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 SIGMA, 2007, Volume 3, 061, 50 pages (Mi sigma187)

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

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.

Keywords: completely integrable systems; Calogero–Moser systems; Toda lattices with boundary conditions

DOI: https://doi.org/10.3842/SIGMA.2007.061

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ArXiv: math-ph/0502028
MSC: 81R12; 70H06
Received: December 14, 2006; in final form March 19, 2007; Published online April 25, 2007
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Citation: Toshio Oshima, “Completely Integrable Systems Associated with Classical Root Systems”, SIGMA, 3 (2007), 061, 50 pp.

Citation in format AMSBIB
\Bibitem{Shi07} \by Toshio Oshima \paper Completely Integrable Systems Associated with Classical Root Systems \jour SIGMA \yr 2007 \vol 3 \papernumber 061 \totalpages 50 \mathnet{http://mi.mathnet.ru/sigma187} \crossref{https://doi.org/10.3842/SIGMA.2007.061} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=134485} \zmath{https://zbmath.org/?q=an:05241574} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000207065200061} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84889234978} 

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This publication is cited in the following articles:
1. Tremblay F., Turbiner A.V., Winternitz P., “An infinite family of solvable and integrable quantum systems on a plane”, J. Phys. A, 42:24 (2009), 242001, 10 pp.
2. Quesne C., “Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane”, Mod. Phys. Lett. A, 25:1 (2010), 15–24
3. Alexander V. Turbiner, “From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model”, SIGMA, 7 (2011), 071, 20 pp.
4. Alexander V. Turbiner, “From Quantum $A_N$ (Sutherland) to $E_8$ Trigonometric Model: Space-of-Orbits View”, SIGMA, 9 (2013), 003, 25 pp.
5. Turbiner A.V., “Particular Integrability and (Quasi)-Exact-Solvability”, J. Phys. A-Math. Theor., 46:2 (2013), 025203
6. van Diejen J.F., Emsiz E., “Integrable Boundary Interactions For Ruijsenaars' Difference Toda Chain”, Commun. Math. Phys., 337:1 (2015), 171–189
7. Sokolov V.V., Turbiner A.V., “Quasi-Exact-Solvability of the a(2)/G(2) Elliptic Model: Algebraic Forms, Sl(3)/G((2)) Hidden Algebra, and Polynomial Eigenfunctions”, J. Phys. A-Math. Theor., 48:15 (2015), 155201
8. van Diejen J.F. Emsiz E., “Spectrum and Eigenfunctions of the Lattice Hyperbolic Ruijsenaars–Schneider System with Exponential Morse Term”, Ann. Henri Poincare, 17:7 (2016), 1615–1629
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