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We present a new technique to obtain polynomial decay estimates for thematrix coefficients of unitary operators. Our approach, based on commutatormethods, applies to nets of unitary operators, unitary representations oftopological groups, and unitary operators given by the evolution group of aself-adjoint operator or by powers of a unitary operator. Our results areillustrated with a wide range of examples in quantum mechanics and dynamicalsystems, as for instance Schr\"odinger operators, Dirac operators, quantumwaveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walksor skew products.

Decay estimates for unitary representations with applications to continuous- and discrete-time models