to Hypothesis \[hyp1.1\] is essentially bounded from below, has subexponential decay at $+\infty $ and arbitrary otherwise. Such functions can grow (arbitrarily fast) at $-\infty $ or look like a stock market (Gaussian white noise on a left half line) but still satisfy our hypothesis as long as they exhibit rapid decay (\[Cond2\]) at $+\infty $. In spectral terms (\[Cond2\]) implies that $\left( 0,\infty \right) $ belongs to the absolutely continuous spectrum of $-\partial _{x}^{2}+q(x)$.

We now state our main results.

\[thm1.2\] Under Hypothesis \[hyp1.1\] with $\varepsilon \geq 1/2$ on the initial data $q$ in , the problem has an analytic in $z$ solution $u(z,t)$ given by $$u(z,t)=-2\partial _{z}^{2}\log \det \left( 1+\mathbb{M}(z,t)\right) ,
\label{det_form}$$where $\mathbb{M}(z,t)$ is a trace class operator-valued function constructed in Proposition \[pr4.1\] below for any $t>0$. Moreover, for any $t>0$

1.  If $\varepsilon>1/2$ then $u(z,t)$ is meromorphic on $\mathbb{C}$.

2.  If $\varepsilon =1/2$ then $u(z,t)$ is meromorphic in the strip $$\left\vert {\operatorname{Im}}z\right\vert <\frac{9\sqrt{2}}{8}c\sqrt{t}  \label{strip}$$where $c$ is as in .

\[thm1.3\] Under Hypothesis \[hyp1.1\] with $0<\varepsilon <1/2$ on the initial data $q$ in , the operator-valued function $\mathbb{M}(x,t)$ given in Proposition \[pr4.1\] is trace class for any real $x$ and $t>0$ and $$\mathbb{M}(x,t)=\mathbb{M}^{\left( 1\right) }(x,t)+\mathbb{M}^{\left(
2\right) }(x,t),$$where $\mathbb{M}^{\left( 