  We develop an algebraic approach to studying the spectral properties of the
stationary Schr\"odinger equation in one dimension based on its high order
conditional symmetries. This approach makes it possible to obtain in explicit
form representations of the Schr\"odinger operator by $n\times n$ matrices for
any $n\in {\bf N}$ and, thus, to reduce a spectral problem to a purely
algebraic one of finding eigenvalues of constant $n\times n$ matrices. The
connection to so called quasi exactly solvable models is discussed. It is
established, in particular, that the case, when conditional symmetries reduce
to high order Lie symmetries, corresponds to exactly solvable Schr\"odinger
equations. A symmetry classification of Sch\"odinger equation admitting
non-trivial high order Lie symmetries is carried out, which yields a hierarchy
of exactly solvable Schr\"odinger equations. Exact solutions of these are
constructed in explicit form. Possible applications of the technique developed
to multi-dimensional linear and one-dimensional nonlinear Schr\"odinger
equations is briefly discussed.
