Past exam of the theoretical physics II course of the University of Cambridge / 2015 / Paper 1 / 2 / d / Solution

The normalized Dirichlet eigenfunctions on $(-L,L)$ and their decay rates areTheir norms are one because the sine-squared integral is $L$. At the initial position zero, ${\phi }_n(0)=L^{-1/2}\sin (n\pi /2)$, so all even $n$ disappear. For $n=2j+1$, the two signs from ${\phi }_n(x)$ and ${\phi }_n(0)$ cancel, yieldingThis holds for $|x|<L$ and is zero outside. The kernel is even in $x$, vanishes at both boundaries and tends weakly to the initial delta. The eigenvalues are decay rates rather than quantum energies unless one makes an additional unit convention.