= Solution
Eliminate $\mu_2$ between the two connection relations and evaluate at $(0,T)$, where $\mu_1=I$. This yields the matrix <global relation>
$$
\boxed{S(k)^{-1}s(k)=e^{2ik^2T\widehat\sigma_3}\mu_3(0,T,k).}
$$
Its upper off-diagonal entry is especially useful. Since $S^{-1}=\begin{pmatrix}A&-B\\-B^\sharp&A^\sharp\end{pmatrix}$, it is
$$
\boxed{A(k)b(k)-B(k)a(k)=e^{4ik^2T}c_T(k),\qquad c_T(k)=[\mu_3(0,T,k)]_{12}.}
$$
The function $c_T$ is analytic in the upper half-plane and is $O(1/k)$ there by the spatial <Volterra integral equation>. This is the <half-line NLS global relation>. It expresses compatibility between initial and boundary scattering and the final state. At finite $T$ the right-hand side is generally nonzero; setting it to zero would silently impose an extra assumption.
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