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We discuss several recent results on spectral and scattering theory for a model which generalizes discrete and continuous Schrödinger operators with short-range and long-range perturbations. Many of these were inspired by old papers by Isozaki and Kitada of 1980's.

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a “two point” local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Then we consider the almost stationary, almost symmetric case. There we establish two main results: The first is a “two point” local energy decay estimate which is valid for a general class of (non-symmetric) almost stationary wave equations which satisfy a certain nonresonance property at zero frequency. The second result, which also requires the almost symmetry condition, is to establish an exponential trichotomy in the energy space via finite dimensional time dependent stable and unstable sub-spaces, with an infinite dimensional complement on which solutions disperse via the usual local energy decay estimate.