Renormalized two-body low-energy scattering

TL;DR

This paper develops a stationary scattering theory for Schrödinger operators with long-range potentials, including Coulomb-like cases, that remains well-defined at zero energy and characterizes eigenfunctions in a generalized Besov space.

Contribution

It introduces a regular at-zero-energy scattering framework for certain long-range potentials, extending previous theories to include ultra-strong Coulomb perturbations.

Findings

Well-defined scattering theory at zero energy for specified potentials

Characterization of generalized eigenfunctions in an adapted Besov space

Use of global solutions to the eikonal equation and radiation bounds

Abstract

For a class of long-range potentials, including ultra-strong perturbations of the attractive Coulomb potential in dimension $d\geq3$, we introduce a stationary scattering theory for Schr\"odinger operators which is regular at zero energy. In particular it is well defined at this energy, and we use it to establish a characterization there of the set of generalized eigenfunctions in an appropriately adapted Besov space generalizing parts of \cite{DS3}. Principal tools include global solutions to the eikonal equation and strong radiation condition bounds.