1210.4515 (Alexander V Turbiner)

From quantum $A_N$ (Sutherland) to $E_8$ trigonometric model    [PDF]

Alexander V Turbiner

A brief review of some integrable and exactly-solvable quantum models with trigonometric potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian given by affine Weyl group, (ii) a number of polynomial eigenfunctions and usually quadratic in quantum numbers eigenvalues, (iii) a factorization property for eigenfunctions, admit (iv) an algebraic form in invariants of a discrete symmetry group (in space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for $A-B-C-D$-series is related with the universal enveloping algebra $U_{gl_n}$ while for the exceptional $G-F-E$-series new infinite-dimensional finitely-generated algebras of differential operators occur.

View original: http://arxiv.org/abs/1210.4515