Equal wavenumber $k_a = k_b = k$

The case $k_a = k_b$ can be reached most easily by setting $k_a = k$ and $k_b = k + \delta$ while taking the limit $\delta \rightarrow 0$. The solution $b$ now has to be spatially scaled (dilated about the arbitrary origin) appropriately for its new wavenumber. Section 6.1.1 (also Appendix I) gives the expansion of the dilation operator at wavenumber $k$ in powers of $\delta$, thus

$$b(k + \delta;{\mathbf r}) \; = \; \left[ 1 + \frac{\delta}{... ...} \cdot \nabla + O(\delta^2) \cdots \right] b(k;{\mathbf r}) .$$ (H.6)

Substitution into (H.5), noting $k_a^2 - k_b^2 = -2 \delta k + O(\delta^2)$, and taking the limit (in which only the lowest order $O(\delta^0)$ survives) gives

$$\left\langle a \left\vert b\right.\right\rangle _{\mathcal{... ...nabla ({\mathbf r}\cdot\nabla b) \rule{0in}{0.2in} \right] .$$ (H.7)

This can be shown to be a generalization of the overlap formula given by Boasman [33] to general $d$ and differing solutions $a$ and $b$. Despite the derivation using scaling, (H.7) is in fact an identity, which can be proved using a messy algebra sequence^H.1. Unlike the LHS, the RHS is not manifestly $a^*\!{\leftrightarrow}b$ symmetric. However, it can be written in the symmetrized form (H.14) derived in the next section. Notice that the choice $k_a = k_b$ has increased the order of derivative required on the boundary by one.

In the case of Dirichlet BCs the second term in (H.7) vanishes, and the replacement ${\mathbf r}\cdot\nabla b = ({\mathbf r}\cdot{\mathbf n}) {\mathbf n}\cdot\nabla b$ (which follows from the fact that $\nabla b$ and ${\mathbf n}$ are parallel), gives

$$\left\langle a \left\vert b\right.\right\rangle _{\mathcal{... ...l_n a^* \partial_n b , \hspace{0.5in} \mbox{(Dirichlet BCs)}.$$ (H.8)

However the knowledge that the LHS is zero for $b \ne a$ (orthogonal eigenstates) shows that (H.8) is an exact boundary orthogonality relation for degenerate Dirichlet eigenstates. Note that the RHS is proportional to the matrix element $(\partial {\mathcal{H}} / \partial x)_{ab}$ of the billiard dilation deformation between degenerate states (Appendix C). On the other hand, choosing $b = a$ we have

$$\left\langle a \left\vert a\right.\right\rangle _{\mathcal{... ...\partial_n a \vert^2 , \hspace{0.5in} \mbox{(Dirichlet BCs)}.$$ (H.9)

This very useful boundary formula for the norm of Dirichlet eigenstates appears to have been found first for $d=2$ by Berry and Wilkinson (appendix of [28]). It was since derived in a different way by Boasman[33], and for general $d$ was derived in our work[14]. However, I believe the new derivation above to be the simplest yet.