On a Quantization of the Classical θ-Functions
The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find corresponding Poisson brackets. Availability of these ingredients a...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2015 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2015
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/147012 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On a Quantization of the Classical θ-Functions / Y.V. Brezhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862537111364698112 |
|---|---|
| author | Brezhnev, Y.V. |
| author_facet | Brezhnev, Y.V. |
| citation_txt | On a Quantization of the Classical θ-Functions / Y.V. Brezhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 19 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find corresponding Poisson brackets. Availability of these ingredients allows us to state the problem of a canonical quantization to these equations and disclose some important problems. In a particular case the problem is completely solvable in the sense that spectrum of the Hamiltonian can be found. The spectrum is continuous, has a band structure with infinite number of lacunae, and is determined by the Mathieu equation: the Schrödinger equation with a periodic cos-type potential.
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| first_indexed | 2025-11-24T11:44:39Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-147012 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-24T11:44:39Z |
| publishDate | 2015 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Brezhnev, Y.V. 2019-02-12T20:34:07Z 2019-02-12T20:34:07Z 2015 On a Quantization of the Classical θ-Functions / Y.V. Brezhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14H70; 33E05; 33E10; 37N20; 37J35; 81S10 DOI:10.3842/SIGMA.2015.035 https://nasplib.isofts.kiev.ua/handle/123456789/147012 The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series. We study this system in the framework of Hamiltonian dynamics and find corresponding Poisson brackets. Availability of these ingredients allows us to state the problem of a canonical quantization to these equations and disclose some important problems. In a particular case the problem is completely solvable in the sense that spectrum of the Hamiltonian can be found. The spectrum is continuous, has a band structure with infinite number of lacunae, and is determined by the Mathieu equation: the Schrödinger equation with a periodic cos-type potential. This paper is a contribution to the Special Issue on Algebraic Methods in Dynamical Systems. The full
 collection is available at http://www.emis.de/journals/SIGMA/AMDS2014.html.
 The author would like to thank Dima Kaparulin and Peter Kazinsky for stimulating discussions
 and my special thanks are addressed to S. Lyakhovich and A. Sharapov for valuable consultations.
 Also, much gratitude is extended to the anonymous referee for helpful suggestions and
 constructive criticism, which resulted in considerable improvement of the final text. The study
 was supported by the Tomsk State University Academic D. Mendeleev Fund Program. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On a Quantization of the Classical θ-Functions Article published earlier |
| spellingShingle | On a Quantization of the Classical θ-Functions Brezhnev, Y.V. |
| title | On a Quantization of the Classical θ-Functions |
| title_full | On a Quantization of the Classical θ-Functions |
| title_fullStr | On a Quantization of the Classical θ-Functions |
| title_full_unstemmed | On a Quantization of the Classical θ-Functions |
| title_short | On a Quantization of the Classical θ-Functions |
| title_sort | on a quantization of the classical θ-functions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147012 |
| work_keys_str_mv | AT brezhnevyv onaquantizationoftheclassicalθfunctions |