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 $n\in \mathbf{N}$ 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.

LaTeX-file, 31 pages, to appear in J.Math.Phys., v.37, N7, 1996

References in corpus (3)

• Conditional Lie-Bäcklund symmetry and reduction of evolution equations.
• Supersymmetric quantum mechanics and the Korteweg-de Vries hierarchy
• Symmetry, Local Linearization, and Gauge Classification of the Doebner-Goldin Equation

Cited by in corpus (7)

• Nonlinear supersymmetry in Quantum Mechanics: algebraic properties and differential representation
• On quasi-exactly solvable matrix models
• Quasi-exactly solvable models based on special functions
• Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics
• On algebraic classification of quasi-exactly solvable matrix models
• Invariant Modules and the Reduction of Nonlinear Partial Differential Equations to Dynamical Systems
• Stationary mKdV hierarchy and integrability of the Dirac equations by quadratures