On a Quantization of the Classical $\theta$-Functions
Mathematical Physics
2015-04-30 v2 math.MP
Quantum Physics
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.
Cite
@article{arxiv.1504.07063,
title = {On a Quantization of the Classical $\theta$-Functions},
author = {Yurii V. Brezhnev},
journal= {arXiv preprint arXiv:1504.07063},
year = {2015}
}