Global representations of the Heat and Schrodinger equation with singular potential

• Main Authors: Jose A. Franco, Mark R. Sepanski
• Format: Article
• Language: English
• Published: Texas State University 2013-07-01
• Series: Electronic Journal of Differential Equations
• Subjects: Schr"{o}dinger equation; heat equation; singular potential; Lie theory; hfillreakindent representation theory; globalization
• Online Access: http://ejde.math.txstate.edu/Volumes/2013/154/abstr.html

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 $widetilde{SL(2,mathbb{R})}imes O(n)$. 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.