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

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 $q' \in L^1$.