2026/03/16 by Joseph Bernstein, Eyal Subag
Mathematics · #math-ph #math.AG #math.MP #math.RT
To every Lorentzian quadratic space (V,q) of even dimension n such that n≥ 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 L2 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)⊂ 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.