Affine Toda-Sutherland Systems

TL;DR

This paper introduces a new class of partially integrable multi-particle systems combining affine Toda and Sutherland models, revealing algebraic equilibrium points and exact solutions related to affine Toda field theories.

Contribution

It presents a novel hybrid integrable system based on affine root systems, extending known models with partial integrability and exact quantum solutions.

Findings

Equilibrium positions are algebraic and proportional to the Weyl vector.

Small oscillation frequencies relate to affine Toda masses.

Exact quantum eigenvalues and eigenfunctions are obtained for certain frequencies.

Abstract

A cross between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system, is introduced for any affine root system. Though it is not completely integrable but partially integrable, or quasi exactly solvable, it inherits many remarkable properties from the parents. The equilibrium position is algebraic, i.e. proportional to the Weyl vector. The frequencies of small oscillations near equilibrium are proportional to the affine Toda masses, which are essential ingredients of the exact factorisable S-matrices of affine Toda field theories. Some lower lying frequencies are integer times a coupling constant for which the corresponding exact quantum eigenvalues and eigenfunctions are obtained. An affine Toda-Calogero system, with a corresponding rational potential, is also discussed.