The aim of this paper is to develop new optimized Schwarz algorithms for the one dimensional Schrödinger equation with linear and nonlinear potential. The classical algorithm is an iterative process. In case of time-independent linear potential, we construct explicitly the interface problem and use direct LU method on the interface problem. The algorithm therefore turns to be a direct process. Thus, the algorithm is independent of transmission condition and the numerical computation is smaller. To our knowledge, this is the first time that the Schwarz algorithm is constructed as direct process. 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.