Exact single-particle kinetic energy functional for general two-level and model n-level one-dimensional systems: Dependence only on electron density and its gradient

From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional $t_s$[ρ] is the density ρ times a function solely of the ratio ρ’/ρ, with ρ’ the first derivative of ρ. For an n-level system with a harmonic-oscillator potential, the form of $t_s$[ρ] is explicitly obtained. Again $t_s$[ρ] involves only ρ and ρ’; no higher derivatives appear.