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

TL;DR

This paper investigates the Brioschi-Halphen equation, exploring its radial distribution and quasi-exact solvability, and derives asymptotic solutions and wave functions using advanced mathematical techniques.

Contribution

It provides new insights into the asymptotic behavior and solutions of the Brioschi-Halphen equation, linking it to spherical functions and canonical polynomials.

Findings

Radial part of BHE for large r derived

Asymptotic wave functions expressed via canonical polynomials

Solutions connected to spherical functions in L^2(G)

Abstract

The Brioschi-Halphen equation (BHE) is a second order complex differential equation obtained by a two step transformation of the Lam\'e equation. The Lam\'e 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 $2\pi$ 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 $L^{2}(G,{\rm d}\mu), G=SL(2,\mathbb{R})$ using point canonical transformation and distributional solution in $\mathscr{C}_{c}^{\infty}(\Omega)$ using Fourier transform method are obtained.