Central European Journal of Mathematics

, Volume 10, Issue 3, pp 927–941

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

Authors

    • Department of MathematicsBaylor University
Research Article

DOI: 10.2478/s11533-012-0040-8

Cite this article as:
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 Vλ(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 \(\widetilde{SL(2,\mathbb{R})}\)H3, where H3 is the three-dimensional Heisenberg group.

Keywords

Schrödinger equationTime-dependent potentialsLie theoryRepresentation theoryGlobalizations

MSC

22E7035Q41

Copyright information

© © Versita Warsaw and Springer-Verlag Wien 2012