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

Serge Richard (1) , Rafael Tiedra de Aldecoa (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 Facultad de Matemáticas [Santiago de Chile]

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.

Mots clés: Manifolds; spectral analysis; scattering theory; conjugate operator

Domaines: Physique mathématique [math-ph]

Serge Richard, Rafael Tiedra de Aldecoa. Spectral analysis and time-dependent scattering theory on manifolds with asymptotically cylindrical ends. Reviews in Mathematical Physics, 2013, 25, pp.1350003-1 - 1350003-40. ⟨hal-00660532⟩