Group algebras acting on the space of solutions of a special double confluent Heun equation
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GROUP ALGEBRAS ACTING ON THE SPACE OF SOLUTIONS OF A SPECIAL DOUBLE CONFLUENT HEUN EQUATION V. M. Buchstaber∗ and S. I. Tertychnyi†
We study properties of the space Ω of solutions of a special double confluent Heun equation closely related to the model of a overdamped Josephson junction. We describe operators acting on Ω and relations in the algebra A generated by them over the real number field. The structure of A depends on parameters. We give conditions under which A is isomorphic to a group algebra and describe two corresponding group structures.
Keywords: special double confluent Heun equation, monodromy operator, solution space symmetry, group algebra DOI: 10.1134/S0040577920080012
Dedicated to the memory of S. Yu. Slavyanov
1. Introduction Heun class equations are the central topic in [1]. We call the following equation the special double confluent Heun equation (DCHE) for a function E = E(z): z 2 E (z) + ((l + 1)z + μ(1 − z 2 ))E (z) + (−μ(l + 1)z + λ)E(z) = 0,
(1)
where λ, μ, and l are constant complex parameters and the prime denotes the derivative with respect to the complex variable z. This equation forms a three-parameter subfamily in the family of DCHEs [1]–[4]. After division by z 2 , we obtain an equation on the Riemann sphere (or CP 1 ) whose coefficients have singularities only at zero and infinity. Both these singular points of Eq. (1) are irregular. It is natural to use the Riemann surface C∗ (the universal cover of C∗ ) as the domain of solutions of Eq. (1). The space C∗ can be identified with C such that the group operation in C∗ corresponds to the operation of adding complex numbers in C. Moreover, the projection C∗ → C∗ is given by the holomorphic function z = eζ , where z ∈ C∗ and ζ ∈ C. In C∗ , there is a multiplicative group structure with respect to which the projection exp is a group homomorphism, and we obtain the exact sequence j
exp
0 −→ Z −→ C∗ −→ C∗ −→ 0. ∗
Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia, e-mail: [email protected].
†
All-Russian Scientific Research Institute for Physical and Radio-Technical Measurements (VNIIFTRI), Mendeleevo, Russia, e-mail: [email protected] (corresponding author). This research was supported in part by the Russian Foundation for Basic Research (Grant No. 17-01-00192). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 204, No. 2, pp. 153–170, August, 2020. Received March 6, 2020. Revised March 6, 2020. Accepted April 6, 2020. c 2020 Pleiades Publishing, Ltd. 0040-5779/20/2042-0967
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The group Z of integers acts by shifts on the plane C of the complex variable. The projection exp allows lifting functions f (z) holomorphic on C∗ to functions f˜(ζ) holomorphic on C, and such functions are invariant under the shift ζ → ζ + 2iπ. Therefore, on the space of functions holomorphic on C∗ , the action of the monodromy operator M shifts the variable ζ by 2iπ. The change of variables z = eζ transforms Eq. (1) into the equation ¨ ˙ E(ζ) + (l − 2μ sinh ζ)E(ζ) + (λ − (l + 1)μeζ )E(ζ) = 0,
(2)
whose co
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