Modified Jost solutions of Schr\"odinger operators with locally $H^{-1}$ potentials

TL;DR

This paper extends the analysis of Schr"odinger operators by studying Jost solutions for potentials with decay measured in a local $H^{-1}$ norm, including non-integrable and oscillatory potentials, establishing existence and asymptotic behavior.

Contribution

It generalizes previous results to potentials with decay in local $H^{-1}$ norm, including non-integrable and oscillatory cases, and proves existence of solutions with WKB asymptotics.

Findings

Established existence of Jost solutions for $H^{-1}$ decay potentials.

Proved WKB asymptotic behavior for solutions at positive energies.

Extended applicability to non-integrable and oscillatory potentials.

Abstract

We study Jost solutions of Schr\"odinger operators with potentials which decay with respect to a local $H^{-1}$ Sobolev norm; in particular, we generalize to this setting the results of Christ--Kiselev for potentials between the integrable and square-integrable rates of decay, proving existence of solutions with WKB asymptotic behavior on a large set of positive energies. This applies to new classes of potentials which are not locally integrable, or have better decay properties with respect to the $H^{-1}$ norm due to rapid oscillations.