behavior at $-\infty $ (approaching either a constant or a periodic function) and fall off at $+\infty$. In [ET11]{}, the interesting case of two finite gap potentials fused together is considered.

We summarize important properties of the Marchenko operator in the following (see [@Ry11] for details).

\[pr4.1\] Assuming Hypothesis \[hyp1.1\], let $\mathbb{M}(z,t)$ be the Marchenko operator associated with $q$ and let $A$ be as in Proposition [pr3.2]{}. Then for any $z\in \mathbb{R}$, $t>0$, $$\mathbb{M}(z,t)=\mathbb{M}_{+}(z,t)+\mathbb{A}(z,t),  \label{eq4}$$where $\mathbb{M}_{+}(z,t)$ is the Marchenko operator associated with $q_{+}=q|_{\mathbb{R}_{+}}$ and $\mathbb{A}(z,t)$ is a Hankel integral operator with the kernel $$\frac{1}{2\pi }\int_{\mathbb{R}+ih}e^{2i\lambda (\cdot )}\zeta
_{z,t}(\lambda )A(\lambda )d\lambda ,\quad h>h_{0}.$$Furthermore, $\mathbb{A}(z,t)$ is an entire operator-valued function of trace class for any complex $z$ and $t>0$, continuous with respect to $q$ in the following sense: If $q_{1},q_{2}$ are two functions subject to Hypothesis \[hyp1.1\] then $$\left\Vert \mathbb{A}_{1}(z,t)-\mathbb{A}_{2}(z,t)\right\Vert _{\mathfrak{S}_{1}}\leq \frac{1}{4\pi h}\left\Vert \zeta _{z,t}(A_{1}-A_{2})\right\Vert
_{L^{1}(\mathbb{R}+ih)}$$for any $z\in \mathbb{C}$, $t>0$.

Note that $\mathbb{M}(z,t)$ depends on $(z,t)$ through $\zeta _{z,t}$.

Proof of the main results
=========================

With all the preparations done in the previous sections, the actual proofs will be quite short. It is convenient to conduct both proofs at a time. Note first that, by a trivial shifting, we may assume without 