Appendix C: Numerical evaluation of quantum band profile in billiards

Here I gather the tools used to estimate the quantum `band profile' $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ given the eigenstates of a billiard. Sections 2.2.2 and 3.1 are required preliminary reading. I start with the very important derivation of the matrix elements.