This paper proposes a new, purely algebraic model of the hydrogen atom. Instead of using ordinary three‑dimensional space, the authors build their quantum model on the cone of a four‑dimensional Lorentzian quadratic form. In that setup they define a natural Hilbert space of square‑integrable functions on the cone and an algebra of differential operators that play the role of observables. When they compute the allowed energies in this model, they recover the familiar hydrogen spectrum seen in physics.

Concretely, the model starts from a four‑dimensional real vector space V with a Lorentzian quadratic form q and the corresponding cone C = {v in V : q(v) = 0}. The Hilbert space H is L2(C0(R), |ω|), the square‑integrable functions on the regular part of that cone with a canonical invariant measure |ω|. Observables come from the algebra D(C) of algebraic differential operators on the cone. Inside H the authors single out a distinguished “Schwartz” subspace H^∞: a space of very smooth, rapidly decaying functions that is invariant under D(C) and on which the operators act in a self‑adjoint way (self‑adjoint meaning the operators can represent real, observable quantities).

The Schrödinger operator of the usual hydrogen model is replaced here by a Schrödinger family S(E) in D(C), a one‑parameter family of algebraic operators labeled by the energy E. The paper gives an explicit form S(E) = H_w + κ + E w, where H_w is a certain second‑order differential operator on the cone, κ is the coupling constant that appears in the physical Coulomb potential, and w is a distinguished linear functional. The authors compute the spectrum of S(E) inside the Schwartz space H^∞ supported on the upper half of the cone and show it matches the standard hydrogen energy levels. They also identify the usual “physical” solution spaces with the kernels of S(E) inside H^∞ on the upper half‑cone.