arXiv subjects

Alexey A. Glutsyuk

Publications and source records attributed to Alexey A. Glutsyuk.

2 recordsLinked to original sources

arXiv2017

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

We study a family of double confluent Heun equations of the form $\mathcal E=0$, where $\mathcal L=\mathcal L_{λ,μ,n}$ is a family of differential operators of order two. They depend on complex parameters $λ$, $μ$, $n$. Its restriction to real parameter domain $λ+μ^2>0$ is a linearization of the family of nonlinear equations on two-torus modeling the Josephson effect in superconductivity. We show that for every $b,n\in\mathbb C$ satisfying a certain "non-resonance condition" and every $λ,μ\in\mathbb C$, $μ\neq0$ there exists an entire function $f_{\pm}:\mathbb C\to\mathbb C$ (unique up to constant factor) such that $z^{-b}\mathcal L(z^b f_{\pm}(z^{\pm1}))=d_{0\pm}+d_{1\pm}z$ for some $d_{0\pm},d_{1\pm}\in\mathbb C$. The constants $d_{j,\pm}$ are expressed as functions of the parameters. This result has several applications. First of all, it gives the description of those $λ$, $μ$, $n$, $b$ for which the monodromy of the Heun equation has eigenvalue $e^{2πi b}$. It also describes those $λ$, $μ$, $n$ for which the monodromy is parabolic: has multiple eigenvalue. We consider the rotation number $ρ$ of the dynamical system on two-torus as a function of parameters restricted to a surface $λ+μ^2=const$. The phase-lock areas are its level sets having non-empty interiors. For general families of dynamical systems the problem to describe the boundaries of the phase-lock areas is known to be very complicated. Here we include the results in this direction obtained by methods of complex variables. In our case the phase-lock areas exist only for integer rotation numbers (quantization effect). The result on parabolic monodromy implies the description of the union of their boundaries by an explicit functional equation. For every $θ\notin\mathbb Z$ we get a description of the set $\{ρ\equiv\pmθ(mod2\mathbb Z)\}$.

math.DS↗

arXiv2016

On determinants of modified Bessel functions and entire solutions of double confluent Heun equations

We investigate the question on existence of entire solutions of well-known linear differential equations that are linearizations of nonlinear equations modeling the Josephson effect in superconductivity. We consider the modified Bessel functions $I_j(x)$ of the first kind, which are Laurent series coefficients of the analytic function family $e^{\frac x2(z+\frac 1z)}$. For every $l\geq1$ we study the family parametrized by $k, n\in\mathbb Z^l$, $k_1>\dots>k_l$, $n_1>\dots>n_l$ of $(l\times l)$-matrix functions formed by the modified Bessel functions of the first kind $a_{ij}(x)=I_{k_j-n_i}(x)$, $i,j=1,\dots,l$. We show that their determinants $f_{k,n}(x)$ are positive for every $l\geq1$, $k,n\in\mathbb Z^l$ as above and $x>0$. The above determinants are closely related to a sequence (indexed by $l$) of families of double confluent Heun equations, which are linear second order differential equations with two irregular singularities, at zero and at infinity. V.M.Buchstaber and S.I.Tertychnyi have constructed their holomorphic solutions on $\mathbb C$ for an explicit class of parameter values and conjectured that they do not exist for other parameter values. They have reduced their conjecture to the second conjecture saying that if an appropriate second similar equation has a polynomial solution, then the first one has no entire solution. They have proved the latter statement under the additional assumption (third conjecture) that $f_{k,n}(x)\neq0$ for $k=(l,\dots,1)$, $n=(l-1,\dots,0)$ and every $x>0$. Our more general result implies all the above conjectures, together with their corollary for the overdamped model of the Josephson junction in superconductivity: the description of adjacency points of phase-lock areas as solutions of explicit analytic equations.

math.DS↗