Semiclassical Moser--Trudinger type inequalities for orthonormal families in $\mathbb{R}^2$

Abstract

For $H^1(\mathbb{R}^2)$-orthonormal families $\{u_n\}_{n=1}^N$ with one-body density $ρ=\sum_{n=1}^N |u_n|^2$, we prove the whole-space endpoint Moser--Trudinger inequality $$ \int_{\mathbb{R}^2}\frac{e^{4πρ(x)}-1-4πρ(x)}{ρ(x)}\,dx \le C\left(N-1+\int_{\mathbb{R}^2}ρ(x)\,dx\right), $$ where the integrand on the left-hand side is defined to be zero on $\{x: ρ(x)=0\}$. We develop a heat-semigroup refinement of Rumin's density method to treat both the endpoint coefficient $4π$ and the whole-space setting. Our inequality is sharp in two respects: the optimality of the coefficient $4π$ and the linear growth in the rank $N$. Moreover, through a semiclassical analysis of the density generated by normalized Dirichlet eigenfunctions, we establish the collective asymptotic behaviors of the quantities appearing in the inequality in various regimes. These semiclassical behaviors show that the rank-dependent term $N-1$ on the right-hand side is indispensable.