  Based on the Schrodinger equation, exact expressions for the non-relativistic
particle energy in the local external field and the external field potential
are derived as inhomogeneous density functionals. On this basis, it is shown
that, when considering more than two noninteracting electrons, the energy of
such a system cannot be an inhomogeneous density functional. The result is
extended for the system of interacting electrons. This means that the
Hohenberg-Kohn lemma which assert that in the ground state to each
inhomogeneous density corresponds only one potential of the external field
cannot be a justification of the existence of the universal density functional
in the general case. At the same time, statements of the density functional
theory remain valid when considering any number of noninteracting ground-state
bosons due to the Bose condensation effect.
