On q-Deformations of the Heun Equation

Abstract

The q-Heun equation and its variants arise as degenerations of Ruijsenaars-van Diejen operators with one particle. We investigate local properties of these equations. In particular, we characterize the variants of the q-Heun equation by using analysis of regular singularities. We also consider the quasi-exact solvability of the q-Heun equation and its variants. Namely, we investigate finite-dimensional subspaces that are invariant under the action of the q-Heun operator or variants of the q-Heun operator.