Fang, Yan-Long;
Strohmaier, Alexander;
(2022)
Trace singularities in obstacle scattering and the Poisson relation for the relative trace.
Annales mathématiques du Québec
, 46
(1)
pp. 55-75.
10.1007/s40316-021-00188-0.
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Abstract
We consider the case of scattering by several obstacles in Rd , d ≥ 2 for the Laplace operator with Dirichlet boundary conditions imposed on the obstacles. In the case of two obstacles, we have the Laplace operators 1 and 2 obtained by imposing Dirichlet boundary conditions only on one of the objects. The relative operator g() − g(1) − g(2) + g(0) 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) = − 1 π Im(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 ξˆ rel is realanalytic 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.
| Type: | Article |
|---|---|
| Title: | Trace singularities in obstacle scattering and the Poisson relation for the relative trace |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s40316-021-00188-0 |
| Publisher version: | https://doi.org/10.1007/s40316-021-00188-0 |
| Language: | English |
| Additional information: | 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/. |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10181222 |
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