Applying splitting methods with complex coefficients to the numerical integration of unitary problems

Abstract

We explore the applicability of splitting methods involving complex coefficients to solve numerically the time-dependent Schr¨odinger equation. We prove that a particular class of integrators are conjugate to unitary methods for sufficiently small step sizes when applied to problems defined in the group SU(2). In the general case, the error in both the energy and the norm of the numerical approximation provided by these methods does not possess a secular component over long time intervals, when combined with pseudo-spectral discretization techniques in space.