On the solutions of multicomponent generalizations of the Lam{\'e} equation

TL;DR

This paper investigates singular solutions to multicomponent Lamé equations linked to elliptic Calogero–Moser systems with spin, providing explicit meromorphic solutions, quantization conditions, and additional integrals of motion.

Contribution

It introduces a new class of solutions for multicomponent Lamé equations, including a special ansatz for meromorphic solutions and extra integrals of motion for three-particle systems.

Findings

Derived meromorphic solutions with two parameters at special coupling values.

Quantization conditions for wave function regularity and boundary conditions.

Identified additional integrals of motion commuting with the Hamiltonian.

Abstract

We describe a class of the singular solutions to the multicomponent analogs of the Lam{\'e} equation, arising as equations of motion of the elliptic Calogero--Moser systems of particles carrying spin 1/2. At special value of the coupling constant we propose the ansatz which allows one to get meromorphic solutions with two arbitrary parameters. They are quantized upon the requirement of the regularity of the wave function on the hyperplanes at which particles meet and imposing periodic boundary conditions. We find also the extra integrals of motion for three-particle systems which commute with the Hamiltonian for arbitrary values of the coupling constant.