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

TL;DR

This paper explores the Hamiltonian dynamics of Jacobi theta-functions, formulates their canonical quantization, and analyzes the resulting spectral properties, revealing a band structure related to the Mathieu equation.

Contribution

It introduces a Hamiltonian framework for theta-functions, develops their canonical quantization, and characterizes the spectrum in terms of Mathieu equations.

Findings

Spectrum is continuous with a band structure.

Spectrum has infinite lacunae.

Spectrum determined by Mathieu equation.

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.