Resolvent Trace Formula and Determinants of Laplacians on Orbifold Riemann Surfaces
For a nonnegative integer, we consider the -Laplacian Δₙ acting on the space of -differentials on a confinite Riemann surface X which has ramification points. The trace formula for the resolvent kernel is developed along the line à la Selberg. Using the trace formula, we compute the regularized...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2021 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2021
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211444 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Resolvent Trace Formula and Determinants of Laplacians on Orbifold Riemann Surfaces. Lee-Peng Teo. SIGMA 17 (2021), 083, 40 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | For a nonnegative integer, we consider the -Laplacian Δₙ acting on the space of -differentials on a confinite Riemann surface X which has ramification points. The trace formula for the resolvent kernel is developed along the line à la Selberg. Using the trace formula, we compute the regularized determinant of Δₙ + ( + 2 − 1), from which we deduce the regularized determinant of Δₙ, denoted by det′Δₙ. Taking into account the contribution from the absolutely continuous spectrum, det′Δₙ is equal to a constant Cₙ times ( ) when ≥ 2. Here ( ) is the Selberg zeta function of . When = 0 or = 1, ( ) is replaced by the leading coefficient of the Taylor expansion of ( ) around = 0 and = 1, respectively. The constants Cn are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but are independent of the moduli parameters.
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| ISSN: | 1815-0659 |