On the structure of multivariate Gabor systems and a result on Gaussian Gabor frames

We introduce an equivalence relation on the set of lattices in $\mathbb{R}^{2d}$ such that equivalent lattices support identical structures of Gabor systems, up to unitary equivalence, a notion we define. These equivalence classes are parameterized by symplectic forms on $\mathbb{R}^{2d}$ and they consist of lattices related by symplectic transformations. This implies that $2d^2 - d$ parameters suffice to describe the possible structures of Gabor systems over lattices in $\mathbb{R}^{2d}$, as opposed to the $4d^2$ degrees of freedom in the choice of lattice. We also prove that (modulo a minor complication related to complex conjugation) symplectic transformations are the only linear transformations of the time-frequency plane which implement equivalences of this kind, thereby characterizing symplectic transformations as the structure-preserving transformations of the time-frequency plane in the context of Gabor analysis. In addition, we investigate the equivalence classes that have separable lattices as representatives and find that the parameter space in this case is $d^2$-dimensional. We provide an explicit example showing that non-separable lattices with irrational lattice points can behave exactly like separable and rational ones. This approach also allows us to formulate and prove a higher-dimensional variant of the Lyubarskii-Seip-Wallst\'en Theorem for Gaussian Gabor frames. This gives us, for a large class of lattices in $\mathbb{R}^{2d}$ (including all symplectic ones), necessary and sufficient conditions for $d$-parameter families of Gaussians to generate Gabor frames.