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

Abstract

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.

Contribution to the Proceedings of the Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March - May 2022