Absorbing boundary conditions for the one-dimensional Schrِdinger equation with an exterior repulsive potential

Mathematical constructions and comparisons of accurate absorbing boundary conditions for the one-dimensional Schrِdinger equation with a general variable repulsive potential are developed. Stable semi-discretization schemes are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbing boundary conditions and show that they yield accurate computations.