Bound states and heat kernels for fractional-type Schrödinger operators with singular potentials

We consider non-local Schrödinger operators $H=-L-V$ in $L^2({\mathbf{R}}^d)$, $d\ge 1$, 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 Schrödinger operator with Coulomb potential.