Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends

Details Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends

2011-08-25

arxiv.org/abs/1108.5018v1

Physics

Mathematical Physics

30 pages

Scientific paper

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud-Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.

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