In this lecture we will study the well posedness problem for the nonlinear Schr\\{o}dinger equation for the magnetic Laplacian on $\\R^{2n}$, corresponding to constant magnetic field, namely the twisted Laplacian on $\\C^n$. We establish the well posednes in certain first order Sobolev spaces associated to the twisted Laplacian, and also in $L^2(\\C^n)$.

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