$\det
    (1+\mathbb{M}(x,t))$ doesn’t vanish for all $x\in \mathbb{R}$. The latter occurs if $1+\mathbb{M}(x,t)$ is invertible on $\mathbb{R}$ for any $t>0$.

Theorem \[thm1.3\] is proven.

Discussions, corollaries, and open problems
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Discussions
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Theorem \[thm1.2\] improves our main result from [@Ry11] where $\mathbb{M}\left( x,t\right) \in \mathfrak{S}_{1}$ was not proven and only real analyticity of $u\left( x,t\right) $ was obtained. The main idea of [@Ry11] is to put together the analytic continuation arguments of [Ryb10]{} to treat initial data on $\mathbb{R}_{-}$ and Tarama’s approach from [@Tarama04] to handle the data on $\mathbb{R}_{+}$. As far as we know the solution to Problem \[pb1\] given in [@Tarama04] was best known back then. The main result of [@Tarama04] says that $u\left( x,t\right) $ is real analytic under the following conditions: $q$ is real and $L_{\limfunc{loc}}^{2}$ such that $$\int_{-\infty }^{\infty }\left( 1+\left\vert x\right\vert \right) \left\vert
q\left( x\right) \right\vert dx<\infty$$and for $x$ large enough there are positive $C_{q},c$ so that$$\int_{x}^{\infty }\left\vert q\right\vert ^{2}\leq C_{q}e^{-cx^{1/2}}.$$Note that these conditions are much stronger than Hypothesis \[hyp1.1\]. The techniques used in [@Tarama04] are also based upon the (classical) IST but his analysis relies on the properties of the Airy function as opposed to ours which is based on analytic and pseudo-analyitc 