  We consider solutions of the eigenvalue equation at zero energy for a class
of non-local Schr\"odinger operators with potentials decreasing to zero at
infinity. Using a path integral approach, we obtain detailed results on the
spatial decay of both $L^2$ and resonance solutions at infinity. We highlight
the interplay of the kinetic term and the potential in these decay behaviours,
and identify the decay mechanisms resulting from specific balances of global
lifetimes with or without the potential.
