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