Electron. J. Diff. Equ., Vol. 2013 (2013), No. 154, pp. 1-16.

Global representations of the Heat and Schrodinger equation with singular potential

Jose A. Franco, Mark R. Sepanski

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 2, 2013.
Math Subject Classifications: 22E70, 35Q41.
Key Words: Schr\"{o}dinger equation; heat equation; singular potential; Lie theory; \hfill\break\indent representation theory; globalization

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Jose A. Franco
University of North Florida
1 UNF Drive, Jacksonville, FL 32082, USA
email: jose.franco@unf.edu
Mark R. Sepanski
Baylor University, One Bear Place # 97328
Waco, TX 76798, USA
email: mark_sepanski@baylor.edu

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