Spectral invariants of integrable polygons
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910616632950784 |
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| author | Mårdby, Gustav Rowlett, Julie |
| author_facet | Mårdby, Gustav Rowlett, Julie |
| contents | An integrable polygon is one whose interior angles are fractions of $π$; that is to say of the form $\frac πn$ for positive integers $n$. We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14391 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral invariants of integrable polygons Mårdby, Gustav Rowlett, Julie Spectral Theory Mathematical Physics Number Theory 35P15, 58J35, 11M36, 53C22 An integrable polygon is one whose interior angles are fractions of $π$; that is to say of the form $\frac πn$ for positive integers $n$. We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation. |
| title | Spectral invariants of integrable polygons |
| topic | Spectral Theory Mathematical Physics Number Theory 35P15, 58J35, 11M36, 53C22 |
| url | https://arxiv.org/abs/2409.14391 |