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We study heat kernels of Schr\"odinger operators whose kinetic terms arenon-local operators built for sufficiently regular symmetric L\'evy measureswith radial decreasing profiles and potentials belong to Kato class. Oursetting is fairly general and novel -- it allows us to treat both heavy- andlight-tailed L\'evy measures in a joint framework. We establish a certainrelative-Kato bound for the corresponding semigroups and potentials. Thisenables us to apply a general perturbation technique to construct the heatkernels and give sharp estimates of them. Assuming that the L\'evy measure andthe potential satisfy a little stronger conditions, we additionally obtain theregularity of the heat kernels. Finally, we discuss the applications to thesmoothing properties of the corresponding semigroups. Our results cover manyimportant examples of non-local operators, including fractional andquasi-relativistic Schr\"odinger operators.

Heat kernels of non-local Schr\"odinger operators with Kato potentials