Josephson effect, Arnold tongues and double confluent Heun equations

In 1973 B. Josephson received Nobel Prize for discovering a new fundamental effect in superconductivity concerning a system of two superconductors separated by a very narrow dielectric (this system is called the Josephson junction): there could exist a supercurrent tunneling through this junction. We will discuss the reduction of the overdamped Josephson junction to a family of first order non-linear ordinary differential equations that defines a family of dynamical systems on two-torus. Physical problems of the Josephson junction led to studying the rotation number of the above-mentioned dynamical system on the torus as a function of the parameters and to the problem on the geometric description of the phase-lock areas: the level sets of the rotation number function $\rho$ with non-empty interiors.
Phase-lock areas were observed and studied for the first time by V.I.Arnold in the so-called Arnold family of circle diffeomorphisms at the beginning of 1970-ths. He has shown that in his family the phase-lock areas (which later became Arnold tongues) exist exactly for all the rational values of the rotation number.
In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). On their complement, which is an open set, the rotation number function $\rho$ is an analytic submersion that induces its fibration by analytic curves. It appears that the family of dynamical systems on torus under consideration is equivalent to a family of second order linear complex differential equations on the Riemann sphere with two irregular singularities, the well-known double confluent Heun equations. This family of linear equations has the form $\mathcal{L} E=0$, where $\mathcal{L}=\mathcal{L}_{\lambda,\mu,n}$ is a family of second order differential operators acting on germs of holomorphic functions of one complex variable. They depend on complex parameters $\lambda$, $\mu$, $n$. The above-mentioned dynamical systems on torus correspond to the equations with real parameters satisfying the inequality $\lambda+\mu^2>0$. The monodromy of the Heun equations is expressed in terms of the rotation number. We show that for all $b,n\in\mathbb{C}$ satisfying a certain “non-resonance condition” and for all parameter values $\lambda,\mu\in\mathbb{C}$, $\mu\neq0$ there exists an entire function $f_{\pm}:\mathbb{C}\to\mathbb{C}$ (unique up to a constant factor) such that $z^{-b}\mathcal{L}(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb{C}$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those values $\lambda$, $\mu$, $n$ and $b$ for which the monodromy operator of the corresponding Heun equation has eigenvalue $e^{2\pi i b}$. It also gives the description of those values $\lambda$, $\mu$, $n$ for which the monodromy is parabolic, i.e., has a multiple eigenvalue; they correspond exactly to the boundaries of the phase-lock areas. This implies the explicit description of the union of boundaries of the phase-lock areas as solutions of an explicit transcendental functional equation. For every $\theta\notin\mathbb{Z}$ we get a description of the set $\{\rho\equiv\pm\theta(mod2\mathbb{Z})\}$.
The talk will be accessible for a wide audience and devoted to different connections between physics, dynamical systems on two-torus and applications of analytic theory of complex linear differential equations.