  In the vast majority of many-body problems, it is the kinetic energy part of
the Hamiltonian that is best known microscopically, and it is the detailed form
of the interactions between the particles, the potential energy term, that is
harder to determine from first principles. An example is the case of high
temperature superconductors: while a tight-binding model captures the kinetic
term, it is not clear that there is superconductivity with only an onsite
repulsion and, thus, that the problem is accurately described by the Hubbard
model alone. Here we pose the question of whether, once the kinetic energy is
fixed, a candidate ground state is {\it groundstatable or not}. The easiness to
answer this question is strongly related to the presence or the absence of a
sign problem in the system. When groundstatability is satisfied, it is simple
to obtain the potential energy that will lead to such a ground state. As a
concrete case study, we apply these ideas to different fermionic wavefunctions
with superconductive or spin-density wave correlations and we also study the
influence of Jastrow factors. The kinetic energy considered is a simple next
nearest neighbor hopping term.
