Factorization of the Riesz-Feller fractional quantum harmonic oscillators

TL;DR

This paper extends the factorization method to Riesz-Feller fractional quantum harmonic oscillators, introducing Riesz-Feller Hermite polynomials and analyzing eigenvalue problems in both k and x spaces.

Contribution

It develops a novel factorization approach for fractional quantum oscillators using Riesz-Feller derivatives and introduces new Hermite polynomials for non-Hermitian eigenvalue problems.

Findings

Derived analytic solutions in k space using Riesz-Feller Hermite polynomials.

Obtained eigenvalues and eigenfunctions in x space via inverse Fourier transform.

Proposed a generalized factorization with two Levy indices.

Abstract

Using the Riesz-Feller fractional derivative, we apply the factorization algorithm to the fractional quantum harmonic oscillator along the lines previously proposed by Olivar-Romero and Rosas-Ortiz, extending their results. We solve the non-Hermitian fractional eigenvalue problem in the k space by introducing in that space a new class of Hermite `polynomials' that we call Riesz-Feller Hermite `polynomials'. Using the inverse Fourier transform in Mathematica, interesting analytic results for the same eigenvalue problem in the x space are also obtained. Additionally, a more general factorization with two different Levy indices is briefly introduced