The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V. M. Buchstaber and S. I. Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are \(l,\lambda ,\mu \in \mathbb {R}\). Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for μ ≠ 0 this happens exactly when \(l\in \mathbb {N}\) and the parameters (λ, μ) lie on an algebraic curve \({\Gamma }_{l}\subset \mathbb {C}^{2}_{(\lambda ,\mu )}\) called the l-spectral curve and defined as zero locus of determinant of a remarkable three-diagonal l × l-matrix. They studied the real part of the spectral curve and obtained important results with applications to model of Josephson junction, which is a family of dynamical systems on 2-torus depending on real parameters (B, A; ω); the parameter ω, called the frequency, is fixed. One of main problems on the above-mentioned model is to study the geometry of boundaries of its phase-lock areas in \(\mathbb {R}^{2}_{(B,A)}\) and their evolution, as ω decreases to 0. An approach to this problem suggested in the present paper is to study the complexified boundaries. We prove irreducibility of the complex spectral curve Γ_l for every \(l\in \mathbb {N}\). We also calculate its genus for \(l\leqslant 20\) and present a conjecture on general genus formula. We apply the irreducibility result to the complexified boundaries of the phase-lock areas of model of Josephson junction. The family of real boundaries taken for all ω > 0 yields a countable union of two-dimensional analytic surfaces in \(\mathbb {R}^{3}_{(B,A,\omega ^{-1})}\). We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four two-dimensional irreducible components, and we describe them. This is done by using the representation of some special points of the boundaries (the so-called generalized simple intersections) as points of the real spectral curves and the above irreducibility result. We also prove that the spectral curve has no real ovals. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits, as ω decreases, and a partial positive result towards its confirmation.

中文翻译：

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约瑟夫森结模型中锁相区域的光谱曲线和复杂边界

本文讨论了由VM Buchstaber和SI Tertychnyi引入并研究的特殊双汇合Heun方程的三参数族，它等效地表示了超导的过阻尼约瑟夫森结模型。参数为\（l，\ lambda，\ mu \ in \ mathbb {R} \）。Buchstaber和Tertychnyi描述了这些参数值，对于这些参数值，相应的方程式具有多项式解。他们表明，对于μ ≠0，这恰好发生在\（l \ in \ mathbb {N} \）和参数（λ，μ）位于代数曲线\（{\ Gamma} _ {l} \ subset \ mathbb {C} ^ {2} _ {（\ lambda，\ mu}} \）称为l光谱曲线定义为显着的三对角l × l-矩阵的行列式的零位。他们研究了光谱曲线的实部，并获得了重要的结果，并将其应用到约瑟夫森结模型中，该模型是一个基于真实参数（B，A，ω）的2托环动力学系统。参数ω（称为频率）是固定的。上述模型的主要问题之一是研究\（\ mathbb {R} ^ {2} _ {（B，A）} \）中其锁相区域边界的几何形状及其演化，如下ω减少到0。本文提出的解决此问题的方法是研究复杂的边界。我们证明了复杂的光谱曲线Γ的不可约_升每\（升\在\ mathbb {N} \） 。我们还计算\（l \ leqslant 20 \）的属，并对一般属公式给出一个猜想。我们将不可约性结果应用于约瑟夫森结模型的锁相区域的复杂边界。对于所有ω > 0采取的实数边界族在\（\ mathbb {R} ^ {3} _ {（B，A，\ omega ^ {-1}）} \\中产生二维分析面的可数并集）。我们发现，出乎意料的是，它的复杂化是一个复杂的分析子集，包括四个二维不可约成分，我们对其进行描述。这是通过使用边界的某些特殊点（所谓的广义简单交点）的表示作为实际光谱曲线的点以及上述不可约性结果来完成的。我们还证明光谱曲线没有真正的椭圆形。当ω减小时，我们就锁相区域肖像的演化提出了单调性猜想，并对其进行了部分肯定的结果。