Approximations to the action of unitary matrix exponentials provide numerical methods for time propagation of spatially discretized linear Schrödinger-type equations. The discretized problem, besides resolving frequencies which are relevant for the given initial state, often includes perturbations in relatively high frequency ranges. Such perturbations can critically affect the convergence of (polynomial) approximations to the action of the matrix exponential. Our approach is to generate unitary rational approximants in barycentric rational form which are accurate in relevant frequency ranges. Due to unitarity, the effect of perturbations is negligible - a property which yields strong advantages in this setting. Relevant frequency ranges (relevant parts of the matrix spectrum) are detected on the run using estimates on the spectral distribution of the initial state. These estimates are based on the Lanczos method and bounds on quadrature weights of Gaussian quadrature formulae.