Global $\widetilde{SL(2,R)}$ representations of the Schr\"{o}dinger equation with time-dependent potentials

We study the representation theory of the solution space of the one-dimensional Schr\"{o}dinger equation with time-dependent potentials that posses $\mathfrak{sl}_2$-symmetry. We give explicit local intertwining maps to multiplier representations and show that the study of the solution space for potentials of the form $V(t,x)=g_2(t)x^2+g_1(t)x+g_0(t)$ reduces to the study of the potential free case. We also show that the study of the time-dependent potentials of the form $V(t,x)=\lambda x^{-2}+g_2(t)x^2+g_0(t)$ reduces to the study of the potential $V(t,x)=\lambda x^{-2}$. Therefore, we study the representation theory associated to solutions of the Schr\"{o}dinger equation with this potential. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.