Recursion representation of gradient expansion for free fermion ground state in one dimension

We consider noninteracting Fermion system in an external field, confined to one-dimensional space. Our main objective is to take advantage of low dimension to find explicitly potential-density profile as well as kinetic energy functional in the ground state. This is accomplished by deriving the complete x-space linear response and a simple recursion representation of the gradient expansion for the kinetic energy. The results can serve as a potential test for higher dimensions, e.g., in exploring convergence properties of the gradient expansion in various regions of the energy spectrum.