Conditional symmetry and spectrum of the one-dimensional Schrödinger equation

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Abstract
We develop an algebraic approach to studying the spectral properties of the stationary Schrödinger equation in one dimension based on its high-order conditional symmetries. This approach makes it possible to obtain in explicit form representations of the Schrödinger operator by n×n matrices for any n∈N and, thus, to reduce a spectral problem to a purely algebraic one of finding eigenvalues of constant n×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, corresponds to exactly solvable Schrödinger equations. A symmetry classification of Schrödinger equation admitting nontrivial high-order Lie symmetries is carried out, which yields a hierarchy of exactly solvable Schrödinger equations. Exact solutions of these are constructed in explicit form. Possible applications of the technique developed to multidimensional linear and one-dimensional nonlinear Schrödinger equations are briefly discussed.
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