Derivatives of Theta Functions and a Problem of Lyubarskii and Nes

Abstract

We characterize all lattices $\Lambda \subset \mathbb{R}^2$ of rational density for which the Gabor system generated by the first Hermite function along $\Lambda$ is a frame for $L^2(\mathbb{R})$. We prove that these are precisely the lattices whose density satisfies $D(\Lambda) = \frac{q}{p} $ where $p,q \in \mathbb{N}$ are coprime with $q \geq p+2$, thereby confirming a conjecture of Lyubarskii and Nes. A formalization of our main result in Lean $4$ is also included.