, Volume 10, Issue 3, pp 927-941
Date: 11 Mar 2012

Global \(\widetilde{SL(2,R)}\) representations of the Schrödinger equation with singular potential

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Abstract

We study the representation theory of the solution space of the one-dimensional Schrödinger equation with singular potential V λ (x) = λx −2 as a representation of \(\widetilde{SL(2,\mathbb{R})}\) . The subspace of solutions for which the action globalizes is constructed via nonstandard induction outside the semisimple category. By studying the subspace of K-finite vectors in this space, a distinguished family of potentials, parametrized by the triangular numbers is shown to generate a global representation of \(\widetilde{SL(2,\mathbb{R})}\) H 3, where H 3 is the three-dimensional Heisenberg group.