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arXiv · 1504.07063

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

Yurii V. Brezhnev
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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 Schr\"odinger equation with a periodic cos-type potential.

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Yurii V. Brezhnev. 2015-04-27. On a Quantization of the Classical $\theta$-Functions. https://doi.org/10.3842/sigma.2015.035

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