Completely Integrable Systems Associated with Classical Root Systems

Details Completely Integrable Systems Associated with Classical Root Systems Completely Integrable Systems Associated with Classical Root Systems

2005-02-08

arxiv.org/abs/math-ph/0502028v4

SIGMA 3 (2007), 061, 50 pages

Physics

Mathematical Physics

This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Sym

Scientific paper

10.3842/SIGMA.2007.061

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.

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