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On a Quantization of the Classical θ-Functions
Репозиторій DSpace/Manakin
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2015, том 11
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On a Quantization of the Classical θ-Functions
Інший ID:
2010 Mathematics Subject Classification: 14H70; 33E05; 33E10; 37N20; 37J35; 81S10
DOI:10.3842/SIGMA.2015.035
DOI:10.3842/SIGMA.2015.035
Посилання:
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.
Підтримка:
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.
Дата:
2015
Переглядів:
505
Завантажень:
253
Анотація:
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.
