Solution space monodromy of a special double confluent Heun equation and its applications

We consider three linear operators determining automorphisms of the solution space of a special double confluent Heun equation of positive integer order ($\mathcal{L}$-operators). We propose a new method for describing properties of the solution space of this equation based on using eigenfunctions of one of the $\mathcal{L}$-operators, called the universal $\mathcal{L}$-operator. We construct composition laws for $\mathcal{L}$-operators and establish their relation to the monodromy transformation of the solution space of the special double confluent Heun equation. We find four functionals quadratic in eigenfunctions of the universal automorphism; they have a property with respect to the considered equation analogous to the property of the first integral. Based on them, we construct matrix representations of the $\mathcal{L}$-operators and also the monodromy operator. We give a method for extending solutions of the special double confluent Heun equation from the subset $\operatorname{Re} z>0$ of a complex plane to a maximum domain on which the solution exists. As an example of its application to the RSJ model theory of overdamped Josephson junctions, we give the explicit form of the transformation of the phase difference function induced by the monodromy of the solution space of the special double confluent Heun equation and propose a way to continue this function from a half-period interval to any given interval in the domain of the function using only algebraic transformations.