  We review two results in which trial states for bosonic Hamiltonians were
discussed. The problem of finding a trial state for a system with a hard-core
potential in the Gross-Pitaevskii regime was recently solved by proving a link
with the problem of finding a trial state for a system with a more regular
potential in a less singular scaling, one of the type
$N^{-1+3\beta}V(N^\beta\cdot)$ with $\beta\in(0,1)$. For both of these models
we present the main result for the upper bound to the ground state energy, and
discuss the key steps in the proof.
