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 |
|---|---|
| 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 |
