Georgia Institute of Technology College of Sciences

A brief overview of some integrable and exactly-solvable Schroedinger equations with trigonometric potentials of Calogero-Moser-Sutherland type is given.All of them are characterized bya discrete symmetry of the Hamiltonian given by the affine Weyl group,a number of polynomial eigenfunctions and eigenvalues which are usually quadratic in the quantum number, each eigenfunction is an element of finite-dimensionallinear space of polynomials characterized by the highest root vector, anda factorization property for eigenfunctions. They admitan algebraic form in the invariants of a discrete symmetry group(in space of orbits) as 2nd order differential operator with polynomial coefficients anda hidden algebraic structure. The hidden algebraic structure for $A-B-C-D$-series is related to the universal enveloping algebra $U_{gl_n}$. For the exceptional $G-F-E$-seriesnew infinite-dimensional finitely-generated algebras of differential operatorswith generalized Gauss decomposition property occur.