The weighted normal operator under a winding window is Fredholm of index zero

Let $S_\infty : \mathbb{R} \to \mathbb{R}^{2\times 2}$ be a smooth symmetric loop and $\delta \in \mathbb{R}$ such that $\delta$ is not an eigenvalue of $A_\infty = -J_0\partial_t - S_\infty$, eigenvalues $\mu < \delta$ have eigenfunctions of winding $\le 0$ and eigenvalues $\mu > \delta$ have eigenfunctions of winding $\ge 1$. Let $S : \mathbb{R}^2 \to \mathbb{R}^{2\times 2}$ be smooth, $1$-periodic in $t$, exponentially asymptotic to $S_\infty$ at some rate $d > 0$ (IsExpAsymptoticTo), and of plane form (IsPlaneCoeff S: $S(s,t) = 4\pi\bar z A(z)$, $z = e^{2\pi(s+it)}$, $A$ smooth on $\mathbb{C}$). Let $\chi$ be smooth with $\chi' = \pi$ on $(-\infty,-1]$ and $\chi' = \delta$ on $[1,\infty)$ (IsSlopeWeight χ π δ), and write $D = \partial_s + J_0\partial_t + S + \chi'(s)$. Let $1 < p < \infty$. Then $D : W^{1,p} \to L^p$ is Fredholm (IsFredholmCR) and its index is $0$ (crIndex).

Role. The index count for the weighted normal operator along a holomorphic plane: the window at $\delta$ at the end $s = +\infty$ and the slope $\pi$ at the removable end $s = -\infty$ balance to index $0$. Combined with injectivity, this gives bijectivity.