Extended Model of Josephson Junction, Linear Systems with Polynomial Solutions, Determinantal Surfaces, and Painlevé III Equations

We consider a three-parameter family of linear special double confluent Heun equations introduced and studied by V. M. Buchstaber and S. I. Tertychniy, which is an equivalent presentation of a model of Josephson junction in superconductivity. Buchstaber and Tertychniy have shown that the set of those complex parameters for which the Heun equation has a polynomial solution is a union of explicit algebraic curves in $\mathbb C^2$, so-called spectral curves, indexed by $\ell \in \mathbb N$. In a joint paper with I. V. Netay, the author showed that each spectral curve is irreducible in the parameter space of the Heun equation (and consists of two irreducible components in the parameter space of the Josephson junction model). Netay discovered numerically and conjectured a genus formula for spectral curves. He reduced it to the conjecture stating that each of the spectral curves is regular in $\mathbb C^2$ outside a coordinate axis. Here we prove Netay's regularity and genus conjectures. To prove them, we study a four-parameter family of linear systems on the Riemann sphere that extends a family of linear systems equivalent to the Heun equations. It yields an equivalent presentation of the extension of the model of Josephson junction introduced by the author in a joint paper with Yu. P. Bibilo. We describe the so-called determinantal surfaces, which consist of linear systems with polynomial solutions, as explicit affine algebraic hypersurfaces in $\mathbb C^3$. The spectral curves are their intersections with the hyperplane corresponding to the initial model. We prove that each determinantal surface is regular outside an appropriate hyperplane and consists of two rational irreducible components. The proofs use the theory of Stokes phenomena, the holomorphic vector bundle technique, and isomonodromic deformations governed by the Painlevé III equation.