  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.
