lambda|^{-3/2}\,w(\lambda) \;. 
\end{align*}
\end{enumerate}
The constant \,$R'$\, can be chosen uniformly for \,$(u,u_y) \in P$\,, where \,$P$\, is a relatively 
compact subset of \,$\mathsf{Pot}_{np}$\,. 
\end{cor}

\begin{proof}
We show the estimates for the function \,$a'$\,, the other functions are handled analogously.
Let \,$\varepsilon>0$\, be given, and 
fix \,$\delta>0$\,. By Theorem~\ref{T:asymp:basic} there exists \,$R>0$\, such that for \,$\lambda\in \C$\, with \,$|\lambda|\geq R$\, we have
$$ |a(\lambda)-\upsilon\,a_0(\lambda)| \leq \frac{\varepsilon\,\delta}{2\,e^\delta}\,w(\lambda) \;. $$
There exists \,$R'>R$\, such that for all \,$\lambda\in \C$\, with \,$|\lambda|\geq R'$\,, the closure \,$\overline{U}$\, of 
\,$U := \Mengegr{\lambda'\in \C}{|\lambda'-\lambda|<\delta\,|\lambda|^{1/2}}$\, is entirely contained in \,$\{|\lambda'|\geq R\}$\,. 
For \,$\lambda'\in \overline{U}$\,, we have \,$w(\lambda') \leq 2\,e^\delta\,w(\lambda)$\, by Proposition~\ref{P:asymp:w}(1),
and therefore by Cauchy's inequality, applied to the holomorphic function \,$a-a_0$\,:
\begin{align*}
|a'(\lambda)-\upsilon\,a_0'(\lambda)| & \leq \frac{1}{\delta\,|\lambda|^{1/2}} \max_{\lambda'\in\partial U} |a(\lambda')-\upsilon\,a_0(\lambda')| \\
& \leq \frac{1}{\delta\,|\lambda|^{1/2}} \cdot \frac{\varepsilon\,\delta}{2\,e^\delta}\cdot 2\,e^\delta\,w(\lambda) = \varepsilon\,|\lambda|^{-1/2}\,w(\lambda) \;. 
\end{align*}
For the case \,$\lambda\to0$\,, we take \,$U:=\Mengegr{\lambda'\in \C^*}{|\lambda'-\lambda|<\delta\,|\lambda|^{3/2}}$\,, and again choose \,$R'>R$\,, such that for \,$|\lambda|\leq \tfrac{1}{R'}$\,,
we have \,$\overline{U} \subset \{|\lambda'|\leq \tfrac{1}{R'}\}$\,. We again have \,$w(\lambda') \leq 2\,e^\delta\,w(\lambda)$\, for \,$\lambda'\in \overline{U}$\,, and therefore
by Cauchy's inequality
\begin{align*}
|a'(\lambda)-\upsilon^{-1}\,a_0'(\lambda)| & \leq \frac{1}{\delta\,|\lambda|^{3/2}} \max_{\lambda'\in\partial U} |a(\lambda')-\upsilon^{-1}\,a_0(\lambda')| \\
& \leq \frac{1}{\delta\,|\lambda|^{3/2}} \cdot \frac{\varepsilon\,\delta}{2\,e^\delta}\cdot 2\,e^\delta\,w(\lambda) = \varepsilon\,|\lambda|^{-3/2}\,w(\lambda) \;. 
\end{align*}
\end{proof}

For Cauchy data \,$(u,u_y)$\, that are once more differentiable, we obtain an even better asymptotic estimate.
For stating this asymptotic estimate, we introduce the space of (non-periodic) once more differentiable potentials
\begin{equation}
\label{eq:asymp:Potnp1}
\mathsf{Pot}_{np}^1 := \Menge{(u,u_y)}{u \in W^{2,2}([0,1]), u_y \in W^{1,2}([0,1])} \;,
\end{equation}
and phrase the statement in terms of the regauged monodromy \,$E$\, from the proof of Theorem~\ref{T:asymp:basic}:

\enlargethispage{3em}

\begin{prop}
\label{P:asymp:more}
Suppose that we have \,$(u,u_y) \in \mathsf{Pot}^1_{np}$\,. Then there exist constants
\,$R,C>0$\,, such that for every \,$\lambda\in \C$\, with \,$|\lambda|\geq R$\, and \,$x\in [0,1]$\,, we have
$$ |E_\lambda(x)-E_{0,\lambda}(x)| \leq \frac{C}{\sqrt{|\lambda|}}\,w(\lambda) \;. $$
If \,$P$\, is a relatively compact subset of \,$\mathsf{Pot}^1_{np}$\,, then 
the constants \,$R,C$\, can be chosen uniformly for \,$(u,u_y) \in P$\,.
\end{prop}

\begin{proof}
We use the objects and notations from the proof of Theorem~\ref{T:asymp:basic}. In particular
we use the regauging described at the beginning at that proof. But now we can use the higher
degree of differentiability of \,$(u,u_y)$\, 
to eliminate the \,$\lambda^0$-component from \,$\widetilde{\alpha}$\, 
by regauging \,$\widetilde{\alpha}$\, resp.~\,$\widetilde{F}$\, once more, with 
\begin{equation}
\label{eq:asymp:more:h}
h := \begin{pmatrix} \lambda^{-1/2}\,u_z & 1 \\ -1 & -\lambda^{-1/2}\,u_z \end{pmatrix} \;. 
\end{equation}
We have
\begin{equation}
\label{eq:asymp:more:h-1}
h^{-1} = \frac{1}{1-\lambda^{-1}\,u_z^2}\begin{pmatrix} -\lambda^{-1/2}\,u_z & -1 \\ 1 & \lambda^{-1/2}\,u_z \end{pmatrix}
\qmq{and} \mathrm{d}h = \begin{pmatrix} \lambda^{-1/2}\,u_{zx} & 0 \\ 0 & -\lambda^{-1/2}\,u_{zx} \end{pmatrix} \;, 
\end{equation}
and from this, we obtain for the again regauged \,$\widetilde{\alpha}$\, by an explicit calculation
\begin{align*}
\overline{\alpha} & := h^{-1}\,\widetilde{\alpha}\,h - h^{-1}\,\mathrm{d}h 
= \widetilde{\alpha}_0 + \lambda^{-1/2}\cdot (\overline{\beta}_++\overline{\beta}_-) + O(|\lambda|^{-1}) \\
& \qquad \text{ with } \overline{\beta}_+ := \begin{pmatrix} 0 & \tfrac12\,u_z^2-\tfrac14\,\cosh(u)+1 \\ -\tfrac12\,u_z^2 + \tfrac14\,\cosh(u)-1 & 0 \end{pmatrix} \\
& \qquad \text{ and } \overline{\beta}_- := \begin{pmatrix} 0 & -u_{zx}-\tfrac14\,\sinh(u) \\ -u_{zx}-\tfrac14\,\sinh(u) & 0 \end{pmatrix} \;. 
\end{align*}
We also consider \,$\overline{E}(x) := h^{-1}(x)\,E(x)\,h(0)$\,. From Equations~\eqref{eq:asymp:E0}, \eqref{eq:asymp:more:h} and \eqref{eq:asymp:more:h-1}, we obtain by an explicit calculation
\begin{equation}
\label{eq:asymp:more:h-1E0h}
h^{-1}\,E_0\,h = E_0 + O(|\lambda|^{-1/2}) \;. 
\end{equation}


By Lemma~\ref{L:asymp:compare}(2)(b) we have \,$\overline{E} = E_0 + \sum_{n=1}^\infty \overline{E}_n$\, with
$$ \overline{E}_n(x) = \lambda^{-n/2}\,\sum_{\varepsilon\in\{\pm 1\}^n} \int_0^x \int_0^{t_1} \cdots \int_0^{t_{n-1}} \!\!\!\!\!\!\!\!\!\! E_0(\xi_\varepsilon(t))\,\overline{\beta}_{\varepsilon_1}(t_1)\,\overline{\beta}_{\varepsilon_2}(t_2)\dotsc\,\overline{\beta}_{\varepsilon_n}(t_n)\,\mathrm{d}^nt + O(|\lambda|^{-1}) \;, $$
where \,$\xi_\