Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 34C / b / i / 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,
${\phi }_{-}=\overline{{\phi }_{+}}$, forcing the transfer matrix to have the formThe Wronskian of two solutions of
$-{\varphi }^{\prime \prime }+u\varphi =k^2\varphi$ is independent of $x$. Translation therefore preserves it, soConsequently