Exact nodal confinement versus quasi-exact solvability in hydrogen: cutoff factors in spherical and parabolic representations

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.