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

Abstract

We study the nn-dimensional Schr\"odinger equation with singular potential Vλ(x)=λ∣x∣−2V_\lambda(x)=\lambda |x|^{-2}. Its solution space is studied as a global representation of SL(2,R)~×O(n)\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 KK-finite vectors is calculated, obtaining conditions for λ\lambda so that this space is non-empty. The direct sum of solution spaces, over such admissible values of λ\lambda is studied as a representation of the 2n+12n+1-dimensional Heisenberg group

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