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

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DOI: 10.2478/s11533-012-0040-8

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- Franco, J. centr.eur.j.math. (2012) 10: 927. doi:10.2478/s11533-012-0040-8

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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.