  We consider the determination of an unknown potential $q(x)$ form a
fractional diffusion equation subject to overposed lateral boundary data. We
show that this data allows recovery of two spectral sequences for the
associated inverse Sturm-Liouville problem and these are sufficient to apply
standard uniqueness results for this case.
  We also look at reconstruction methods and in particular examine the issue of
stability of the solution with respect to the data. The outcome shows the
inverse problem to be severely ill-conditioned and we consider the differences
between the cases of fractional and of classical diffusion.
