On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

TL;DR

This paper investigates the existence and behavior of ground states in a pseudo-relativistic Fermi system at the critical $L^2$ level, establishing conditions for existence and analyzing the limiting behavior as the attractive strength approaches a critical constant.

Contribution

It proves the existence of ground states based on the attractive strength and analyzes their limiting behavior near the critical constant, also exploring properties of the dual fractional Lieb-Thirring inequality optimizers.

Findings

Ground states exist if and only if the attraction strength is below a critical value.

The limiting behavior of ground states is characterized as the strength approaches the critical value.

Qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are studied.

Abstract

We consider ground states of a pseudo-relativistic Fermi system in the $L^2$-critical case. We prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<D_{4/3,2}$, where $D_{4/3,2}\in(0, \infty)$ is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as $a\nearrow D_{4/3,2}$. As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.