Asymptotic boundary forms for tight Gabor frames and lattice localization domains

We consider Gabor localization operators $G_{ϕ,Ω}$ defined by two parameters, the generating function $ϕ$ of a tight Gabor frame $\{ϕ_λ\}_{λ\in Λ}$, parametrized by the elements of a given lattice $Λ\subset \Bbb{R}^2$, i.e. a discrete cocompact subgroup of $\Bbb{R}^2$, and a lattice localization domain $Ω\subset \Bbb{R}^2$ with its boundary consisting of line segments connecting points of $Λ$. We find an explicit formula for the boundary form $BF(ϕ,Ω)=\text{A}_Λ\lim_{R\rightarrow \infty}\frac{PF(G_{ϕ,RΩ})}{R}$, the normalized limit of the projection functional $PF(G_{ϕ,Ω})=\sum_{i=0}^{\infty}λ_i(G_{ϕ,Ω})(1-λ_i(G_{ϕ,Ω}))$, where $λ_i(G_{ϕ,Ω})$ are the eigenvalues of the localization operators $G_{ϕ,Ω}$ applied to dilated domains $RΩ$, $R$ is an integer and $\text{A}_Λ$ is the area of the fundamental domain of the lattice $Λ$.

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