Thermodynamic limit and $L^\infty$-convergence rate for the cubic-quintic Schr\"{o}dinger model

We investigate the thermodynamic limit for the cubic-quintic Schr\"{o}dinger model as the size of the domain tends to infinity with fixed density $\rho= N/|\mathcal{D}|$, 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 $0<\rho\leq \frac{3}{4}$, while $-\left(\frac{1}{2}-\frac{\rho}{3}\right)\frac{\rho}{2}$ for $\frac{3}{4}< \rho\leq 1$. When $0<\rho<1$ and $\mathcal{D}$ is a spherical domain, we further show that, up to a scaling, the ground state of the cubic-quintic Schr\"{o}dinger 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 $0<\rho<3/4$ 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.