Research Article

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

  • Jose FrancoAffiliated withDepartment of Mathematics, Baylor University Email author 

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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.

Keywords

Schrödinger equation Time-dependent potentials Lie theory Representation theory Globalizations

MSC

22E70 35Q41