  In this paper, we apply the Schwarz Waveform Relaxation (SWR) method to the
one dimensional Schr{\"o}dinger equation with a general linear or a nonlinear
potential. We propose a new algorithm for the Schr{\"o}dinger equation with
time independent linear potential, which is robust and scalable up to 500
subdo-mains. It reduces significantly computation time compared with the
classical algorithms. Concerning the case of time dependent linear potential or
the non-linear potential, we use a preprocessed linear operator for the zero
potential case as preconditioner which lead to a preconditioned algorithm. This
ensures high scalability. Besides, some newly constructed absorbing boundary
conditions are used as the transmission condition and compared numerically.
