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

TL;DR

This paper reviews recent advances in constructing trial states for bosonic Hamiltonians, linking singular and regular potentials, and provides bounds on ground state energies in different scaling regimes.

Contribution

It introduces a novel approach connecting trial states for hard-core and regular potentials in bosonic systems, with new energy bounds in various scalings.

Findings

Established an upper bound for the ground state energy in the Gross-Pitaevskii regime.

Linked trial states for singular and regular potentials in bosonic Hamiltonians.

Outlined key steps in proving energy bounds for different potential scalings.

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.