Nonlinear Schr\"{o}dinger equation for the twisted Laplacian

We establish the local well posedness of solution to the nonlinear Schr\"{o}dinger equation associated to the twisted Laplacian on $\C^n$ 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 Schr\"{o}dinger equation in the plane with magnetic potential $A(z)=iz, z\in\C$.