Integral transformation of Heun's equation and some applications

Abstract

It is known that the Fuchsian differential equation which produces the sixth Painlevé equation corresponds to the Fuchsian differential equation with different parameters via Euler's integral transformation, and Heun's equation also corresponds to Heun's equation with different parameters, again via Euler's integral transformation. In this paper we study the correspondences in detail. After investigating correspondences with respect to monodromy, it is demonstrated that the existence of polynomial-type solutions corresponds to apparency (non-branching) of a singularity. For the elliptical representation of Heun's equation, correspondence with respect to monodromy implies isospectral symmetry. We apply the symmetry to finite-gap potentials and express the monodromy of Heun's equation with parameters which have not yet been studied.

43 pages

References in corpus (7)

• The Heun equation and the Calogero-Moser-Sutherland system I: the Bethe Ansatz method
• Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
• The Heun equation and the Calogero-Moser-Sutherland system II: perturbation and algebraic solution
• Middle Convolution and Heun's Equation
• Complex Periodic Potentials with a Finite Number of Band Gaps
• Integral representation of solutions to Fuchsian system and Heun's equation
• Integral transformation and Darboux transformation

Cited by in corpus (3)

• Elementary Integral Series for Heun Functions. With an Application to Black-Hole Perturbation Theory
• A Riemann-Hilbert Approach to the Heun Equation
• Series solutions of confluent Heun equations in terms of incomplete Gamma-functions