= Solution
Translation by $2\pi$ preserves the scattering equation, so it acts linearly on its two-dimensional solution space. Because $u$ is real and the initial conditions are conjugate,
$\varphi_-=\overline{\varphi_+}$, forcing the transfer <matrix> to have the form
$$
\widehat T=\begin{pmatrix}a&b\\\bar b&\bar a\end{pmatrix}.
$$
The Wronskian of two solutions of
$-\phi''+u\phi=k^2\phi$ is independent of $x$. Translation therefore preserves it, so
$$
\det\widehat T=1.
$$
Consequently
$$
\boxed{|a|^2-|b|^2=1}.
$$
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