  In this article we prove short time local well-posedness in low-regularity
Sobolev spaces for large data general quasilinear Schr\"odinger equations with
a non-trapping assumption. These results represent improvements over the small
data regime considered by the authors in previous works, as well as the
pioneering works by Kenig-Ponce-Vega and Kenig-Ponce-Rolvung-Vega, where
viscosity methods were used to prove existence of solutions for localized data
in high regularity spaces. Our arguments here are purely dispersive. The
function spaces in which we show existence are constructed in ways motivated by
the results of Mizohata, Ichinose, Doi, and others, including the authors.
