Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2017 / S18 / 1 / 6

For $E=-{\hslash }^2{\kappa }^2/(2m)$, the decaying Green's function is $m{e}^{-\kappa |x-x^{\prime }|}/(\kappa {\hslash }^2)$, with the same derivative jump as before. A normalizable negative-energy state has no incident homogeneous part. Applying the inverse to $L\psi =-U\psi$ givesThe condition $\kappa >0$ is necessary for decay on both sides.