Real plane waves (RPWs)

This is the default basis set:

$$\bar{\phi}_{2i-1}(\bar{{\mathbf r}}) = \cos ({\mathbf n}_i \cdot \bar{{\mathbf r}})$$ (J.3)

$$\bar{\phi}_{2i}(\bar{{\mathbf r}}) = \sin ({\mathbf n}_i \cdot \bar{{\mathbf r}}),$$ (J.4)

for $i = 1 \cdots N/2$, where ${\mathbf n}_i$ is the unit direction vector at angle $\theta_i$. The angular distribution can be uniform in the range $[0,\pi]$, giving $\theta_i = 2\pi(i- \mbox{\small$\frac{1}{2}$})/N$. A slight improvement results if there is more concentration in the directions perpendicular to the billiard's largest distances from the origin. In the 2-by-4 stadium (following [195]) I used $\theta_i = {\textstyle\frac{1}{4}}[5 - 4i/N]\cdot 2\pi(i- \mbox{\small$\frac{1}{2}$})/N$ .

Note that Li[137] claims he can half the RPW basis set size required, by randomizing phases (removing the need for both cos and sin functions). From our considerations in Section 5.3.1, it seems unlikely that he has somehow defeated the semiclassical basis size requirement by a factor of 2, however this is an area for investigation.

Figure J.1: Basis functions in the stadium billiard. Left: Real plane wave, and Right: Evanescent plane wave (showing geometry). Both are symmetrized, and are shown at $k=30$.