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We study the fractional Schr\"odinger equation with quasilocal perturbations.These are a family of nonlocal perturbations vanishing at infinity, whichinclude e.g. convolutions against Schwartz functions. We show that thequalitative unique continuation and Runge approximation properties hold in theassumption of sufficient decay. Quantitative versions of both results are alsoobtained via a propagation of smallness analysis for the Caffarelli-Silvestreextension. The results are then used to show uniqueness in the inverse problemof retrieving a quasilocal perturbation from DN data under suitable geometricassumptions. Our work generalizes recent results regarding the locallyperturbed fractional Calder\'on problem.

Uniqueness for the fractional Calder\'on problem with quasilocal perturbations