The curl operator on odd-dimensional manifolds
- We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions, its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. We compute the Weyl asymptotics and study the zeta-function. We give a sharp lower eigenvalue bound for positively curved manifolds and analyze the equality case. Finally, we compute the spectrum for flat tori, round spheres, and 3-dimensional spherical space forms. Published under license by AIP Publishing.
Author details: | Christian BärORCiDGND |
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DOI: | https://doi.org/10.1063/1.5082528 |
ISSN: | 0022-2488 |
ISSN: | 1089-7658 |
Title of parent work (English): | Journal of mathematical physics |
Publisher: | American Institute of Physics |
Place of publishing: | Melville |
Publication type: | Article |
Language: | English |
Date of first publication: | 2019/03/07 |
Publication year: | 2019 |
Release date: | 2021/03/23 |
Volume: | 60 |
Issue: | 3 |
Number of pages: | 16 |
Organizational units: | Mathematisch-Naturwissenschaftliche Fakultät / Institut für Mathematik |
DDC classification: | 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik |
Peer review: | Referiert |
Publishing method: | Open Access / Green Open-Access |