Completely Integrable Systems Associated with Classical Root Systems

TL;DR

This paper investigates quantum integrable systems linked to classical root systems, constructing integrals for non-invariant systems, and proposing a conjecture about their invariants and integrability.

Contribution

It introduces new integrals for non-invariant quantum systems associated with classical root systems and formulates a conjecture relating these to invariants of the Weyl group.

Findings

Constructed integrals for non-invariant systems

Established complete integrability for certain systems

Proposed a conjecture relating integrals to Weyl group invariants

Abstract

We study integrals of completely integrable quantum systems associated with classical root systems. We review integrals of the systems invariant under the corresponding Weyl group and as their limits we construct enough integrals of the non-invariant systems, which include systems whose complete integrability will be first established in this paper. We also present a conjecture claiming that the quantum systems with enough integrals given in this note coincide with the systems that have the integrals with constant principal symbols corresponding to the homogeneous generators of the $B_n$-invariants. We review conditions supporting the conjecture and give a new condition assuring it.