Magnus-Lanczos methods with simplified commutators for the Schrödinger equation with a time-dependent potential

Abstract: The computation of the Schr\"odinger equation featuring time-dependent potentials is of great importance in quantum control of atomic and molecular processes. These applications often involve highly oscillatory potentials and require inexpensive but accurate solutions over large spatio-temporal windows. In this work we develop Magnus expansions where commutators have been simplified. Consequently, the exponentiation of these Magnus expansions via Lanczos iterations is significantly cheaper than that for traditional Magnus expansions. At the same time, and unlike most competing methods, we simplify integrals instead of discretising them via quadrature at the outset -- this gives us the flexibility to handle a variety of potentials, being particularly effective in the case of highly oscillatory potentials, where this strategy allows us to consider significantly larger time steps.