Strohmaier, A orcid.org/0000-0002-8446-3840 and Fang, Y (2022) Trace Singularities in Obstacle Scattering and the Poisson Relation for the Relative Trace. Annales mathématiques du Québec, 46 (1). pp. 55-75. ISSN 2195-4755
Abstract
We consider the case of scattering by several obstacles in Rd, d≥2 for the Laplace operator Δ with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators Δ1 and Δ2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g(Δ)−g(Δ1)−g(Δ2)+g(Δ0) was introduced in Hanisch, Waters and one of the authors in (A relative trace formula for obstacle scattering. arXiv:2002.07291, 2020) and shown to be trace-class for a large class of functions g, including certain functions of polynomial growth. When g is sufficiently regular at zero and fast decaying at infinity then, by the Birman–Krein formula, this trace can be computed from the relative spectral shift function ξrel(λ)=−1πIm(Ξ(λ)), where Ξ(λ) is holomorphic in the upper half-plane and fast decaying. In this paper we study the wave-trace contributions to the singularities of the Fourier transform of ξrel. In particular we prove that ξ^rel is real-analytic near zero and we relate the decay of Ξ(λ) along the imaginary axis to the first wave-trace invariant of the shortest bouncing ball orbit between the obstacles. The function Ξ(λ) is important in the physics of quantum fields as it determines the Casimir interactions between the objects.
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Item Type: | Article |
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Copyright, Publisher and Additional Information: | © The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | Casimir energy; Spectral Shift function; Trace-formula; Wave-trace invariant |
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) > Applied Mathematics (Leeds) 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: | 01 Nov 2021 15:46 |
Last Modified: | 25 Jun 2023 22:48 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s40316-021-00188-0 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:179728 |
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