Scattering length for stable processes

Abstract

Let $α\in(0,2)$ and $X_t$ be a symmetric $α$-stable process. We define the scattering length $Γ(v)$ of the positive potential $v$ and prove several of its basic properties. We use the scattering length to findestimates for the first eigenvalue of the Schrödinger operator of the ``Neumann'' fractional Laplacian in a cube with potential $v$.