A note on the Schr\"odinger operator with a long-range potential

TL;DR

This paper develops a spectral and scattering theory for one-dimensional Schr"odinger operators with long-range potentials using a new approach that requires less restrictive conditions on the potential's derivatives.

Contribution

It introduces a novel Ansatz that enables spectral and scattering analysis under the minimal assumption that the potential's derivative is integrable.

Findings

Spectral theory established for potentials with $q' \

,

,

Abstract

Our goal is to develop spectral and scattering theories for the one-dimensional Schr\"odinger operator with a long-range potential $q(x)$, $x\geq 0$. Traditionally, this problem is studied with a help of the Green-Liouville approximation. This requires conditions on the first two derivatives $q' (x)$ and $q'' (x)$. We suggest a new Ansatz that allows us to develop a consistent theory under the only assumption $q' \in L^1$.