Sommerfeld Radiation Condition for Helmholtz Equations with long-range Potentials

Abstract

We study the electric Helmholtz equation $\Delta u + Vu + \lambda u =f$ and show that, for certain potentials, the solution $u$ given by the limited absorption principle obeys a Sommerfeld radiation condition. We use a non-spherical approach based on the solution $K$ of the eikonal equation $|\nabla K|^2=1 + \frac{p}{\lambda}$ to improve previous results in that area and extend them to long-range potentials which decay like $|x|^{-2-\alpha}$ at infinity, with $\alpha > 0$.