Comment on "Bose-Einstein condensation with a finite number of particles in a power-law trap"

TL;DR

This paper critically examines the finite-size expansion for Bose-Einstein condensation temperature in power-law traps, highlighting potential inaccuracies and differences from established results, especially in harmonic traps.

Contribution

It provides a critical analysis of previous analytical expansions, clarifying their limitations and discrepancies for specific trap geometries.

Findings

The expansion may not improve upon the first order correction for harmonic traps.

Results differ at first order for some non-harmonic traps.

Caution is advised when applying the previous expansion to finite systems.

Abstract

In Jaouadi et al. [Phys. Rev. A 83, 023616 (2011)] the authors derive an analytical finite-size expansion for the Bose-Einstein condensation critical temperature of an ideal Bose gas in a generic power-law trap. In the case of a harmonic trap, this expansion adds higher order terms to the well- known first order correction. We point out a delicate point in connection to these results, showing that the claims of Jaouadi et al. should be treated with caution. In particular, for a harmonic trap, the given expansion yields results that, depending on what is considered to be the critical temperature of the finite system, do not generally improve on the established first order correction. For some non-harmonic traps, the results differ at first order from other results in the literature.