A new model for the quantum mechanics of the Hydrogen atom

To every Lorentzian quadratic space $(V,q)$ of even dimension $n$ such that $n\geq 4$ we attach a canonical algebraic quantum mechanical model for a corresponding generalized hydrogen atom system.
In our model the configuration space is the regular null cone $C$ of the quadratic space. The Hilbert space $H$ is a canonical $L^2$ space on the cone $C$, and observables are realized in the algebra $D(C)$ of algebraic differential operators on $C$. We also construct a distinguished Schwartz space $S(H)\subset H$, which carries a self-adjoint action of $D(C)$ and encodes the boundary conditions of the standard theory. The role of the Schrödinger operator is played by a one-parameter Schrödinger family of operators in $D(C)$. We explain how the model relates to the realization of the minimal representation of $O(n,2)$ on $H$.
For $n=4$, which corresponds to the physical hydrogen atom system, we prove that the spectrum of the Schrödinger family on the upper-half component $S(H)_+$ coincides with the usual spectrum of the hydrogen atom and that the corresponding solution spaces recover the standard physical solutions.
The spectrum of the Schrödinger family on the lower-half component $S(H)_-$ gives additional positive-energy solution spaces not present in the usual formulation.