later.

We have also had some flexibility in choosing $r$ and $\delta $ in subject to $r>1$, $1/r<\delta <1$. The range for $Q=2\widetilde{Q}$ given by is $$0<Q<\frac{2(\varepsilon -1)(3\varepsilon )^{\frac{2\varepsilon +1}{1-\varepsilon }}}{(2\varepsilon +1)^{\frac{1+\varepsilon }{1-\varepsilon }}}\cdot c^{\frac{1}{1-\varepsilon }}$$which is inessential to what follows but the borderline case $\varepsilon
=1/2$. For this case, $$0<Q<\frac{3^{3}}{2^{7}}c^{2}.  \label{eq7.1}$$

The Marchenko integral operator and the inverse scattering transform
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The integral operator we are concerned with in this section appears to have been introduced by Marchenko and received a comprehensive treatment in his classical book [@Mar86]. To acknowledge Marchenko’s profound contribution to the subject, we denote this operator by $\mathbb{M}$ (Marchenko used $\mathbb{F}$) but otherwise try to retain as much of his original notation as possible.

We call the integral operator $\mathbb{M}:L^{2}\left( \mathbb{R}_{+}\right)
\rightarrow L^{2}\left( \mathbb{R}_{+}\right)$ Marchenko if $$\begin{aligned}
\left( \mathbb{M}f\right) (x)& =\int_{0}^{\infty }M(x+y)f(y)dy,\quad f\in
L^{2}\left( \mathbb{R}_{+}\right) ,  \notag \\
M(\cdot )& =\int_{0}^{\infty }e^{-(\cdot )\lambda }d\rho (\lambda
)+\int_{-\infty }^{\infty }e^{2i(\cdot )\lambda }R(\lambda )\frac{d\lambda }{2\pi },  \label{eq4.1}\end{aligned}$$where $\rho $ is a finite nonnegative measure and $R$ is such that for a.e. $\lambda \in \mathbb{R}$ $$R(-\lambda )=\overline{R(\lambda )}\quad ,\quad \left\vert R(\lambda
)\right\vert \leq 1.$$

The operator $\mathbb{M}$ introduced this way is clearly a Hankel operator (the kernel depends on 