Exact nodal confinement versus quasi-exact solvability in hydrogen: cutoff factors in spherical and parabolic representations
Year: 2026
Type: article
Abstract: Hard walls placed on nodal surfaces of an unconfined hydrogenic orbital select an exact state at unchanged energy. We distinguish this nodal confinement from quasi-exact solvability (QES), which requires a multidimensional boundary-compatible polynomial space invariant under the reduced Hamiltonian of a fixed cavity. Turbiner and Escobar-Ruiz [Phys. Lett. A 468, 128738 (2023)] identified the hidden g (2) algebra of free hydrogen in the O( 2Building on this result, we construct an exact spherical-conical family whose minimal polynomial cutoff factor F (r, u) has weighted degree six. Four confocal paraboloids yield a second exact family with a degree-four cutoff factor in the aff(2) = gl(2) ⋉ R 2 representation acting on (ξ, η) = (r + z, rz). For a parabolic cell invariant under z ↦ → -z, the variables r = (ξ + η)/2 and u = ξη reduce the affine representation to the same hidden g (2) structure despite the different walls. Within a free Coulomb O(4) shell, the axial Runge-Lenz component A z is diagonal in the affine parabolic basis and exchanges the even and odd g (2) partners. Displacing one parabolic wall by δ shifts the ground-state energy E 0 (δ) of the deformed cavity and drives the separation eigenvalue α(δ) associated with A z away from zero. Their variations measure the energetic response and the loss of z-reflection symmetry, respectively. Complementing conventional cavity-parameter scans, exact nodal cells provide structural reference points for identifying which solvability and symmetry structures survive confinement.
Institutions: Tecnológico de Monterrey, Universidad Autónoma Metropolitana
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,325
OpenAlex ID: W7225800884
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