Ground States of Attractive Fermi Schr\"{o}dinger Systems with Ring-Shaped Potentials

TL;DR

This paper investigates the existence, nonexistence, and mass concentration of ground states in attractive N-coupled Fermi nonlinear Schrödinger systems with ring-shaped potentials, utilizing a finite-rank Lieb-Thirring inequality.

Contribution

It establishes conditions for the existence and nonexistence of ground states and analyzes their behavior near critical interaction strength, advancing understanding of such quantum systems.

Findings

Ground states exist for interaction strength below a critical value a_N^*.

No minimizers exist when the interaction strength exceeds or equals a_N^* for some N.

Ground states exhibit mass concentration behavior as the interaction approaches the critical value.

Abstract

As an application of the finite-rank Lieb-Thirring inequality established in [R. L. Frank, D. Gontier and M. Lewin, Comm. Math. Phys., 2021], we study ground states of mass-critical N-coupled Fermi nonlinear Schr\"{o}dinger systems with attractive interactions in $\mathbb{R}^3$, which are trapped in ring-shaped potentials. For any given $N\in\mathbb{N}^+$, we prove that ground states exist if $0<a<a_N^*$, where $a$ denotes the strength of attractive interactions in the system, and $a_N^*$ is the best constant of a finite-rank Lieb-Thirring inequality. Moreover, for some $N\in\mathbb{N}^+$, we also prove the nonexistence of minimizers for the system as soon as $a\geq a_N^*$. Applying the energy estimates and the blow-up analysis, we further analyze the mass concentration behavior of ground states for the system as $a\nearrow a_N^*$.