On a condition for type-I Bose-Einstein condensation in random potentials in $d$ dimensions

TL;DR

This paper establishes a sufficient gap condition for the occurrence of type-I Bose-Einstein condensation in random potentials across various dimensions, demonstrated through the Luttinger-Sy model.

Contribution

It introduces a general gap condition criterion for type-I BEC in disordered systems and applies it to the Luttinger-Sy model to show macroscopic ground state occupation.

Findings

The gap condition guarantees BEC in probability and in the $r$th mean.

In the Luttinger-Sy model, BEC occurs when particle density exceeds a critical value.

Only the ground state is macroscopically occupied above the critical density.

Abstract

In this paper we discuss Bose-Einstein condensation (BEC) in systems of pairwise non-interacting bosons in random potentials in $d$ dimensions. Working in a rather general framework, we provide a "gap condition" which is sufficient to conclude existence of type-I BEC in probability and in the $r$th mean. We illustrate our results in the context of the well-known (one-dimensional) Luttinger-Sy model. Here, whenever the particle density exceeds a critical value, we show in addition that only the ground state is macroscopically occupied.