Global representations of the Heat and Schrödinger equation with singular potential

We study the $n$-dimensional Schrödinger equation with singular potential $V_λ(x)=λ|x|^{-2}$. Its solution space is studied as a global representation of $\widetilde{SL(2,\R)}\times O(n)$. 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 $λ$ so that this space is non-empty. The direct sum of solution spaces, over such admissible values of $λ$ is studied as a representation of the $2n+1$-dimensional Heisenberg group.