Global Symmetries of Time-Dependent Schrodinger Equations

Susumu Okubo

doi:10.1063/1.1591992

paper

Global Symmetries of Time-Dependent Schrodinger Equations

arXiv:math-ph/0303017 · doi:10.1063/1.1591992

Abstract

Some symmetries of time-dependent Schrödinger equations for inverse quadratic, linear, and quadratic potentials have been systematically examined by using a method suitable to the problem. Especially, the symmetry group for the case of the linear potential turns out to be a semi-direct product $S L (2, R) x T_{2} (R)$ of the $S L (2, R)$ with a two-dimensional real translation group $T_{2} (R)$ . Here, the time variable $t$ transforms as $t \to t^{'} = (c t + d) / (a t + b)$ for real constants $a, b, c$ , and $d$ satisfying $b c - a d = 1$ with an accompanying transformation for the space coordinate $x$ .

33 pages, no figures