JournalsjstVol. 7, No. 1pp. 235–267

Scattering theory of the Hodge–Laplacian under a conformal perturbation

  • Francesco Bei

    Humboldt-Universität zu Berlin, Germany
  • Batu Güneysu

    Humboldt-Universität zu Berlin, Germany
  • Jörn Müller

    Humboldt-Universität zu Berlin, Germany
Scattering theory of the Hodge–Laplacian under a conformal perturbation cover
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Abstract

Let gg and g~\tilde{g} be Riemannian metrics on a noncompact manifold MM, 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 Δg\Delta_g and Δg~\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 jj-forms for a large class of warped product metrics.

Cite this article

Francesco Bei, Batu Güneysu, Jörn Müller, Scattering theory of the Hodge–Laplacian under a conformal perturbation. J. Spectr. Theory 7 (2017), no. 1, pp. 235–267

DOI 10.4171/JST/162