Global Symmetries of Time-Dependent Schrodinger Equations

TL;DR

This paper systematically analyzes the symmetries of time-dependent Schrödinger equations with various potentials, revealing the structure of their symmetry groups, especially for linear potentials, using a suitable method.

Contribution

It provides a systematic examination of symmetries for Schrödinger equations with different potentials, identifying the specific symmetry group structure for linear potentials.

Findings

Symmetry groups depend on the potential type.

The linear potential case has a symmetry group of $SL(2,R) \ltimes T_2(R)$.

Time transformations follow a Möbius transformation form.

Abstract

Some symmetries of time-dependent Schr\"odinger 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 $SL(2,R) x T_2(R)$ of the $SL(2,R)$ with a two-dimensional real translation group $T_2(R)$. Here, the time variable $t$ transforms as $t \to t^\prime = (ct+d)/(at+b)$ for real constants $a, b, c$, and $d$ satisfying $bc - ad =1$ with an accompanying transformation for the space coordinate $x$.