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

Yurii V. Brezhnev

Tomsk State University, 36 Lenin Ave., Tomsk 634050, Russia

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.

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

MSC: 14H70; 33E05; 33E10; 37N20; 37J35; 81S10

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

Language: English

DOI: 10.3842/SIGMA.2015.035

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ArXiv: 1504.07063