Differential dilation matrix elements: $\hat{O} = {\mathbf r} \cdot \nabla$

For $k_a \neq k_b$ we have for the differential dilation operator,

$$\left\langle a\right\vert{\mathbf r}\cdot\nabla\left\vert b\right\rangle _{\mathcal{D}} = \frac{1}{\varepsilon }\oint_\Gamma \!\! d{\mathbf s} \, \left\{ (... ...}{\varepsilon } (a^* \partial_n b - b \partial_n a^*) \rule{0in}{0.3in} \right.$$

$$\hspace{1in} \left. + \; d \,b \partial_n a^* + a^* {\mathbf n}\c... ...bf r}\cdot\nabla)\nabla b + a^*\!{\leftrightarrow}b\rule{0in}{0.3in} \right\} ,$$ (H.24)

where again $\varepsilon \equiv k_a^2 - k_b^2$. This follows from row 2 of the matrix ${\mathcal{M}}^{-1}$. In the Dirichlet BC case this becomes,

$$\left\langle a\right\vert{\mathbf r}\cdot\nabla\left\vert b... ...al_n a^* \partial_n b, \hspace{0.5in} \mbox{(Dirichlet BCs)}.$$ (H.25)

The integral is proportional to the matrix element $(\partial {\mathcal{H}} / \partial x)_{ab}$ of the billiard dilation deformation.

No general form for $k_a = k_b$ has been found. The corresponding unit vector $e_\alpha = \delta_{\alpha 2}$ does not lie in the row space of ${\mathcal{M}}^{-1}$, indicating that other boundary derivatives are needed as input. However, for the particular case of Dirichlet BCs and $b = a$ there exists the simple formula

$$\left\langle a\right\vert{\mathbf r}\cdot\nabla\left\vert a... ...\rangle _{\mathcal{D}},\hspace{0.5in} \mbox{(Dirichlet BCs)},$$ (H.26)

which can be derived by expanding $\nabla\cdot({\mathbf r} \vert a\vert^2)$ and integrating over the domain.