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

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