Asymptotic safety regions for Gabor frames generated by Hermite functions

The aim of this paper is to establish new regions in the frame sets of Hermite functions $h_n$. A classical result of Gröchenig and Lyubarskii shows that the Gabor system $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame for $L^2(\mathbb{R})$ whenever the lattice density exceeds $n+1$. We show that, for every $\eta >0$ and all sufficiently large $n$, the same Gabor system forms a frame whenever $ab\le n^{-\frac{2}{3}-\eta }$. Moreover, we obtain an asymptotically sharp result near the coordinate axes, i.e., when one of the parameters $a$ or $b$ is small. Namely, for every $\delta >0$ and $\rho \in (0,\frac{1}{2})$ and all sufficiently large $n$ we prove that if $\min \{a,b\}\le n^{-\frac{1}{2}-\delta }$ and $ab\le \frac{1}{2}-\rho$ then $\mathcal{G}(h_n,a\mathbb{Z}\times b\mathbb{Z})$ forms a frame.