Completely Integrable Systems Associated with Classical Root Systems

2005/02/28 by Toshio Oshima · 2 citations
Chemistry · Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Molecular spectroscopy and chirality #math-ph #math.DS #math.MP #msc:70H06 #msc:81R12 #nlin.SI

paper · pdf · doi:10.3842/sigma.2007.061

published as SIGMA 3 (2007), 061, 50 pages · This is a contribution to the Vadim Kuznetsov Memorial Issue on Integrable Systems and Related Topics, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/

arxiv created 2007/04/25 · openalex publication_date 2007/04/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Separation of variables for the quantum relativistic Toda lattices
• Classical and quantum-mechanical systems of Toda lattice type. I
• Solution of the One-Dimensional N-Body Problems with Quadratic and/or Inversely Quadratic Pair Potentials
• Classical and quantum mechanical systems of Toda-lattice type

Cited by

• Quasi-exact-solvability of the A₂/G₂ elliptic model: algebraic forms, sl(3)/g⁽²⁾ hidden algebra, and polynomial eigenfunctions
• Particular integrability and (quasi)-exact-solvability