Sommerfeld radiation condition at threshold

TL;DR

This paper establishes Besov space bounds for the resolvent at low energies for a broad class of potentials, including Coulomb, without symmetry assumptions, and characterizes zero-energy boundary values via radiation conditions.

Contribution

It introduces zero-energy Sommerfeld radiation conditions and proves resolvent bounds for non-spherical, negative potentials at low energies.

Findings

Resolvent bounds in Besov spaces at low energies

Characterization of zero-energy boundary values

Extension to Coulomb and similar potentials

Abstract

We prove Besov space bounds of the resolvent at low energies in any dimension for a class of potentials that are negative and obey a virial condition with these conditions imposed at infinity only. We do not require spherical symmetry. The class of potentials includes in dimension $\geq3$ the attractive Coulomb potential. There are two boundary values of the resolvent at zero energy which we characterize by radiation conditions. These radiation conditions are zero energy versions of the well-known Sommerfeld radiation condition.