Using new Nash Moser techniques for Fréchet spaces by M. Poppenberg the local existence, uniqueness and continuous dependence of smooth solutions of a special quasilinear evolutionary Schrödinger equation is proved; as basic function space H∞(IRn), the intersection of all Sobolev spaces Hk(IRn), is used. The method consists in finding an appropriate linearization of the given nonlinear Schrödinger equation, and proving that this linear Schrödinger equation admits a strongly continuous evolution operator which provides the necessary a priori estimates for any derivative; this is shown by a transformation procedure using a time dependent metric which overcomes the difficulty arising from nondissipativity of the linearized Schrödinger equation