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Families of Symmetries and the Hydrogen Atom

We study a new type of symmetry for the hydrogen atom involving algebraic families of groups parametrized by the energy value in the time-independent Schrödinger equation. We construct an algebraic family of Harish-Chandra modules from the solutions of the Schrödinger equation, and we characterize this family. We show that the subspaces of physical states may be obtained from our algebraic family using a Jantzen filtration, and we relate our algebraic methods with spectral theory and scattering theory using the limiting absorption principle