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 ${\text{C}}^n$ 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 $A(z)=iz,z\in \text{C}$.