  We prove Hohenberg-Kohn theorems for several models of quantum mechanics.
First, we show that the pair correlation function of any ground state contains
the information of the interactions and of the external potentials. Then, in
the presence of the Zeeman interaction, a strong constraint on external fields
is derived for systems having the same ground state densities and
magnetizations. Moreover, we provide a counterexample in a setting involving
non-local potentials. Next, we prove that the density and the entropy of a
ground state contain the information of both the imposed external potential and
temperature. Eventually, we conclude that at positive temperature,
Hohenberg-Kohn theorems generically hold.
