Commutator Estimates for Low-Temperature Fermi Gases

TL;DR

This paper analyzes the semiclassical regularity of low-temperature Fermi gases in harmonic potentials, focusing on Schatten norm estimates of commutators and their dependence on physical parameters.

Contribution

It provides new asymptotic estimates for commutators of one-body operators in low-temperature regimes, including magnetic field effects.

Findings

Derived asymptotic behavior of Schatten norms for low-temperature Fermi gases.

Established upper bounds for magnetic field cases with the Fock-Darwin Hamiltonian.

Identified different regimes based on Planck constant, temperature, and magnetic field strength.

Abstract

We investigate the semiclassical regularity of thermal equilibria in the presence of a harmonic potential at low temperature; that is, we obtain the asymptotic behavior of the Schatten norms of commutators of the one-body operators associated with these equilibria and the position and momentum operators. We also obtain upper bounds in the magnetic field case for the Fock-Darwin Hamiltonian. Our estimates, in particular, allow us to observe several regimes depending on the joint behavior of the Planck constant, the temperature, and the strength of the magnetic field.