  We study an inverse problem for the fractional Schr\"odinger equation (FSE)
with a local perturbation by a linear partial differential operator (PDO) of
order smaller than the order of the fractional Laplacian. We show that one can
uniquely recover the coefficients of the PDO from the Dirichlet-to-Neumann (DN)
map associated to the perturbed FSE. This is proved for two classes of
coefficients: coefficients which belong to certain spaces of Sobolev
multipliers and coefficients which belong to fractional Sobolev spaces with
bounded derivatives. Our study generalizes recent results for the zeroth and
first order perturbations to higher order perturbations.
