Strohmaier, A orcid.org/0000-0002-8446-3840 and Waters, A (2020) Geometric and obstacle scattering at low energy. Communications in Partial Differential Equations, 45 (11). pp. 1451-1511. ISSN 0360-5302
Abstract
We consider scattering theory of the Laplace Beltrami operator on differential forms on a Riemannian manifold that is Euclidean at infinity. The manifold may have several boundary components caused by obstacles at which relative boundary conditions are imposed. Scattering takes place because of the presence of these obstacles and possible non-trivial topology and geometry. Unlike in the case of functions eigenvalues generally exist at the bottom of the continuous spectrum and the corresponding eigenforms represent cohomology classes. We show that these eigenforms appear in the expansion of the resolvent, the scattering matrix, and the spectral measure in terms of the spectral parameter λ near zero, and we determine the first terms in this expansion explicitly. In dimension two an additional cohomology class appears as a resonant state in the presence of an obstacle. In even dimensions the expansion is in terms of λ and log λ. The theory of Hahn holomorphic functions is used to describe these expansions effectively. We also give a Birman-Krein formula in this context. The case of one-forms with relative boundary conditions has direct applications in physics as it describes the scattering of electromagnetic waves.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2020 Taylor & Francis Group, LLC. This is an author produced version of an article published in Communications in Partial Differential Equations. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Birman-Krein formula, cohomology, generalised eigenfunctions, geometric scattering, low energy resolvent, obstacle scattering |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Funding Information: | Funder Grant number Leverhulme Trust RPG-2017-329 |
Depositing User: | Symplectic Publications |
Date Deposited: | 27 May 2020 13:05 |
Last Modified: | 05 Jun 2021 00:38 |
Status: | Published |
Publisher: | Taylor & Francis |
Identification Number: | 10.1080/03605302.2020.1774898 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:160937 |