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TMF, 2019, Volume 201, Number 1, Pages 17–36 (Mi tmf9731)  

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

S. I. Tertychnyi

All-Russian Scientific Research Institute of Physical-Technical and Radiotechnical Measurements, Mendeleevo, Moscow Oblast, Russia

Abstract: 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.

Keywords: double confluent Heun equation, solution space automorphism, monodromy, composition law, matrix representation, solution continuation, RSJ model of Josephson junction.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-01-00192
This research is supported in part by the Russian Foundation for Basic Research (Grant No. 17-01-00192).


DOI: https://doi.org/10.4213/tmf9731

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English version:
Theoretical and Mathematical Physics, 2019, 201:1, 1426–1441

Bibliographic databases:

PACS: 74.50.+r
MSC: 33E30, 33C47, 34A05, 34A25, 34M35
Received: 12.04.2019
Revised: 20.05.2019

Citation: S. I. Tertychnyi, “Solution space monodromy of a special double confluent Heun equation and its applications”, TMF, 201:1 (2019), 17–36; Theoret. and Math. Phys., 201:1 (2019), 1426–1441

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  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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