On Radial Distribution and Quasi-exact Solvability of Brioschi-Halphen Equation

TL;DR

This work derives the radial part of the Brioschi-Halphen equation (BHE) by combining asymptotic separation, gauge/point canonical transformations, and Fourier-transform–based distributional analysis. By embedding the radial operator into $sl(2,\mathbb{R})$, it establishes quasi-exact solvability and, at a special parameter choice, exact solvability, with eigenfunctions expressible in terms of canonical polynomials and Jacobi functions after appropriate transformations. It also constructs a distributional, Fourier-based solution in the lemniscatic case, illustrating a Gel'fand triple perspective for the radial problem. The results demonstrate how the BHE's radial part can be treated within a unifying algebraic-analytic framework, yielding explicit eigenfunctions and accessory parameter spectra. These findings clarify the connections between algebraic solvability, spectral theory, and distributional solutions for a class of Fuchsian differential operators.

Abstract

The Brioschi-Halphen equation (BHE) is a second order complex differential equation obtained by a two step transformation of the Lamé equation. The Lamé equation is an equation in Astronomical physics used in the study of motion of planetary bodies. In this seminar, the radial part of the BHE for sufficiently large $r$ and the argument limit $2π$ is obtained. The asymptotic radial wave function associated with BHE is obtained in terms of canonical polynomials $\mathscr{P}_{n+1},$ and spherical function in $L^{2}(G,{\rm d}μ), G=SL(2,\mathbb{R})$ using point canonical transformation and distributional solution in $\mathscr{C}_{c}^{\infty}(Ω)$ using Fourier transform method are obtained.

Theorems & Definitions (5)

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