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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.
The model differs from the standard textbook picture in three clear ways. First, the configuration space is the cone C rather than R3, so the geometric symmetry group of the configuration space is O(3,1) (the Lorentz‑type orthogonal group) rather than the Euclidean group of R3. Second, all operators used are algebraic and free of point singularities (so the model avoids the singular 1/r Coulomb term at the origin and the non‑algebraic square‑root that appears in radial coordinates). Third, the usual boundary conditions are not imposed by hand. Instead they are encoded in the choice of the Schwartz subspace H^∞.
Important caveats are stated in the paper. The model is mathematical and algebraic, not an experimental proposal, and certain solutions appear that are not part of the standard physical description. In particular, the spectrum of the Schrödinger family in the lower half of the cone equals the positive real line (0, ∞). Those lower‑half solutions do not appear in the usual description of the hydrogen atom, and the authors do not claim a physical interpretation for them. More generally, the paper emphasizes representation‑theoretic reasons for the construction: the Schrödinger family can be seen inside the universal enveloping algebra of SL2(R), and the spectrum calculation reduces to known discrete‑series representations. The work therefore gives a clean algebraic account of the familiar hydrogen spectrum, while leaving open questions about the physical meaning of the extra mathematical solutions.