Wave function and external potential from constrained search in density-functional theory

It is proven that in Levy's constrained search or Lieb's ensemble constrained search, if the minimum is achieved by wave functions that satisfy the steadiness condition for the current density kernel and have continuous second derivatives, the minimizing wave functions are eigenstates of some external potential. If the excited states of this potential can only achieve stationarity and cannot achieve a minimum in the constrained search, the minimizing wave functions are ground states and the corresponding electron density is v-representable or ensemble v-representable. This is also true for the ensemble system with a varying electron number. In time-dependent density-functional theory, if the stationarity of the action integral is achieved in the constrained search by a wave function that satisfies the continuity equation for the current density kernel and has a continuous first derivative with respect to the time coordinate and continuous second derivatives with respect to the space coordinates, the stationary wave function satisfies the time-dependent Schrodinger equation of some time-dependent external potential and the corresponding electron density is time-dependent v-representable.