📝 SummarXiv

Abstract

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 $\mathcal{D}\subset\mathbb{R}^d$ ($d=1,2,3$). For density satisfying \(0<\rho\leq 3/4\), we prove that the thermodynamic limit of the ground state energy for the cubic-quintic Schr\"{o}dinger model exists, and is actually equal to the corresponding Thomas-Fermi ground state energy. Further more, when \(\mathcal{D}\) is a spherical domain, we prove that, up to a scaling, the ground state minimizer of the cubic-quintic Schr\"{o}dinger energy will converges to a Thomas-Fermi minimizer strongly in $L^2\cap L^6$, and then we obtain the $L^\infty$-convergence rate by developing a novel energy method, which may be applicable to other general nonlinearities. As a by-product, we study the Thomas-Fermi limit for this model in the whole space \(\mathbb{R}^d\) as the particle number \(N\) tends to infinity.