Dynamical properties of models of physical systems distinguished in having two essential length scales are derived. The method used is non-perturbative. It is cast in terms of systems of orthogonal polynomials, whose properties are reviewed together with recent work on the spectrum and algebraic curves associated with periodic difference operators. Response functions observed in experiments, which can probe spatial and temporal features, are calculated for several models of current interest, including spin excitations in a modulated magnet and the vibrations of a modulated harmonically coupled chain of atoms.
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