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Article overview
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On monodromy eigenfunctions of Heun equations and boundaries of phase-lock areas in a model of overdamped Josephson effect | Victor M. Buchstaber
; Alexey A. Glutsyuk
; | Date: |
1 Sep 2016 | Abstract: | We study a family of double confluent Heun equations that are linearizations
of nonlinear equations on two-torus modeling the Josephson effect in
superconductivity. They have the form $mathcal L E=0$, where $mathcal
L=mathcal L_{lambda,mu,n}$ is a family of differential operators of order
two acting on germs of holomorphic functions in one complex variable. They
depend on parameters $lambda,ninmathbb R$, $mu>0$, $lambda+mu^2equiv
const>0$. We show that for every $binmathbb C$ and $ninmathbb R$ satisfying
a certain "non-resonance condition" and every parameter values $lambda$, $mu$
there exists a unique entire function $f_{pm}:mathbb C omathbb C$ (up to
multiplicative constant) such that $z^{-b}mathcal L(z^b
f_{pm}(z^{pm1}))=d_{0pm}+d_{1pm}z$ for some $d_{0pm},d_{1pm}inmathbb
C$. The latter $d_{j,pm}$ 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. | Source: | arXiv, 1609.0244 | Services: | Forum | Review | PDF | Favorites |
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