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

SIGMA 7 (2011), 071, 20 pages      arXiv:1106.5017
Contribution to the Special Issue “Symmetry, Separation, Super-integrability and Special Functions (S4)”

From Quantum AN (Calogero) to H4 (Rational) Model

Alexander V. Turbiner
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico

Received February 28, 2011, in final form July 12, 2011; Published online July 18, 2011

A brief and incomplete review of known integrable and (quasi)-exactly-solvable quantum models with rational (meromorphic in Cartesian coordinates) potentials is given. All of them are characterized by (i) a discrete symmetry of the Hamiltonian, (ii) a number of polynomial eigenfunctions, (iii) a factorization property for eigenfunctions, and admit (iv) the separation of the radial coordinate and, hence, the existence of the 2nd order integral, (v) an algebraic form in invariants of a discrete symmetry group (in space of orbits).

Key words: (quasi)-exact-solvability; rational models; algebraic forms; Coxeter (Weyl) invariants, hidden algebra.

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