Zero-energy bound state decay for non-local Schr\"odinger operators

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.