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

TL;DR

This paper investigates how solutions to zero-energy eigenvalue equations for non-local Schrödinger operators decay at infinity, revealing the influence of the kinetic term and potential on decay mechanisms using a path integral approach.

Contribution

It provides detailed analysis of decay behaviors of solutions to non-local Schrödinger operators at zero energy, highlighting the interplay between kinetic and potential terms.

Findings

Decay rates depend on the balance between kinetic and potential contributions.

Path integral methods effectively analyze spatial decay of solutions.

Results distinguish decay mechanisms with or without potential influence.

Abstract

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.