Global Symmetries of Time-Dependent Schrodinger Equations

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 $x$.