  We consider non-local Schr\"odinger operators $H=-L-V$ in
$L^2(\mathbf{R}^d)$, $d \geq 1$, where the kinetic terms $L$ are
pseudo-differential operators which are perturbations of the fractional
Laplacian by bounded non-local operators and $V$ is the fractional Hardy
potential. We prove pointwise estimates of eigenfunctions corresponding to
negative eigenvalues and upper finite-time horizon estimates for heat kernels.
We also analyze the relation between the matching lower estimates of the heat
kernel and the ground state near the origin. Our results cover the relativistic
Schr\"odinger operator with Coulomb potential.
