Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 107 / 1 / b

The required estimate follows from one-dimensional barriers on a sufficiently narrow slab. Put $\kappa =\sqrt{\tau }$. On $|x_1|<d_1<\pi /(2\kappa )$, the functionsatisfies $h^{\prime \prime }+\tau h=0$, equals one at $x_1=\pm d_1$, and obeysChoose $\delta (\epsilon ,\tau )$ so thatand so that the analogous solution of $k^{\prime \prime }+\tau k=-1$ with zero endpoint values is bounded by $e^{2d_1}-1$. Comparing $u$ and $-u$ withand using $c\le \tau$ givesThe permitted width is of order ${\tau }^{-1/2}$, so it can be chosen with $\delta (\epsilon ,\tau )\to \infty$ as $\tau \downarrow 0$.