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We study the $n$-dimensional Schr\"odinger equation with singular potential$V_\lambda(x)=\lambda |x|^{-2}$. Its solution space is studied as a globalrepresentation of $\widetilde{SL(2,\R)}\times O(n)$. A special subspace ofsolutions for which the action globalizes is constructed via nonstandardinduction outside the semisimple category. The space of $K$-finite vectors iscalculated, obtaining conditions for $\lambda$ so that this space is non-empty.The direct sum of solution spaces, over such admissible values of $\lambda$ isstudied as a representation of the $2n+1$-dimensional Heisenberg group.

Global representations of the Heat and Schr\"odinger equation with singular potential