  We present a new technique to obtain polynomial decay estimates for the
matrix coefficients of unitary operators. Our approach, based on commutator
methods, applies to nets of unitary operators, unitary representations of
topological groups, and unitary operators given by the evolution group of a
self-adjoint operator or by powers of a unitary operator. Our results are
illustrated with a wide range of examples in quantum mechanics and dynamical
systems, as for instance Schr\"odinger operators, Dirac operators, quantum
waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walks
or skew products.
