In this article we review first some of the possibilities in which the notions of Fo lner sequences and quasidiagonality have been applied to spectral approximation problems. We construct then a canonical Fo lner sequence for the crossed product of a concrete C* -algebra and a discrete amenable group. We apply our results to the rotation algebra (which contains interesting operators like almost Mathieu operators or periodic magnetic Schrödinger operators on graphs) and the C* -algebra generated by bounded Jacobi operators.