A Simple Proof of Sklyanin's Formula for Canonical Spectral Coordinates of the Rational Calogero-Moser System

TL;DR

This paper simplifies the proof of Sklyanin's formula for spectral coordinates in the rational Calogero-Moser system using Hamiltonian reduction, confirming a conjecture and deriving the formula as a consequence.

Contribution

It provides a simplified, Hamiltonian reduction-based proof of Sklyanin's formula, verifying a prior conjecture and clarifying the derivation.

Findings

Verification of Falqui and Mencattini's conjecture

Derivation of Sklyanin's formula as a corollary

Simplification of the proof process

Abstract

We use Hamiltonian reduction to simplify Falqui and Mencattini's recent proof of Sklyanin's expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of Falqui and Mencattini, and to obtain Sklyanin's formula as a corollary.