Perturbative numerical methods to solve the Schroedinger equation

The perturbation theory was thought for long as an efficient tool to describe the physical phenomena in terms of analytic formulas, thus avoiding (or postponing at least) the numerical approach. The complementary attitude, that is to use the perturbation theory as the very tool for the computation itself, came at light only about ten years ago. The resulted perturbative algorithms are nowadays more efficient than the classical ones and thus they open the possibility to treat problems which earlier seemed prohibited by the amount of computation. The purpose of these notes is to review the field in its main lines. We restrict ourselves to the case of the one-dimensional Schroedinger equation and show: (i) how can the perturbation theory be used to generate new algorithms; (ij) how can it be used to improve the efficiency of the classjcal algorithms. (author)