Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 34D / b / Solution

The scattering data for the Schrodinger operator consist of the reflection coefficient $R(k,t)$ on the continuous spectrum, the negative discrete eigenvalues $-{\kappa }_n^2$, and their norming constants $c_n$. Isospectrality givesComparing the asymptotic plane waves in ${\psi }_t=A\psi$ gives