Semi-classical asymptotics for scattering length of symmetric stable processes

Abstract

In this paper we study the problem of semi-classical asymptotics for the scattering length of non-negative potentials with infinite range. It was proved analytically by Tamura (1993) that the scattering length Γ(V) of a non-negative V induced by 3-dimensional Brownian motion obeys Γ(ε−2V)∼ε−2∕(ρ−2) in the semi-classical limit ε→0, if V(x) behaves like |x|−ρ,ρ>3 at infinity. We will extend this result probabilistically for the scattering length of non-negative potentials including a jumping function under the framework of symmetric stable processes.