Potentials for non-local Schrodinger operators with zero eigenvalues

In this paper we propose a systematic description of potentials decaying to zero at infinity, generating eigenvalues at the edge of the continuous spectrum when combined with non-local operators. Introducing Holder-Zygmund type spaces we study these non-local Schrodinger operators via integrals containing singular kernels. We obtain conditions under which the potentials decay at all, and are bounded continuous. Next we derive decay rates for operators with regularly varying and exponentially light Levy jump kernels. We show situations in which no decay occurs, implying absence of zero-energy eigenfunctions. Then we obtain results on the sign of potentials at infinity, which is also crucial for the occurrence or absence of zero eigenvalues. Finally, we analyze a delicate interplay between the pinning effect resulting from a well at zero combined with decay and sign at infinity, as a main mechanism in the formation of zero-energy bound states. We develop a purely analytic approach. (C) 2022 Elsevier Inc. All rights reserved.