Spectral and analytic properties of some non-local Schrödinger operators and related jump processes

We discuss recent developments in the spectral theory of non-local SchrOdinger operators via a Feynman-Kac-type approach. The processes we consider are subordinate Brownian motion and a class of jump Levy processes under a Kato-class potential. We discuss some explicitly soluble specific cases, and address the spatial decay properties of eigenfunctions and the number of negative eigenvalues in the general framework of the processes we introduce.