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SIGMA 9 (2013), 003, 25 pages arXiv:1210.4515
https://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 (A–B–C–D)-models, both rational and
trigonometric, is related to the universal enveloping algebra Ugln. 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 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.
pdf (529 kb)
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