The Heun equation and the Calogero-Moser-Sutherland system II: perturbation and algebraic solution

TL;DR

This paper develops a perturbation method for the $BC_1$ Inozemtsev model, proving convergence of eigenvalues and eigenfunctions, and explores the relationship between $L^2$ space and elliptic functions.

Contribution

It introduces a novel perturbation approach for the Inozemtsev model and establishes convergence and holomorphy results for eigenvalues and eigenfunctions.

Findings

Proved the holomorphy of perturbation in the Inozemtsev model.

Established convergence of eigenvalues and eigenfunctions as formal power series.

Explored the connection between $L^2$ space and finite-dimensional elliptic function spaces.

Abstract

We apply a method of perturbation for the $BC_1$ Inozemtsev model from the trigonometric model and show the holomorphy of perturbation.Consequently, the convergence of eigenvalues and eigenfuncions which are expressed as formal power series is proved. We investigate also the relationship between $L^2$ space and some finite dimensional space of elliptic functions.