Generally, the spectrum of a non-local Schrödinger operator may be rather intricate, even when they are self-adjoint operators. In this talk I plan to discuss some explicit cases when positive or zero eigenvalues occur, and also address the problem more generally, aiming to describe potentials which can give rise to zero eigenvalues for massive or massless (fractional) relativistic Schrodinger operators. If time permits, I intend to explain how random processes with jumps can be used to analyse such properties.