Mass concentration and ground states of Fermi systems in Hartree--Fock model with potentials

Abstract

Understanding the onset and mechanism of collapse at critical attractive interaction is a fundamental problem in the variational theory of finite fermionic systems. In this paper, we study this problem within the Hartree--Fock framework under various trapping potentials. By establishing a finite-rank Hartree-type Gagliardo--Nirenberg--Sobolev inequality, we identify a critical threshold $\mathcal{K}^{(N)}$ governing the existence of ground states. We then determine the concentration behavior of ground states as the interaction parameter approaches $\mathcal{K}^{(N)}$. To reveal how the geometry of the trap influences this critical phenomenon, we consider three representative classes of potentials: polynomial-type potentials, ring-shaped potentials, and potentials whose zero set has nonempty interior. For each class, we derive the concentration scale and sharp asymptotics of the least energy. The resulting regimes are qualitatively different, showing that both the location and the rate of concentration are selected by the local geometry of the potential near its minimum set. In contrast to scalar Hartree problems, the nonlocal interaction is coupled here with pairwise $L^2$-orthogonality constraints, and their interplay creates substantial difficulties in the compactness and blow-up analysis.