!- keywords for search engines><!- colors of text and background, "bgcolor" is redundant> We investigate exponential-based adaptive numerical time integrators for time-dependent systems of linear ordinary differential equations of Schrödinger type. Applications in the study of the design of novel solar cells motivate the interest in finding efficient adaptive time integration methods for this task. We consider commutator-free Magnus-type methods, classical Magnus integrators and novel integrators based on a splitting approach. In all the methods, efficient time-stepping is realized based on defect-based estimators for the local error constructed especially for the task. We show the asymptotical correctness of the error estimators and demonstrate the advantages of adaptive time-stepping.