Diamagnetic expansions for perfect quantum gases II: uniform bounds

TL;DR

This paper establishes uniform bounds for magnetic susceptibilities of a perfect quantum gas, enabling the application of Vitali's Convergence Theorem to confirm their thermodynamic limit across a range of parameters.

Contribution

It provides the final step in a series proving uniform bounds on magnetic susceptibilities, crucial for understanding quantum gas behavior in magnetic fields.

Findings

Proved uniform bounds on magnetic susceptibilities for all admissible fugacities.

Confirmed the thermodynamic limit of generalized magnetic susceptibilities.

Extended previous pointwise results to uniform bounds on compact sets.

Abstract

Consider a charged, perfect quantum gas, in the effective mass approximation, and in the grand-canonical ensemble. We prove in this paper that the generalized magnetic susceptibilities admit the thermodynamic limit for all admissible fugacities, uniformly on compacts included in the analyticity domain of the grand-canonical pressure. The problem and the proof strategy were outlined in \cite{3}. In \cite{4} we proved in detail the pointwise thermodynamic limit near $z=0$. The present paper is the last one of this series, and contains the proof of the uniform bounds on compacts needed in order to apply Vitali's Convergence Theorem.