across 1D, 2D, and 3D. By a two-step approach—first absorbing perturbations through coordinate shifts/rotations to map to a pure AHO, then applying the RE construction via exceptional Hermite polynomials and finally transforming back—the authors derive explicit RE potentials and eigenfunctions, along with conditions for real spectra and degeneracies. They treat real and purely imaginary perturbations, revealing PT-symmetric structures and showing that, in many cases, real spectra persist under specific constraints and frequency ratios become rational for degeneracy. The results generalize previous RE-QHO developments to perturbed, higher-dimensional oscillators and highlight open questions about broader perturbations and spherically symmetric systems, with potential applications in exactly solvable quantum models and spectral engineering.

Abstract

We present a comprehensive study of the rational extension of the quantum anisotropic harmonic oscillator (QAHO) potentials with linear and/or quadratic perturbations. For the one-dimensional harmonic oscillator plus imaginary linear perturbation ($iλx$), we show that the rational extension is possible not only for the even but also for the odd co-dimensions $m$. In two-dimensional case, we construct the rational extensions for QAHO potentials with quadratic ($λ\, xy$) perturbation both when $λ$ is real or imaginary and obtain their solutions. Finally, we extend the discussion to the three-dimensional QAHO with linear and quadratic perturbations and obtain the corresponding rationally extended potentials. For all these cases, we obtain the conditions under which the spectrum remains real and also when there is degeneracy in the system.

Rational Extension of Quantum Anisotropic Oscillator Potentials with Linear and/or Quadratic Perturbations

TL;DR

This work extends the rational extension (RE) framework to quantum anisotropic oscillator (QAO) potentials with linear and/or quadratic perturbations across 1D, 2D, and 3D. By a two-step approach—first absorbing perturbations through coordinate shifts/rotations to map to a pure AHO, then applying the RE construction via exceptional Hermite polynomials and finally transforming back—the authors derive explicit RE potentials and eigenfunctions, along with conditions for real spectra and degeneracies. They treat real and purely imaginary perturbations, revealing PT-symmetric structures and showing that, in many cases, real spectra persist under specific constraints and frequency ratios become rational for degeneracy. The results generalize previous RE-QHO developments to perturbed, higher-dimensional oscillators and highlight open questions about broader perturbations and spherically symmetric systems, with potential applications in exactly solvable quantum models and spectral engineering.

Abstract

We present a comprehensive study of the rational extension of the quantum anisotropic harmonic oscillator (QAHO) potentials with linear and/or quadratic perturbations. For the one-dimensional harmonic oscillator plus imaginary linear perturbation (), we show that the rational extension is possible not only for the even but also for the odd co-dimensions . In two-dimensional case, we construct the rational extensions for QAHO potentials with quadratic () perturbation both when is real or imaginary and obtain their solutions. Finally, we extend the discussion to the three-dimensional QAHO with linear and quadratic perturbations and obtain the corresponding rationally extended potentials. For all these cases, we obtain the conditions under which the spectrum remains real and also when there is degeneracy in the system.

Paper Structure

This paper contains 16 sections, 76 equations, 12 figures, 4 tables.

Table of Contents

  1. Introduction
  2. One-dimensional harmonic oscillator with linear perturbation
  3. Rational extension: Real $\lambda_0$ case
  4. Rational extension: Imaginary $\lambda_0$ case
  5. Two-dimensional QAHO potential with quadratic perturbation
  6. Rational Extension
  7. Conditions for Degeneracy
  8. Example of Degeneracy in Case $\lambda$ is Real
  9. Example of Degeneracy in Case $\lambda$ is Imaginary
  10. Three-dimensional QAHO potentials
  11. Combination of a linear and a quadratic perturbations
  12. Rational Extension of $\tilde{V}^{lq}(\tilde{x}, \tilde{y}, \tilde{z})$
  13. 3D QAHO Potential with Combinations of Quadratic Perturbations
  14. Case I: $\mathbf{\lambda_1=0}$ and $\mathbf{\lambda_2\ne \lambda_3}$
  15. Case II: $\mathbf{\lambda_1\ne0}$ and $\mathbf{\lambda_2=\lambda_3}$
  16. ...and 1 more sections

Figures (12)

  • Figure 1: Plots of real and imaginary components of $V^l_{RE,m}(x)$ vs $x$ for $m=0$ to $5$ with $\omega_1=2$.
  • Figure 2: Plots of real and imaginary components of $\psi^l_{RE,m,0}(x)$ vs $x$ for $m=0$ to $5$ and $\omega_1=2$.
  • Figure : $V_{RE,,m_1=0,m_2=2}(x,y)$
  • Figure : $Re[V_{RE,m_1=0,m_2=2}(x,y)]$
  • Figure : $V_{RE,,m_1=0,m_2=2}(x,y)$
  • ...and 7 more figures