Quantum Calogero-Moser Models: Integrability for all Root Systems

TL;DR

This paper proves the integrability of quantum Calogero-Moser models for all root systems, constructing conserved quantities, eigenfunctions, and excited states, extending known results beyond the (A) series.

Contribution

It establishes the integrability of quantum Calogero-Moser models for all root systems, including explicit conserved quantities and eigenfunctions, generalizing previous (A) series results.

Findings

Constructed quantum conserved quantities for all root systems.

Proved Liouville integrability of the models.

Defined generalized Jack polynomials as eigenfunctions.

Abstract

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.