Integral transformation of Heun's equation and some applications

TL;DR

This paper explores the integral transformation relationships between Heun's equation and other Fuchsian equations, analyzing monodromy, polynomial solutions, and symmetries, with applications to finite-gap potentials and spectral theory.

Contribution

It provides a detailed study of the integral transformations of Heun's equation, revealing new insights into monodromy, polynomial solutions, and isospectral symmetries.

Findings

Correspondence between Heun's equation and Fuchsian equations via Euler's integral transformation.

Polynomial solutions relate to apparent singularities and non-branching.

Identification of isospectral symmetry in the elliptical representation of Heun's equation.

Abstract

It is known that the Fuchsian differential equation which produces the sixth Painlev\'e 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.