Abstract: The Calogero-Sutherland (CS) quantum integrable models are characterized by rather special feature of being exactly solvable. Their eigenvalues are given by closed analytic expressions and their eigenfunctions can be found by linear algebra methods. These models emerge from the Hamiltonian reduction method, associated with different root spaces and admit a Lie-algebraic interpretation.
It can be shown that all Calogero-Sutherland models associated with classical A-B-C-D root spaces emerge from a single quadratic polynomial in generators of the maximal affine subalgebra of the gl(n)-algebra but unusually realized by the first order differential operators. The memory about their A-B-C-D origin is kept in coefficients of the polynomial. For the case of models related to the exceptional root spaces some unknown infinite-dimensional Lie algebras admitting finite-dimensional irreps appear.
Lie-algebraic theory allows us to construct the 'quasi-exactly-solvable' generalizations of the above Hamiltonians where a finite number of eigenstates is known exactly (algebraically). A general notion of (quasi)-exactly-solvable spectral problem is introduced.