Thermodynamic limit and $L^\infty$-convergence rate for the cubic-quintic Schrödinger model
Deke Li, Yuan Li, Qingxuan Wang
TL;DR
This work analyzes the thermodynamic limit for the cubic-quintic Schrödinger model on expanding domains at fixed density, establishing explicit bulk energy limits and a Thomas–Fermi limit. It develops a novel framework combining energy and gradient estimates to obtain an $L^{\infty}$-convergence rate for ground states in the diluted regime $0<\rho<3/4$, and proves strong convergence to TF profiles in both bounded spherical domains and the full space. The results connect microscopic particle models to macroscopic TF theory, providing sharp energy asymptotics and convergence rates that are robust to domain geometry and potentially extend to more general nonlinearities. The methods offer a versatile toolkit for analyzing nonlinear dispersive models beyond the cubic-quintic case.
Abstract
We investigate the thermodynamic limit for the cubic-quintic Schrödinger model as the size of the domain tends to infinity with fixed density $ρ= N/|\mathcal{D}|$, where $N$ denotes particle number and $|\mathcal{D}|$ denotes the volume of the bounded domain $\mathcal{D}\subset\mathbb{R}^d$ ($d=1,2,3$). We firstly prove the existence of thermodynamic limit, which is equal to $-\frac{3}{32}$ for \(0<ρ\leq \frac{3}{4}\), while $-\left(\frac{1}{2}-\fracρ{3}\right)\fracρ{2}$ for $\frac{3}{4}< ρ\leq 1$. When \(0<ρ<1\) and \(\mathcal{D}\) is a spherical domain, we further show that, up to a scaling, the ground state of the cubic-quintic Schrödinger energy will converge strongly to a Thomas-Fermi ground state in $L^2\cap L^6$. Finally, we obtain the $L^\infty$-convergence rate of ground states for \(0<ρ<3/4\) by developing a novel method, including some iterative techniques, uniform energy estimates and gradient estimates. We believe this method is applicable to other general nonlinearities.