Conditional symmetry and spectrum of the one-dimensional Schrödinger equation

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 $n\times n$ 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 Schrödinger equations. A symmetry classification of Schödinger equation admitting non-trivial 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 multi-dimensional linear and one-dimensional nonlinear Schrödinger equations is briefly discussed.