Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 035, 11 pages      arXiv:1504.07063      https://doi.org/10.3842/SIGMA.2015.035
Contribution to the Special Issue on Algebraic Methods in Dynamical Systems

On a Quantization of the Classical $\theta$-Functions

Yurii V. Brezhnev
Tomsk State University, 36 Lenin Ave., Tomsk 634050, Russia

Received January 31, 2015, in final form April 17, 2015; Published online April 28, 2015

Abstract
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.

Key words: Jacobi theta-functions; dynamical systems; Poisson brackets; quantization; spectrum of Hamiltonian.

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