A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functions Article Swipe
Alexander Adam
,
Anke Pohl
·
YOU?
·
· 2018
· Open Access
·
· DOI: https://doi.org/10.1017/etds.2018.51
YOU?
·
· 2018
· Open Access
·
· DOI: https://doi.org/10.1017/etds.2018.51
Over the last few years Pohl (partly jointly with coauthors) has developed dual ‘slow/fast’ transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ with cusps and all finite-dimensional unitary representations $\unicode[STIX]{x1D712}$ of $\unicode[STIX]{x1D6E4}$ . The eigenfunctions with eigenvalue 1 of the fast transfer operators determine the zeros of the Selberg zeta function for $(\unicode[STIX]{x1D6E4},\unicode[STIX]{x1D712})$ . Further, if $\unicode[STIX]{x1D6E4}$ is cofinite and $\unicode[STIX]{x1D712}$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue 1 of the slow transfer operators characterize Maass cusp forms for $\unicode[STIX]{x1D6E4}$ . Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $\unicode[STIX]{x1D6E4}\backslash \mathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $\unicode[STIX]{x1D712}$ of the Hecke triangle group $\unicode[STIX]{x1D6E4}$ . In particular, we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Möller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.
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- article
- Language
- en
- Landing Page
- https://doi.org/10.1017/etds.2018.51
- OA Status
- green
- Cited By
- 3
- References
- 73
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W2765655039
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https://openalex.org/W2765655039Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1017/etds.2018.51Digital Object Identifier
- Title
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A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functionsWork title
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articleOpenAlex work type
- Language
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enPrimary language
- Publication year
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2018Year of publication
- Publication date
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2018-08-10Full publication date if available
- Authors
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Alexander Adam, Anke PohlList of authors in order
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https://doi.org/10.1017/etds.2018.51Publisher landing page
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/1606.09109Direct OA link when available
- Concepts
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Selberg trace formula, Mathematics, Automorphic form, Pure mathematics, Eigenfunction, Riemann zeta function, Hecke operator, Cusp (singularity), Transfer operator, Unitary representation, Eigenvalues and eigenvectors, Mathematical analysis, Modular form, Lie group, Geometry, Physics, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
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3Total citation count in OpenAlex
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2023: 1, 2020: 2Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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| referenced_works | https://openalex.org/W2080253405, https://openalex.org/W1966848242, https://openalex.org/W2593370177, https://openalex.org/W2077608758, https://openalex.org/W2964347540, https://openalex.org/W2089647072, https://openalex.org/W4255042285, https://openalex.org/W2042968368, https://openalex.org/W2033580296, https://openalex.org/W4253412396, https://openalex.org/W2963150239, https://openalex.org/W2006552368, https://openalex.org/W2167375677, https://openalex.org/W1967936406, https://openalex.org/W6760631224, https://openalex.org/W2052819073, https://openalex.org/W4211107600, https://openalex.org/W2963668783, https://openalex.org/W2168651023, https://openalex.org/W2962925377, https://openalex.org/W2092748042, https://openalex.org/W6635236187, https://openalex.org/W6600773470, https://openalex.org/W2040404114, https://openalex.org/W1626196745, https://openalex.org/W132516776, https://openalex.org/W2962747702, https://openalex.org/W4246070966, https://openalex.org/W2015302289, https://openalex.org/W2963773897, https://openalex.org/W1520848524, https://openalex.org/W2020939759, https://openalex.org/W2950891784, https://openalex.org/W1985857248, https://openalex.org/W4242350006, https://openalex.org/W2111482760, https://openalex.org/W1537552285, https://openalex.org/W1970909788, https://openalex.org/W2010410868, https://openalex.org/W2962853814, https://openalex.org/W2013872780, https://openalex.org/W2090111652, https://openalex.org/W2325015015, https://openalex.org/W2118402228, https://openalex.org/W2090736834, https://openalex.org/W2021890214, https://openalex.org/W1495027777, https://openalex.org/W2036138621, https://openalex.org/W2331228503, https://openalex.org/W6662996866, https://openalex.org/W3183602334, https://openalex.org/W2964161446, https://openalex.org/W3104158215, https://openalex.org/W3103966093, https://openalex.org/W1968665623, https://openalex.org/W3099148112, https://openalex.org/W3101800758, https://openalex.org/W2017874208, https://openalex.org/W2049909177, https://openalex.org/W2111552383, https://openalex.org/W4230742884, https://openalex.org/W4394640298, https://openalex.org/W18679530, https://openalex.org/W3105620089, https://openalex.org/W1591355866, https://openalex.org/W1527758730, https://openalex.org/W3102752390, https://openalex.org/W2261496677, https://openalex.org/W1493483184, https://openalex.org/W2950864842, https://openalex.org/W3212292137, https://openalex.org/W2920142520, https://openalex.org/W4251479369 |
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