CP methods for the Schrِdinger equation

After a short survey over the efforts in the direction of solving the Schrödinger equation by using piecewise approximations of the potential function, the paper focuses on the piecewise perturbation methods in their CP implementation. The presentation includes a short list of problems for which CP versions are available, a sketch of the derivation of the CPM formulae, a description of various ways to construct or identify a certain version and also the main results of the error analysis. One of the most relevant results of the latter is that the energy dependence of the error is bounded, a fact which places these methods on a special position among the numerical methods for differential equations. A numerical illustration is also included in which a CPM based code for the regular Sturm–Liouville problem is compared with some other, well-established codes.