Ground States of One-Dimensional Fermionic Schr\"{o}dinger Systems Near a Critical Exponent

TL;DR

This paper investigates the existence and behavior of ground states in a one-dimensional fermionic nonlinear Schrödinger system near the critical exponent p=2, revealing nonexistence for p approaching 2 from above and detailed profile structure.

Contribution

It proves the nonexistence of ground states as p approaches 2 from above and analyzes the limiting profile, addressing a conjecture and revealing the structure of the density.

Findings

No ground state exists for p approaching 2 from above.

The density profile has exactly two bumps with increasing separation as p nears 2.

The results confirm and refine aspects of a conjecture in prior work.

Abstract

We study ground states of the fermionic nonlinear Schr\"{o}dinger system $J_2(p)$ in $\R$, where $p>1$ denotes a polynomial exponent of the nonlinear term. It is known that the system $J_2(p)$ admits ground states for any $1<p<2$, while there is no ground state for $J_2(2)$. We prove that there is no ground state of $J_2(p)$ as $p\searrow 2$, which addresses the special case of Conjecture 5 in [D. Gontier, M. Lewin and F. Q. Nazar, ARMA, 2021]. The refined limiting profile of ground states for $J_2(p)$ is also analyzed as $p\nearrow 2$, which shows that the corresponding density admits exactly two bumps whose distance goes up to infinity as $p\nearrow 2$.