Nonlinear Schrödinger equation for the twisted Laplacian

We establish the local well posedness of solution to the nonlinear Schrödinger equation associated to the twisted Laplacian on Cn in certain first order Sobolev space. Our approach is based on Strichartz type estimates, and is valid for a general class of nonlinearities including power type. The case n=1 represents the magnetic Schrödinger equation in the plane with magnetic potential A(z)=iz, z∈C.