Semi-Fredholm property of Cauchy–Riemann operators with a large antilinear zeroth-order term

Let $n=1$, let $S^\pm$ be continuous loops of symmetric $2\times2$ matrices, and let $S:\mathbb{R}\times S^1\to\mathbb{R}^{2\times2}$ be continuous with $S(s,\cdot)\to S^\pm$ uniformly as $s\to\pm\infty$. Let $\beta:\mathbb{R}\times S^1\to\mathbb{C}$ be smooth with bounded first derivative, such that $|\beta|$ and $1/|\beta|$ are bounded for large $|s|$: there are $R$ and $c>0$ with $c\le|\beta(s,t)|\le 1/c$ whenever $|s|\ge R$. For $\sigma\in\mathbb{R}$ let $D_\sigma\eta=\partial_s\eta+J_0\partial_t\eta+S\eta+\sigma\beta\bar\eta$, acting from $W^{1,2}(\mathbb{R}\times S^1,\mathbb{C})$ to $L^2(\mathbb{R}\times S^1,\mathbb{C})$.

Then there is $\sigma_0$ such that for every $\sigma\ge\sigma_0$ the operator $D_\sigma$ is semi-Fredholm: its kernel is finite-dimensional and its range is closed in $L^2$.

No nondegeneracy of the asymptotic operators of $S^\pm$ is assumed, and nothing is assumed about the zeros of $\beta$ in the region $|s|<R$. This is the setting of Wendl's § 5.5 (operators $D_S+\sigma\beta\,\overline{\,\cdot\,}$ with a large antilinear zeroth-order term); closedness of the range is what allows the cokernel of $D_\sigma$ to be identified with the annihilator of its range.

Formalization Note $\mathbb{C}=\mathbb{R}^2$ via $(x,y)\mapsto x+iy$, and $\eta\mapsto\beta\bar\eta$ is the matrix antilinearMat β, so $D_\sigma$ is $D_{S_\sigma}$ with $S_\sigma=S+\sigma\cdot$antilinearMat ∘ β. Points of the cylinder are pairs $(s,t)$ and $\beta$ is $1$-periodic in $t$. $W^{1,2}$, the kernel crKernel 2 and the range crRange 2 are those of Definitions.Def_CauchyRiemannIndex_Setting. The hypotheses on $S$, $S^\pm$ and $\beta$ are a subset of those of the milestone CauchyRiemannIndex.large_antilinear_perturbation_index (Wendl, Proposition 5.27). The exponent is $p=2$.