Integrable many-body systems of Calogero-Ruijsenaars type

TL;DR

This thesis advances the understanding of Calogero-Ruijsenaars integrable systems by deriving explicit formulas, exploring dualities, and constructing new deformations and Lax pairs for various models within this class.

Contribution

It provides explicit spectral coordinates, explores dualities, and constructs deformations and Lax pairs for multiple Calogero-Ruijsenaars type systems, enhancing their mathematical understanding.

Findings

Explicit spectral coordinates for rational Calogero-Moser system.

Action-angle duality for BC(n) Sutherland system established.

Poisson-Lie deformation of BC(n) Sutherland system derived.

Abstract

This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic Ruijsenaars-Schneider system with two couplings. 5. We present an explicit construction of compactified trigonometric and elliptic Ruijsenaars-Schneider systems.