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

Symmetry Integrability and Geometry-Methods and Applications

article

Yurii V. Brezhnev¹ 

[1] Tomsk State University

关键词: Jacobi theta-functions;    dynamical systems;    Poisson brackets;    quantization;    spectrum of Hamiltonian;   

DOI  :  10.3842/SIGMA.2015.035

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown