On the convergence of the Born series for Coulomb potentials

TL;DR

This paper proves the convergence of the Born series for Coulomb potentials on asymptotically conic manifolds at high energy, accommodating multiple singularities and long-range interactions, and establishes a resonance-free region.

Contribution

It provides a novel proof of Born series convergence for complex Coulomb potentials using advanced semiclassical estimates and Sobolev spaces.

Findings

Convergence of Born series at high energy for Coulomb potentials.

Existence of a resonance-free region for Hamiltonians with Coulomb singularities.

Handling of long-range Coulomb potentials using anisotropic semiclassical Sobolev spaces.

Abstract

We provide a short proof of the convergence of the Born series on asymptotically conic manifolds, at sufficiently high energy. The potential is allowed to have multiple Coulomb singularities. This is handled using powerful semiclassical estimates recently proven by Hintz for the case of a single dipole (or better) singularity. The potential is also allowed to be long-range, like the actual Coulomb potential $1/r$; long-range potentials are handled using anisotropic semiclassical (sc-) Sobolev spaces. As a consequence of the above estimates, we show the existence of a resonance-free region for Hamiltonians with multiple screened Coulomb singularities.