On the Calogero-Moser solution by root-type Lax pair

TL;DR

This paper investigates the root-type Lax pair solution for the rational Calogero-Moser system, showing that the solution is uniquely determined by inner product values up to automorphisms, and introduces an algebraic indicator for boundary crossings.

Contribution

It demonstrates the uniqueness of the solution determined by inner product values and develops an algebraic method to detect boundary crossings in the root system.

Findings

Solution determined by inner products up to automorphisms

Counterexample showing non-uniqueness of q for same inner products

Algebraic indicator function for boundary crossing detection

Abstract

The `root type Lax pair' for the rational Calogero-Moser system for any simply-laced root system yields not a solution for the path q(t), but for the values of the inner products (\alpha,q(t)), where \alpha\ ranges over all roots of the root system. It does not, however, tell us which value of the inner product corresponds to which root. We show that the solution is indeed uniquely determined by these values (up to root system automorphisms) at almost all times. We show by counterexample that it is possible for two different values of q to yield the same set of values for the inner products (\alpha,q). The indeterminacy introduced by the root system automorphisms gives rise to the question when the path crosses the boundary of the fundamental domains. We present an algebraic approach for constructing an indicator function containing this information.