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

TL;DR

This paper explores the representation theory of the solution space of the 1D Schrödinger equation with time-dependent potentials exhibiting sl(2) symmetry, reducing complex cases to simpler, well-understood potentials and constructing solutions via nonstandard induction.

Contribution

It provides explicit local intertwining maps and reduces the analysis of certain time-dependent potentials to the study of simpler, canonical cases, advancing the understanding of symmetries in quantum mechanics.

Findings

Reduction of complex potentials to free or inverse-square potentials

Explicit construction of intertwining maps for solution spaces

Development of nonstandard induction methods for globalizing solutions

Abstract

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.