Joint compact-window construction

Choose real $\chi \in {C_c}^{\infty }((-2,2))$, equal to one on $[-1,1]$, and choose $\psi \in {C_c}^{\infty }((-2,2))$ with $\int \psi =1$. For $L>0$, put

Let $V_L:{\mathbb{C}}^2\to L^2(\mathbb{R},\mathrm{d}t)$ have columns $v_{i,L}$. For sufficiently large $L$, $G_L={V_L}^{†}{V}_L$ is positive definite. First orthonormalize:

Let $H:{\mathbb{C}}^2\to L^2(\mathbb{R},\mathrm{d}\omega )$ have columns $h_1,h_2$. Write the left polar decomposition

and define

The columns of $F_L$ are denoted ${\hat{f}}_{1,L},{\hat{f}}_{2,L}$.