  External potentials play a crucial role in modelling quantum systems, since,
for a given inter- particle interaction, they define the system Hamiltonian. We
use the metric space approach to quantum mechanics to derive, from the energy
conservation law, two natural metrics for potentials. We show that these
metrics are well defined for physical potentials, regardless of whether the
system is in an eigenstate or if the potential is bounded. In addition, we
discuss the gauge freedom of potentials and how to ensure that the metrics
preserve physical relevance. Our metrics for potentials, together with the
metrics for wavefunctions and densities from [I. D'Amico, J. P. Coe, V. V.
Franca, and K. Capelle, Phys. Rev. Lett. 106, 050401 (2011)] paves the way for
a comprehensive study of the two fundamental theorems of Density Functional
Theory. We explore these by analysing two many- body systems for which the
related exact Kohn-Sham systems can be derived. First we consider the
information provided by each of the metrics, and we find that the density
metric performs best in distinguishing two many-body systems. Next we study for
the systems at hand the one-to-one relationships among potentials, ground state
wavefunctions, and ground state densities defined by the Hohenberg-Kohn theorem
as relationships in metric spaces. We find that, in metric space, these
relationships are monotonic and incorporate regions of linearity, at least for
the systems considered. Finally, we use the metrics for wavefunctions and
potentials in order to assess quantitatively how close the many-body and
Kohn-Sham systems are: We show that, at least for the systems analysed, both
metrics provide a consistent picture, and for large regions of the parameter
space the error in approximating the many-body wavefunction with the Kohn-Sham
wavefunction lies under a threshold of 10%.
