Factorization of the Quantum Fractional Oscillator

TL;DR

This paper extends the factorization method to quantum fractional-differential Hamiltonians, deriving fractional energy eigenvalues and wave-functions for a fractional quantum oscillator, revealing power-law relations with momentum.

Contribution

It introduces a novel factorization approach where the factorization energy becomes a fractional-differential operator, applied to the quantum oscillator.

Findings

Energy eigenvalues are power-laws of momentum.

Wave-functions are derived for the fractional quantum oscillator.

The method generalizes the factorization technique to fractional operators.

Abstract

The applicability of the factorization method is extended to the case of quantum fractional-differential Hamiltonians. In contrast with the conventional factorization, it is shown that the `factorization energy' is now a fractional-differential operator rather than a constant. As a first example, the energies and wave-functions of a fractional version of the quantum oscillator are determined. Interestingly, the energy eigenvalues are expressed as power-laws of the momentum in terms of the non-integer differential order of the eigenvalue equation.