Distributional Solution and Spectral Shift Function of Heun Differential Equation

TL;DR

This paper explores the algebraic structure of the Heun differential operator within the Lie algebra framework of SL(2,C), deriving its Green function and spectral shift function using Fourier analysis.

Contribution

It introduces a Lie algebraic approach to analyze the Heun operator and computes its Green and spectral shift functions through Fourier transform methods.

Findings

Heun operator expressed in the universal enveloping algebra of sl(2,C)

Green function derived via Fourier transform over SL(2,C)

Spectral shift function obtained from algebraic analysis

Abstract

In this work, the Heun operator is written as an element in the universal enveloping algebra of the Lie algebra $\mathscr{G}=\mathscr{L}(G)$ of the Lie group $G=SL(2,\mathbb{C})$. The Green function and the spectral shift function of the exactly solvable Heun operator from the resulting Lie algebraic equation are obtained via Fourier transform over $G$.