Finite-basis-set technique for solving time-dependent Schrödinger equations

Time-dependent Schr\"odinger (TDS) equations can be solved efficiently by expanding the solution on a finite basis set. For strongly time-dependent regimes, a set of complex static vectors is proposed as a natural basis. Proofs of completeness and optimization prescriptions are given in a general way, suggesting that this technique can be used to solve a broad class of TDS equations. As an illustration, we use this optimal basis-set expansion to solve a simplified two-dimensional free-electron laser model that can be formulated in terms of a TDS equation. This physical application allows us to prove the accuracy of this basis-set technique by comparison with a standard mesh technique.