Conditional symmetry and spectrum of the one-dimensional Schr\"odinger equation

TL;DR

This paper introduces an algebraic method leveraging high order conditional symmetries to analyze the spectral properties of one-dimensional Schr"odinger equations, enabling explicit matrix representations and classification of exactly solvable models.

Contribution

It develops a novel algebraic approach based on conditional symmetries to explicitly represent Schr"odinger operators and classify exactly solvable equations.

Findings

Representation of Schr"odinger operators by n×n matrices

Connection between high order symmetries and exactly solvable models

Explicit construction of solutions for classified equations

Abstract

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.