Gabor Frame Sets of Invariance - A Hamiltonian Approach to Gabor Frame Deformations

In this work we study families of pairs of window functions and lattices which lead to Gabor frames which all possess the same frame bounds. To be more precise, for every generalized Gaussian $g$, we will construct an uncountable family of lattices $\{{\mathrm{\Lambda }}_{\tau }\}$ such that each pairing of $g$ with some ${\mathrm{\Lambda }}_{\tau }$ yields a Gabor frame, and all pairings yield the same frame bounds. On the other hand, for each lattice we will find a countable family of generalized Gaussians $\{g_i\}$ such that each pairing leaves the frame bounds invariant. Therefore, we are tempted to speak about “Gabor Frame Sets of Invariance”.