For a harmonic potential, which is the special shape-invariant potential discussed in our paper [1], the relation between the ground state and the first-excited-state wave functions [Eq. (2)] should be . . . Therefore, we have shown that one cannot find a closed form of the kinetic-energy density for the harmonic oscillator.
From the explicit solution of a two-level problem previously obtained, it is proved that the single-particle kinetic energy density functional ts[\ensuremath{\rho}] is the density \ensuremath{\rho} times a function solely of the ratio \ensuremath{\rho}'/\ensuremath{\rho}, with \ensuremath{\rho}' the first derivative of \ensuremath{\rho}. For an n-level system with a harmonic-oscillator potential, the form of ts[\ensuremath{\rho}] is explicitly obtained. Again ts[\ensuremath{\rho}] involves only \ensuremath{\rho} and \ensuremath{\rho}'; no higher derivatives appear.