Trial states for Bose gases: singular scalings and non-integrable potentials

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.