We design and study Schwarz Waveform relaxation algorithms for the linear Schrödinger equation with a potential in one dimension. We show that the overlapping algorithm with Dirichlet exchanges of informations on the boundary is slowly convergent, and we introduce two new classes of algorithms: the optimized Robin algorithm and the quasi-optimal algorithm. We study the well-posedness and convergence, in the overlapping and the non overlapping case, for constant or non constant potentials. We then design a discrete algorithm, based on a finite volumes approach, which permits to obtain convergence results through discrete energies. We also present a quasi-optimal discrete algorithm, based on the transparent discrete boundary condition of Arnold and Ehrhardt. Numerical results illustrate the performances of the methods, even in the case where no convergence result is at hand.
