Josephson effect and double confluent Heun equation

Victor Buchstaber, Russian Academy of Sciences. Part of the applied mathematics seminar series.

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 equation that defines a family of dynamical systems on two-torus. Physical problems of the Josephson junction led to studying the rotation number of this dynamical system 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.

In our case the phase-lock areas exist only for integer rotation numbers (quantization effect), and the complement to them is an open set. On their complement 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 depends on complex parameters lambda, mu, n. Our 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.

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 Z we get a description of the set { rho equiv pm theta (mod2Z) }.

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.

Victor Buchstaber, Russian Academy of Sciences