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Based on the Schrodinger equation, exact expressions for the non-relativisticparticle energy in the local external field and the external field potentialare derived as inhomogeneous density functionals. On this basis, it is shownthat, when considering more than two noninteracting electrons, the energy ofsuch a system cannot be an inhomogeneous density functional. This means thatthe Hohenberg-Kohn lemma which assert that in the ground state to eachinhomogeneous density corresponds only one potential of the external fieldcannot be a justification of the existence of the universal density functionalin the general case. At the same time, statements of the density functionaltheory remain valid when considering any number of noninteracting ground-statebosons due to the Bose condensation effect.

Exact Density Functional for the Non-Relativistic Particle Energy in the Local External Field