Reference : Scattering theory for the Hodge-Laplacian
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Scattering theory for the Hodge-Laplacian
[en] Scattering theory for the Hodge-Laplacian
[de] Zur Streutheorie des Hodge-de Rham Laplace-Operators
[fr] Sur la théorie dispersion du laplacien de Hodge-de Rham
Baumgarth, Robert mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
[en] Scattering theory, Wave operators ; Bismut type derivative formulae ; Hodge-Laplacian
[en] We prove using an integral criterion the existence and completeness of the wave operators corresponding to the Hodge-Laplacians acting on differential p-forms, induced by two quasi-isometric Riemannian metrics g and h on a complete open smooth manifold M. In particular, this result provides a criterion for the absolutely continuous spectra to coincide. The proof is based on gradient estimates obtained by probabilistic Bismut-type formulae for the heat semigroup defined by spectral calculus. By these localised formulae, the integral criterion only requires local curvature bounds and some upper local control on the heat kernel acting on functions, but no control on the injectivity radii. A consequence is a stability result of the absolutely continuous spectrum under a Ricci flow. As an application we concentrate on the important case of conformal perturbations.

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