Schrödinger Equations in Electromagnetic Fields: Symmetries and Noncommutative Integration

Abstract

We study symmetry properties and the possibility of exact integration of the time-independent Schrödinger equation in an external electromagnetic field. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra central extensions. Based on the well-known classification of the subalgebras of the algebra e(3), we classify all electromagnetic fields for which the corresponding time-independent Schrödinger equations admit first-order symmetry algebras. Moreover, we select the integrable cases, and for physically interesting electromagnetic fields, we reduced the original Schrödinger equation to an ordinary differential equation using the noncommutative integration method developed by Shapovalov and Shirokov.