 Solution

Eliminate ${\mu }_2$ between the two connection relations and evaluate at $(0,T)$, where ${\mu }_1=I$. This yields the matrix global relationIts upper off-diagonal entry is especially useful. Since $S^{-1}=\left(\begin{array}{cc}A& -B\\ -B^{♯}& A^{♯}\end{array}\right)$, it isThe 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.