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## Собственные функции монодромии уравнений Гойна и границы зон фазового захвата в модели сильношунтированного эффекта Джозефсона

Abstract—We study a family of double confluent Heun equations of the form LE = 0, where

L = L(λ,μ,n) is a family of second-order differential operators acting on germs of holomorphic

functions of one complex variable. They depend on complex parameters λ, μ, and n. The

restriction of the family to real parameters satisfying the inequality λ + μ^2>0 is a linearization

of the family of nonlinear equations on the two-torus that model the Josephson effect in

superconductivity. The main result of the paper gives the description of those values λ, μ, n, and b for which the monodromy

operator of the corresponding Heun equation has eigenvalue exp(2πib). It also gives the description

of those values λ, μ, and n for which the monodromy is parabolic, i.e., has a multiple eigenvalue.

We consider the rotation number ρ of the dynamical system on the two-torus as a function of

parameters restricted to a surface λ + μ^2 = const. The phase-lock areas are its level sets with

nonempty interior. For general families of dynamical systems, the problem of describing the

boundaries of the phase-lock areas is known to be very complicated. In the present paper

we include the results in this direction that were obtained by methods of complex variables.

In our case the phase-lock areas exist only for integer rotation numbers (quantization effect),

and their complement is an open set. On their complement the rotation number function is

an analytic submersion that induces its fibration by analytic curves. The above-mentioned

result on parabolic monodromy implies the explicit description of the union of boundaries of

the phase-lock areas as solutions of an explicit transcendental functional equation. For every non-integer

θ we get a description of the set {ρ ≡ ±θ (mod 2Z)}.