Modified Jost solutions of Schr\"odinger operators with locally H⁻¹ potentials

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.