Basis `badness' $\gamma$

Generally the best (in the least squares sense in the linear Helmholtz function space) coefficient vector ${\mathbf x}$ able to represent a true (domain-normalized) eigenstate has exponentially large coefficients. This can be expressed (following Vergini [194]) in terms of a basis `badness',

$$\gamma \ \equiv \ \vert{\mathbf x}\vert^2 ,$$ (J.2)

which diverges exponentially once the basis set size $N$ is increased beyond $N_{sc}$. Numerically, because of the null-space truncation of Section 5.3.2, $\gamma$ has an upper limit of $\epsilon^{-1} \sim 10^14$. This issue is identical to the divergence of basis coefficients found by Dietz et al. when attempting to expand a `cake billiard' eigenfunction in terms of plane waves (or angular-momentum states) [63]. The problem is that of representing evanescent components, which are generally present in billiard eigenfunctions, by distributions of real plane waves. As Berry has shown [26], rapidly-oscillating distributions are required which grow exponentially with the number of wavelengths across the system and with the evanescence parameter. This was well understood by Vergini, who demonstrated that improving the basis by adding a small number of evanescent plane waves (in the case of the stadium only) enabled $\gamma$ to reach $O(1)$. This improved the basis tension error $\epsilon_{{\mbox{\tiny typ}}}$ from about $10^{-6}$ to about $10^{-11}$.

In general the requirements of a good basis are small $\epsilon_{{\mbox{\tiny typ}}}$ and, if possible, $\gamma \sim O(1)$ for most states. I first give the real plane waves, which work well in many billiard shapes, then give Vergini's evanescent plane waves which currently are well-adapted only to the stadium billiard (and shapes close to it).