Abstract

A non-local Schrödinger operator arises as a sum of a non-local operator (e.g., fractional Laplacian) and a multiplication operator called potential. Such operators also have an interesting probabilistic connection, relating with Lévy-type/Feller processes whose features depend on the position in space, and jumps of given size are encouraged or suppressed in specific directions according to the values taken by the potential. The spectral analysis of such operators is a challenging question, relevant for both pure and applied purposes. Dependent on the properties of the potential, the spectrum may or may not contain a discrete component apart from the absolutely continuous part. In this talk we aim to describe what are the properties of potentials generating point spectrum at the spectral edge, i.e., an eigenvalue embedded at the bottom of the continuous spectrum. This delicate borderline situation has several interesting consequences and applications. This talk is based on joint work with Jozsef Lörinczi.