Electronic Journal of Differential Equations,
Vol. 2013 (2013), No. 154, pp. 1-16.
Title: Global representations of the Heat and Schrodinger equation with
singular potential
Authors: Jose A. Franco (Univ. of North Florida, Jacksonville, FL, USA)
Mark R. Sepanski (Baylor Univ., Waco, TX, USA)
Abstract:
The n-dimensional Schrodinger equation with a singular potential
$V_\lambda(x)=\lambda \|x\|^{-2}$ is studied.
Its solution space is studied as a global representation of
$\widetilde{SL(2,\mathbb{R})}\times O(n)$. A special subspace of solutions
for which the action globalizes is constructed via nonstandard
induction outside the semisimple category. The space of K-finite
vectors is calculated, obtaining conditions for $\lambda$ so that this
space is non-empty. The direct sum of solution spaces over such admissible
values of $\lambda$ is studied as a representation of the (2n+1)-dimensional
Heisenberg group.
Submitted February 28, 2013. Published July 02, 2013.
Math Subject Classifications: 22E70, 35Q41.
Key Words: Schr\"{o}dinger equation; heat equation; singular potential;
Lie theory; \hfill\break\indent representation theory; globalization