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

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 onedimensional 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 twodimensional case, we construct the rational extensions for QAHO potentials with quadratic (λxy) 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.