  We study heat kernels of Schr\"odinger operators whose kinetic terms are
non-local operators built for sufficiently regular symmetric L\'evy measures
with radial decreasing profiles and potentials belong to Kato class. Our
setting is fairly general and novel -- it allows us to treat both heavy- and
light-tailed L\'evy measures in a joint framework. We establish a certain
relative-Kato bound for the corresponding semigroups and potentials. This
enables us to apply a general perturbation technique to construct the heat
kernels and give sharp estimates of them. Assuming that the L\'evy measure and
the potential satisfy a little stronger conditions, we additionally obtain the
regularity of the heat kernels. Finally, we discuss the applications to the
smoothing properties of the corresponding semigroups. Our results cover many
important examples of non-local operators, including fractional and
quasi-relativistic Schr\"odinger operators.
