  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.
