Complete asymptotic analysis of low energy scattering for Schrodinger operators with a short-range potential

TL;DR

This paper completes the asymptotic analysis of low-energy scattering for Schrödinger operators with short-range potentials across all regimes, dimensions, and geometries, extending previous partial results.

Contribution

It refines Hintz's algorithm to provide full asymptotic expansions in all regimes and dimensions, including multipole expansions, for Schrödinger operators.

Findings

Full asymptotic expansions in all regimes and dimensions

Application to Schrödinger, wave, and Klein-Gordon equations

Enhanced understanding of low-energy scattering behavior

Abstract

Recent work by Hintz--Vasy provides a partial asymptotic analysis of the low-energy limit of scattering for Schr\"odinger operators with a short-range potential. Using a slight refinement of Hintz's algorithm, we complete the asymptotic analysis by providing full asymptotic expansions in every possible asymptotic regime. Moreover, the analysis is done in any dimension $d\geq 3$, for any asymptotically conic manifold, and we keep track of partial multipole expansions. Applications include full asymptotic analyses of the Schr\"odinger, wave, and Klein--Gordon equations, one of these being described in a companion paper. Using previous work, only partial asymptotic analyses were possible.