  In this work we obtain boundedness results for fractional operators
associated with Schr\"odinger operators $\ \mathcal{L}=-\Delta+V$ on weighted
variable Lebesgue spaces. These operators include fractional integrals and
their respective commutators. Particularly, we obtain weighted inequalities of
the type $L^{p(\cdot)}$-$L^{q(\cdot)}$ and estimates of the type
$L^{p(\cdot)}$-Lipschitz variable integral spaces. For this purpose, we
developed extrapolation results that allow us to obtain boundedness results of
the type described above in the variable setting by starting from analogous
inequalities in the classical context.
