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
method, associated with different root spaces and admit a Lie-algebraic
It can be shown that all Calogero-Sutherland models associated with
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
For the case of models related to the exceptional root spaces some unknown
infinite-dimensional Lie algebras admitting finite-dimensional irreps
Lie-algebraic theory allows us to construct the 'quasi-exactly-solvable'
generalizations of the above Hamiltonians where a finite number of
is known exactly (algebraically).
A general notion of (quasi)-exactly-solvable spectral problem is introduced.