Ground States of Fermionic Nonlinear Schr\"{o}dinger Systems with Coulomb Potential I: The $L^2$-Subcritical Case

TL;DR

This paper proves the existence of ground states for coupled fermionic nonlinear Schrödinger systems with Coulomb potential in the $L^2$-subcritical case and analyzes their concentration behavior as the attractive parameter grows large.

Contribution

It establishes the existence of ground states for all positive attraction parameters and describes their limiting concentration behavior at singular points.

Findings

Ground states exist for any $\alpha>0$.

Ground states concentrate at Coulomb potential singularities as $\alpha o \infty$.

The analysis applies to the $L^2$-subcritical regime.

Abstract

We consider ground states of the $N$ coupled fermionic nonlinear Schr\"{o}dinger systems with the Coulomb potential $V(x)$ in the $L^2$-subcritical case. By studying the associated constraint variational problem, we prove the existence of ground states for the system with any parameter $\alpha>0$, which represents the attractive strength of the non-relativistic quantum particles. The limiting behavior of ground states for the system is also analyzed as $\alpha\to\infty$, where the mass concentrates at one of the singular points for the Coulomb potential $V(x)$.