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

This article is cited in 8 scientific papers (total in 8 papers)

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
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\by Toshio Oshima
\paper Completely Integrable Systems Associated with Classical Root Systems
\jour SIGMA
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\papernumber 061
\totalpages 50
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    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.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Alexander V. Turbiner, “From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model”, SIGMA, 7 (2011), 071, 20 pp.  mathnet  crossref  mathscinet
    4. Alexander V. Turbiner, “From Quantum $A_N$ (Sutherland) to $E_8$ Trigonometric Model: Space-of-Orbits View”, SIGMA, 9 (2013), 003, 25 pp.  mathnet  crossref  mathscinet
    5. Turbiner A.V., “Particular Integrability and (Quasi)-Exact-Solvability”, J. Phys. A-Math. Theor., 46:2 (2013), 025203  crossref  mathscinet  adsnasa  isi  elib  scopus
    6. van Diejen J.F., Emsiz E., “Integrable Boundary Interactions For Ruijsenaars' Difference Toda Chain”, Commun. Math. Phys., 337:1 (2015), 171–189  crossref  mathscinet  zmath  adsnasa  isi  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi  scopus
  • Symmetry, Integrability and Geometry: Methods and Applications
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