Distributional Solution and Spectral Shift Function of Heun Differential Equation

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 $G$.