On Bose–Einstein Condensation in One-Dimensional Noninteracting Bose Gases in the Presence of Soft Poisson Obstacles

Abstract

We study Bose–Einstein condensation (BEC) in one-dimensional noninteracting Bose gases in Poisson random potentials on \(\mathbb R\) with single-site potentials that are nonnegative, compactly supported, and bounded measurable functions in the grand-canonical ensemble at positive temperatures and in the thermodynamic limit. For particle densities larger than a critical one, we prove the following: With a probability arbitrarily close to one when choosing the fixed strength of the random potential sufficiently large, BEC where only the ground state is macroscopically occupied occurs. If the strength of the Poisson random potential converges to infinity in a certain sense but arbitrarily slowly, then this kind of BEC occurs in probability and in the rth mean, \(r \ge 1\). Furthermore, in Poisson random potentials of any fixed strength a probability arbitrarily close to one for type-I g-BEC to occur is also obtained, but our upper bound for the number of macroscopically occupied one-particle states may be large in this case.