Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 003, 25 pages      arXiv:1210.4515      http://dx.doi.org/10.3842/SIGMA.2013.003
Contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”

From Quantum AN to E8 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

Received September 21, 2012, in final form January 11, 2013; Published online January 17, 2013

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 (ABCD)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (GFE)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC1≡(Z2)⊕T symmetry. In particular, the BC1 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)⊕sl(2).

Key words: (quasi)-exact-solvability; space of orbits; trigonometric models; algebraic forms; Coxeter (Weyl) invariants; hidden algebra.

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