Kinetic-energy density functional for a special shape-invariant potential of a one-dimensional two-level system

By introducing the concept of supersymmetry to quantum mechanics, several authors have shown that exactly soluble potentials, including all those well known, are easily constructed. Is the kinetic-energy density functional corresponding to these potentials a simple form? We show that the answer is no, even to the simplest one, the harmonic potential, if one builds the kinetic-energy density functional from the reduced density matrix.