the sum of the arguments). We say that $\mathbb{M}$ is associated with a potential $q$ if $R$ is a reflection coefficient from $q $ and $\rho $ characterizes the negative spectrum of $H_{q}$. For short-range $q$’s, the measure $\rho $ is purely discrete and ($\delta $ is the Dirac $\delta $-function) $$d\rho (\lambda )=\sum_{n=1}^{N}c_{n}^{2}\delta (\lambda -\varkappa
_{n})d\lambda$$where $\varkappa _{n}$’s are such that $\left\{ -\varkappa _{n}^{2}\right\}
_{n=1}^{N}$ are the negative bound states of $H_{q}$ and $\left\{c_{n}\right\}_{n=1}^N$ are the associated norming constants. Of course, the kernel $M$ (which we also call Marchenko) is nothing but the sum of the Laplace transform of the (finite, positive) measure $\rho $ and the Fourier transform of the (symmetric, bounded) function $R$. This is the main feature of the Marchenko operator resulting in the decomposition $$\mathbb{M}=\mathbb{M}_{1}+\mathbb{M}_{2},  \label{eq4.2}$$where $\mathbb{M}_{1}\geq 0$ and ($\chi =\chi _{+}$, $\mathcal{F}$ is the Fourier transform) $$\mathbb{M}_{2}=\chi \mathcal{F}^{-1}R\mathcal{F}^{-1}$$and is selfadjoint.

Note that the Marchenko operator is not typically studied in the context of Hankel operators. We have found in [@Ry11] that the language of Hankel operators is very convenient in the IST formalism. In this language, $\rho
,R $ are called the measure and symbol of $\mathbb{M}_{1},\mathbb{M}_{2}$ respectively (see e.g. [@Nik2001]). In the language 