On Spectrum Generating Algebra of the Heun Operator

TL;DR

This paper explores the algebraic structure of the Heun operator, showing how it can be represented using an $su(1,1)$ algebra under certain conditions, and demonstrates spectrum generation with examples.

Contribution

It introduces a spectrum generating algebra for the Heun operator using $su(1,1)$ elements, revealing conditions for elementary singularities and illustrating spectrum generation.

Findings

Heun operator can be expressed via $su(1,1)$ algebra elements.

Regular singularities at 0 and infinity are elementary under certain parameters.

Spectrum generation demonstrated through specific examples.

Abstract

The Heun operator has been cast, in terms of the elements of an underlying $su(1,1)$ algebra, under specific parametric conditions, for the purpose of spectrum generation. These elements are differential operators of \emph{degrees} $\pm 1/2$ and $0$. It is found that the regular singularities at $0$ and $\infty$ of the general Heun equation must be \emph{elementary} under the required parametric conditions. The spectrum generation has been demonstrated through a set of examples.