Exponential decay for linear Cauchy–Riemann equations on an end of a cylinder

Let $s_1\in\mathbb{R}$, let $S(s,t)$ be real $2n\times2n$ matrices depending smoothly on $(s,t)\in(s_1,\infty)\times\mathbb{R}$ and $1$-periodic in $t$ there, and let $S_\infty$ be a smooth $1$-periodic loop of symmetric matrices with $S(s,t)\to S_\infty(t)$ in $C^\infty$, uniformly in $t$, as $s\to+\infty$. Suppose the asymptotic operator $-J_0\partial_t-S_\infty$ is nondegenerate. Let $\zeta:\mathbb{R}^2\to\mathbb{R}^{2n}$ be smooth on $(s_1,\infty)\times\mathbb{R}$, $1$-periodic in $t$ there, with

and $\zeta(s,t)\to0$ uniformly in $t$. Then $\zeta$ decays exponentially: there are $\delta>0$, $C$ and $s_2$ with $\|\zeta(s,t)\|\le Ce^{-\delta s}$ for all $s\ge s_2$ and all $t$.

This is the exponential decay statement for linear Cauchy–Riemann equations on a half-cylinder in the form in which it is applied on an end, where the equation and its coefficients are only given for $s$ beyond some $s_1$.

Formalization Note $C^\infty$ convergence of matrices is entrywise IsCInftyAsymptoticTo; uniform convergence is TendstoZeroUnif. Mat n and Vec n are $2n\times2n$ real matrices and $\mathbb{R}^{2n}$, crOp S ζ is $\partial_s\zeta+J_0\partial_t\zeta+S\zeta$, a loop of symmetric matrices is IsSymLoop (continuous, $1$-periodic, symmetric), and nondegeneracy of $-J_0\partial_t-S_\infty$ is CauchyRiemannIndex.IsNondegenerate ($1$ is not an eigenvalue of the time-$1$ map $\Psi(1)$ of $\Psi'=J_0S_\infty\Psi$, $\Psi(0)=1$).