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SIGMA 3 (2007), 061, 50 pages math-ph/0502028
https://doi.org/10.3842/SIGMA.2007.061
Contribution to the Vadim Kuznetsov Memorial Issue
Completely Integrable Systems Associated with Classical Root Systems
Toshio Oshima
Graduate School of Mathematical Sciences, University of
Tokyo, 3-8-1, Komaba, Meguro-ku, Tokyo 153-8914, Japan
Received December 14, 2006, in final form March 19, 2007; Published online April 25, 2007
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 Bn-invariants. We review
conditions supporting the conjecture and give a new condition
assuring it.
Key words:
completely integrable systems; Calogero-Moser systems; Toda lattices with boundary conditions.
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