  In a first part of this paper we investigate the continuity (stability) of
the spectrum of a class of non-local Schr\"odinger operators on varying the
potentials. By imposing conditions of different strength on the convergence of
the sequence of potentials, we give either direct proofs to show the strong or
norm resolvent convergence of the so-obtained sequence of non-local
Schr\"odinger operators, or via $\Gamma$-convergence of the related positive
forms for more rough potentials. In a second part we use these results to show
via a sequence of suitably constructed approximants that the ground states of
massive or massless relativistic Schr\"odinger operators with spherical
potential wells are radially decreasing functions.
