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

We study the representation theory of the solution space of the one-dimensional Schrödinger 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 $V(t,x)=λx^{-2}+g_2(t)x^2+g_0(t)$ reduces to the study of the potential $V(t,x)=λx^{-2}$. Therefore, we study the representation theory associated to solutions of the Schrödinger equation with this potential. The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category.