We refer the interested reader to a particularly influential 1977 paper [@Air77] by Airault-McKean-Moser and recent Bona-Weissler [Bona09]{} and the literature cited therein. More complicated examples of explicit solutions include algebraric, rational, meromorphic simply periodic, elliptic, etc. (see, e.g. [@AblSatJMP78], [@Birnir87], [@GUW06], [@AktMee06] and the literature cited therein). All these examples are of course very specific and in addition those $q$’s are already meromorphic (i.e. smooth on the real line). Although Problem \[pb1\] is not addressed in those papers but they demonstrate the importance of meromorphic solutions.

Through the paper we deal with initial data subject to

\[hyp1.1\] $q$ is real and $L_{\limfunc{loc}}^{1}$ such that

1.  (semiboundedness from below) $$\label{Cond1}
    \inf\func{Spec}\left(-\partial_x^2+q(x)\right)=-h_0^2$$ with some $h_0\ge0$.

2.  (subexponential decay at $+\infty $) For $x$ large enough $$\int_{x}^{\infty }\left\vert q\right\vert \leq C_{q}e^{-cx^{\varepsilon }}
    \label{Cond2}$$with some positive $C_{q},c,\varepsilon $.

We assume that the constants $c,\varepsilon $ in are chosen optimal.

Note that the set of such functions is very large. Indeed, in terms of $q$ itself, Condition is satisfied if $$\limfunc{Sup}\limits_{x}\int_{x-1}^{x}\max \left( -q,0\right) <\infty ,
\label{Cond_q}$$i.e. $q$ is essentially bounded from below [@Glazman66]. The condition (\[Cond\_q\]) cannot be improved since (\[Cond\_q\]) becomes also necessary for (\[Cond1\]) if $q$’s are negative. Therefore, any $q$ subject 