Hanisch, F, Strohmaier, A orcid.org/0000-0002-8446-3840 and Waters, A (2022) A relative trace formula for obstacle scattering. Duke Mathematical Journal. ISSN 0012-7094
Abstract
We consider the case of scattering by several obstacles in RdRd for d≥2d≥2. In this setting, the absolutely continuous part of the Laplace operator Δ with Dirichlet boundary conditions and the free Laplace operator Δ0Δ0 are unitarily equivalent. For suitable functions that decay sufficiently fast, we have that the difference g(Δ)−g(Δ0)g(Δ)−g(Δ0) is a trace-class operator and its trace is described by the Krein spectral shift function. In this article, we study the contribution to the trace (and hence the Krein spectral shift function) that arises from assembling several obstacles relative to a setting where the obstacles are completely separated. In the case of two obstacles, we consider the Laplace operators Δ1Δ1 and Δ2Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. Our main result in this case states that then g(Δ)−g(Δ1)−g(Δ2)+g(Δ0)g(Δ)−g(Δ1)−g(Δ2)+g(Δ0) is a trace-class operator for a much larger class of functions (including functions of polynomial growth) and that this trace may still be computed by a modification of the Birman–Krein formula. In case g(x)=x12g(x)=x12, the relative trace has a physical meaning as the vacuum energy of the massless scalar field and is expressible as an integral involving boundary layer operators. Such integrals have been derived in the physics literature using nonrigorous path integral derivations and our formula provides both a rigorous justification as well as a generalization.
Metadata
Item Type: | Article |
---|---|
Authors/Creators: |
|
Copyright, Publisher and Additional Information: | © 2022 Duke University Press. This is an author produced version of an article published in Duke Mathematical Journal. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Birman–Krein formula , Casimir energy , obstacle scattering , trace formula |
Dates: |
|
Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 06 Aug 2021 15:03 |
Last Modified: | 29 Jul 2022 22:30 |
Status: | Published online |
Publisher: | Duke University Press |
Identification Number: | 10.1215/00127094-2022-0053 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:176832 |