arXiv subjects

Deke Li

Publications and source records attributed to Deke Li.

3 recordsLinked to original sources

arXiv2026

Condensation and Collapse in the Mean-Field Limit of Rotating 2D Bose Gases with Two-Body and Three-Body Interactions

We consider a system of $N$ interacting bosons in a rotating harmonic trap in $\mathbb{R}^2$, where the two-body interaction is attractive and scaled as $N^{2\alpha}U(N^\alpha x)$ with $0<\alpha<1/12$, and the three-body interaction is repulsive and scaled as $N^{4\beta}W(N^\beta x,N^\beta y)$ with $0<\beta<1/24$. In the mean-field limit, the ground state energy is effectively described by a rotating cubic-quintic nonlinear Schr\"{o}dinger functional. We analyze the collapse regime where the two-body coupling $a$ approaches the critical value $a_*$ and the three-body coupling $b$ tends to zero. The NLS ground states blow up with a universal profile given by the optimizer of the Gagliardo-Nirenberg inequality, and the energy satisfies $E^{\mathrm{NLS}} = (1-\zeta/4+o(1))\mathcal{Q}{\rm{pot}}\ell_n^2$ with $\zeta\geqslant0$ determined by the relative rates of $a_n\to a_*$ and $b_n\searrow0$. From the many-body theory, we rigorously justify this effective description: the quantum ground state energy converges to the NLS energy with the same asymptotic expansion in the collapse regime, and the many-body ground states exhibit complete Bose-Einstein condensation onto the universal blow-up profile.

math-ph↗

arXiv2024

The Ground State of a Cubic-quintic Nonlinear Schr\"{o}dinger Equation with Radial Potential in the Thomas-Fermi Limit

We focus on the ground state of the cubic-quintic nonlinear Schr\"{o}dinger energy functional \begin{gather*} \begin{aligned} {E}(\varphi)=\frac{1}{2}\int_{\mathbb{R}^d}\left(|\nabla \varphi|^2+V(x)|\varphi|^2\right)\,dx \pm\frac{1}{4}\int_{\mathbb{R}^d}|\varphi|^4\,dx +\frac{1}{6}\int_{\mathbb{R}^d}|\varphi|^6\,dx, (d=1,2,3) \end{aligned} \end{gather*} under the mass constraint $\int_{\mathbb{R}^d}|\varphi|^2\,dx=N$, where $N$ can be viewed as particle number, and $V(x)$ behaves like $C|x|^p (p\geq 2)$ as $|x|\rightarrow +\infty$, including the harmonic potential. When $N\rightarrow +\infty$, we show that up to a suitable scaling the ground state $\varphi_N$ would convergence strongly in some $L^q(\mathbb{R}^d)$ space to a Thomas-Fermi minimizer, this limit can be referred to as the \emph{Thomas-Fermi limit}. The limit Thomas-Fermi profile has compact support, given by $u^{TF}(x)=\left[\mu^{TF}-C_0|x|^p\right]^{\frac{1}{4}}_{+}$, where $\mu^{TF}$ is a suitable Lagrange multiplier with exact value. We find that, similar to the asymptotic analysis in [J. Funct. Anal. 260 (2011), 2387-2406.] and [Arch. Ration. Mech. Anal. 217 (2015), 439-523.] for Gross-Pitaevskii energy in the Thomas-Fermi limit where a small parameter $\varepsilon$ tends to 0, there also has a steep \emph{corner layer} near the boundary of compact support of $u^{TF}(x)$, in which the ground state has irregular behavior as $N\rightarrow +\infty$. Finally, we establish a new energy method to obtain the $L^\infty$-convergence rates of ground states $\varphi_N$ inside the corner layer and outside corner layer respectively, this method may be applicable to other general nonlinearities.

math.AP↗

arXiv2024

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.

math.AP↗