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

Abstract

We study a family of double confluent Heun equations that are lin-earizations of nonlinear equations on two-torus modeling the Joseph-son effect in superconductivity. They have the form LE = 0, where L = L λ,µ,n is a family of differential operators of order two acting on germs of holomorphic functions in one complex variable. They depend on parameters λ, n ∈ R, µ > 0, λ + µ 2 ≡ const > 0. We show that for every b ∈ C and n ∈ R satisfying a certain non-resonance condition and every parameter values λ, µ there exists a unique entire function f ± : C → C (up to multiplicative constant) such that z −b L(z b f ± (z ±1)) = d 0± + d 1± z for some d 0± , d 1± ∈ C. The latter d j,± are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those parameter values for which the monodromy operator of the corresponding Heun equation has given eigenvalues. This yields the description of the non-integer level curves of the rotation number of the family of equations on two-torus as a function of parameters. In the particular case, when the monodromy is parabolic (has multiple eigenvalue), we get the complete description of those parameter values that correspond to the boundaries of the phase-lock areas: integer level sets of the rotation number, which have non-empty interiors.