discovered in the 1960s. In the general short-range case ($R\neq 0$), we are not sure whom it should be attributed to but it was systemically studied by Pöppe in the 1980s (see, e.g. [@P84] and also [@P89] for the Kadomtsev-Petviashvili and  [@P83] for the sine-Gordon equations. In the context of nondecaying initial data, it appears first in [@Ry11]. Note that the sense in what the determinant in is understood is an issue which doesn’t seem to be fully addressed in the literature. It is typically defined by the Fredholm formula through an absolutely convergent (Fredholm) series. We actually show that $\mathbb{M}(z,t)$ is trace class for any $z$ (even complex) and $t>0$ under Hypothesis \[hyp1.1\]. This means that $\det
\left( 1+\mathbb{M}(z,t)\right) $ is an invariant, i.e. it produces the same value in any basis in $L^{2}\left( \mathbb{R}_{+}\right) $.

In the setting of step-like potentials, the Marchenko operator has been intensively studied in the Kharkov mathematical school by Hruslov, Kotlyarov and their students[^5] (see, e.g. [@Hruslov76], [@KhrKot94]).

We also refer to Cohen  [@Cohen1984], Kappeler [@Kappeler86], Venakides [@Ven86] (and the literature cited therein), and recent Egorova-Teschl [@ET11]. In all the above papers save [@ET11], $q$’s are assumed to have a specific type of 