Principal results

• Characterization of the quasi-exact solvability of the scalar Schroedinger operator on line in terms of its high order conditional symmetry (see the item B-37 of the Publication list ).
• Classification of Hermitian quasi-exactly solvable matrix Shroedinger operators on line. The classification procedure relies heavily upon a special (new) realization of the algebra sl(2, R) by matrix-differential operators. For a number of representatives of the so constructed families of quasi-exactly solvable Schroedinger operators, square integrable eigenfunctions are explicitly constructed (see the items B-42,43,49 of the Publication list ).