  The aim of this paper is to develop new optimized Schwarz algorithms for the
one dimensional Schr{\"o}dinger equation with linear or nonlinear potential.
After presenting the classical algorithm which is an iterative process, we
propose a new algorithm for the Schr{\"o}dinger equation with time-independent
linear potential. Thanks to two main ingredients (constructing explicitly the
interface problem and using a direct method on the interface problem), the new
algorithm turns to be a direct process. Thus, it is free to choose the
transmission condition. Concerning the case of time-dependent linear potential
or nonlinear potential, we propose to use a pre-processed linear operator as
preconditioner which leads to a preconditioned algorithm. Numerically , the
convergence is also independent of the transmission condition. In addition,
both of these new algorithms implemented in parallel cluster are robust,
scalable up to 256 sub domains (MPI process) and take much less computation
time than the classical one, especially for the nonlinear case.
