Past exam of the mathematics course of the University of Cambridge / 2023 / ib / Paper 3 / 6D / b / Solution

As $|x|\to \infty$, the equation approachesIf $E>U_0$, the asymptotic solutions are oscillatory and do not decay, so there are no bound states. If $0<E<U_0$, the asymptotic solutions are exponential; one decaying branch can be selected at each end. Bound states are therefore possible, although only energies satisfying the global matching conditions actually occur.