Coercivity for particle number in mass-critical systems

Establish a coercivity estimate for the particle number in reduced Hartree–Fock grand-potential problems for systems that are mass critical at a specified chemical potential and inverse temperature, namely systems for which finiteness of the entropy trace after an energy shift holds only at zero shift.

Background

The paper explains that, at positive temperature, finiteness of the grand potential is characterized by the finiteness of an entropy trace involving the one-particle Hamiltonian. A shifted version of this condition yields a lower bound controlling the particle number, but the condition may fail for every positive shift in mass-critical systems.

For such systems, the usual argument does not provide coercivity of the particle number along minimizing sequences. Resolving this issue would strengthen the variational analysis of positive-temperature reduced Hartree–Fock grand potentials without imposing a mass-noncriticality assumption.