A note on the Schrödinger operator with a long-range potential

Abstract

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