  This work extends the applications of Anderson-type Hamiltonians to include
transport characterized by anomalous diffusion. Herein, we investigate the
transport properties of a one-dimensional disordered system that employs the
discrete fractional Laplacian, $(-\Delta)^s,\ s\in(0,2),$ in combination with
results from spectral and measure theory. It is a classical mathematical result
that the standard Anderson model exhibits localization of energy states for all
nonzero disorder in one-dimensional systems. Numerical simulations utilizing
our proposed model demonstrate that this localization effect is enhanced for
sub-diffusive realizations of the operator, $s\in (1,2),$ and that the
super-diffusive realizations of the operator, $s\in (0,1),$ can exhibit energy
states with less localized features. These results suggest that the proposed
method can be used to examine anomalous diffusion in physical systems where
strong interactions, structural defects, and correlated effects are present.
