A new model for the quantum mechanics of the Hydrogen atom

Abstract

To every Lorentzian quadratic space $(V,q)$ of even dimension $n$ such that $n\ge 4$ 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.

In this version the abstract and Sections 7.1.3 and 7.2.6 were revised. In addition, the spectrum is now defined using the hermitian dual