We will discuss the Cauchy Problem for a class of quasilinear
Schr\"odinger equations in which the second order terms do not need to be
elliptic. We will outline the local in time existence results for initial
data in appropriate Sobolev spaces. This work build on the work on previous
work as well as the papers of C. Kenig, G. Ponce, L. Vega, and C. Roulvung.
We show that under appropriate conditions it is not necessary to require the
initial data to lie in weighted Sobolev spaces in order to have local in