Condensation in interacting canonical ensembles with fixed harmonic confinement

Determine whether canonical ensembles of interacting bosons confined by a fixed harmonic potential develop condensates in the limit of infinitely many particles, specifically whether they satisfy the proper-condensate condition and exhibit an Onsager–Penrose condensate as suggested by the behavior of scaled resolvents.

Background

The paper discusses canonical ensembles of n particles confined by a fixed harmonic potential and interacting through two-body forces. A cited result establishes that these ensembles approach equilibrium states satisfying the KMS condition as n tends to infinity, but the occurrence and type of condensation in this limit are not established.

The proposed investigation concerns both attractive and repulsive interactions. For attractive short-range forces, the particles are heuristically expected to form a cluster near the minimum of the harmonic potential, while scaled resolvents supported near that minimum might indicate an Onsager–Penrose condensate and resolvents supported far from it might establish a proper condensate. The paper presents these as expectations rather than proven conclusions.