# On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect

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## 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. We show that for all *b*, *n* ∈ ℂ satisfying a certain “non-resonance condition” and for all parameter values *λ*, *μ* ∈ ℂ, *μ* ≠ 0, there exists an entire function *f* _{±}: ℂ → ℂ (unique up to a constant factor) such that *z* ^{−b } *L*(*z* ^{ b } *f* _{±}(*z* ^{±1})) = *d* _{0±} + *d* _{1±z } for some *d* _{0±}, *d* _{1±} ∈ ℂ. The constants *d* _{ j,±} are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those values *λ*, *μ*, *n*, and *b* for which the monodromy operator of the corresponding Heun equation has eigenvalue *e* ^{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 *θ* ∉ ℤ we get a description of the set {*ρ* ≡ ±*θ* (mod 2ℤ)}.

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