CONDITIONAL SYMMETRY AND SPECTRUM OF THE ONE-DIMENSIONAL SCHRODINGER-EQUATION

We develop an algebraic approach to studying the spectral properties o f the stationary Schrodinger equation in one dimension based on its hi gh-order conditional symmetries. This approach makes it possible to ob tain in explicit form representations of the Schrodinger operator by n x n matrices for any n is an element of N and, thus, to reduce a spec tral problem to a purely algebraic one of finding eigenvalues of const ant n x n matrices. The connection to so-called quasiexactly solvable models is discussed. It is established, in particular, that the case, when conditional symmetries reduce to high-order Lie symmetries, corre sponds to exactly solvable Schrodinger equations. A symmetry classific ation of Schrodinger equation admitting nontrivial high-order Lie symm etries is carried out, which yields a hierarchy of exactly solvable Sc hrodinger equations. Exact solutions of these are constructed in expli cit form. Possible applications of the technique developed to multidim ensional linear and one-dimensional nonlinear Schrodinger equations ar e briefly discussed. (C) 1996 American Institute of Physics.