On a quantization of the classical \(\theta\)-functions (Q2353215)

The Jacobi theta-functions admit a definition through the autonomous differential equations (dynamical system); not only through the famous Fourier theta-series \[ \theta(z|\tau)=\sum_{-\infty}^\infty e^{\pi i(k^2\tau+2kz)}. \] In this paper, the author studies this system in the framework of Hamiltonian dynamics systems and finds corresponding Poisson brackets. Availability of these ingredients allows him 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 well known Mathieu equation: the Schrödinger equation with a periodic cos-type potential.