Journal of Spectral Theory


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Volume 7, Issue 1, 2017, pp. 235–267
DOI: 10.4171/JST/162

Published online: 2017-03-22

Scattering theory of the Hodge–Laplacian under a conformal perturbation

Francesco Bei[1], Batu Güneysu[2] and Jörn Müller[3]

(1) Humboldt-Universität zu Berlin, Germany
(2) Humboldt-Universität zu Berlin, Germany
(3) Humboldt-Universität zu Berlin, Germany

Let $g$ and $\tilde{g}$ be Riemannian metrics on a noncompact manifold $M$, which are conformally equivalent. We show that under a very mild first order control on the conformal factor, the wave operators corresponding to the Hodge–Laplacians $\Delta_g$ and $\Delta_{\tilde{g}}$ acting on differential forms exist and are complete. We apply this result to Riemannian manifolds with bounded geometry and more specically, to warped product Riemannian manifolds with bounded geometry. Finally, we combine our results with some explicit calculations by Antoci to determine the absolutely continuous spectrum of the Hodge–Laplacian on $j$-forms for a large class of warped product metrics.

Keywords: Scattering theory, wave operators, Hodge–Laplacian, conformal perturbations

Bei Francesco, Güneysu Batu, Müller Jörn: Scattering theory of the Hodge–Laplacian under a conformal perturbation. J. Spectr. Theory 7 (2017), 235-267. doi: 10.4171/JST/162