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Description

Exact solutions of non-trivial Schroedinger equations are crucially important for applications. Almost unique source of these solutions is Olshanetsky-Perelomov quantum Hamiltonians (rational and trigonometric) emerging in the Harish-Chandra theory. However, an alternative Lie-algebraic theory of these solutions can be developed. It can be shown that all A-B-C-D Olshanetsky-Perelomov Hamiltonians (rational and trigonometric) come from a single quadratic polynomial in generators of the maximal affine subalgebra of the gl(n)-algebra of differential operators. The memory about A-B-C-D origin is kept in coefficients of the polynomial. Lie-algebraic theory allows to construct the 'quasi-exactly-solvable' generalizations of the above Hamiltonians where a finite number of eigenstates is known exactly. A general notion of (quasi)-exactly-solvable spectral problem is introduced.