Complex eigenenergy spectrum of the Schrodinger equation using Lanczos' tau method

Lanczos' tau method provides a powerful and accurate tool for the resolution of the one-dimensional Schrodinger equation where the potential energy is given in rational form or can be brought into such a form after an appropriate change of variable. The eigenfunction is constructed on two intervals where the different boundary conditions are taken into account by searching, in each case, for a local solution as a product of a polynomial by a function describing the corresponding asymptotic behaviour. The matching of the eigenfunction and its first derivative at the common boundary of the intervals leads to a quantization condition from which the complex poles of the scattering amplitude (i.e. bound states, resonances, growing and virtual states) are obtained. A complete implementation of the method is provided together with a few checks on the bound states of the Coulomb potential and the resonances and virtual states of the exponential potential.