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

Each discrete term satisfiesand the same identity follows for the Fourier integral from $R_t=8i{k}^{3}R$. HenceApply ${\partial }_x^2-{\partial }_y^2-u(x,t)$ to the Gelfand-Levitan-Marchenko equation. Differentiating under the integral and integrating the $z$ derivatives by parts, the boundary terms combine to $u=-2{\partial }_{x}K(x,x,t)$. The terms containing $F$ cancel, leavingUniqueness of the Marchenko equation then also gives $G=0$.