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This article is cited in 7 scientific papers (total in 7 papers)
From Quantum $A_N$ (Sutherland) to $E_8$ Trigonometric Model: Space-of-Orbits View
Alexander V. Turbiner Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico
Abstract:
A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries of the metric in the Laplace–Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for $(A{-}B{-}C{-}D)$-models, both rational and trigonometric, is related to the universal enveloping algebra $U_{gl_n}$. For the exceptional $(G{-}F{-}E)$-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with $BC_1\equiv(\mathbb{Z}_2)\oplus T$ symmetry. In particular, the $BC_1$ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra $sl(2)\oplus sl(2)$.
Keywords:
(quasi)-exact-solvability; space of orbits; trigonometric models; algebraic forms; Coxeter (Weyl) invariants;
hidden algebra.
Received: September 21, 2012; in final form January 11, 2013; Published online January 17, 2013
Citation:
Alexander V. Turbiner, “From Quantum $A_N$ (Sutherland) to $E_8$ Trigonometric Model: Space-of-Orbits View”, SIGMA, 9 (2013), 003, 25 pp.
Linking options:
https://www.mathnet.ru/eng/sigma786 https://www.mathnet.ru/eng/sigma/v9/p3
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