Small-energy asymptotics for the Schrödinger equation on the line

The one-dimensional Schrödinger equation is considered when the potential is real valued and integrable and has a finite first moment. The small-energy asymptotics of the logarithmic spatial derivative of the Jost solutions are established. Some consequences of these asymptotics are presented, such as the small-energy limits of the scattering coefficients and a simplified characterization of the scattering data for the inverse scattering problem. When the potential also has a finite second moment, some improved results are given on the small-energy asymptotics of the scattering coefficients and the logarithmic spatial derivatives of the Jost solutions.