Convergence in practice

Following [137], the number of samples per wavelength I will call $b \equiv L/M\lambda_{{\mbox{\tiny B}}}$. A practical choice of $b$ will depend on what type of wavefunction and billiard we have, as well as the required accuracy of $F$.

I found $b=6$ is sufficient for most work involving the stadium billiard and evanescent waves components of $2k$ or less. In this shape, the convergence with $b$ does not follow any obvious power-law, but for $b=6$ I found the fraction error on $F$ is $< 10^{-6}$ for wavefunctions composed of random (real) plane waves. If highly evanescent waves are present (up to $3k$), $b$ needs to be slightly higher (about 10). Certainly the $b$ of 30 quoted by [137] is excessive, considering the fact that their PWDM basis set only includes real plane waves. In the scaling method, I find that errors due to a smaller $b$ cause spurious states (Section 6.3.3) to enter earlier than is necessary. However, the accuracy of the valid states found does not seem very dependent on $b$.

Further study on the accurate evaluation of these integrals is necessary; in particular a way to handle corners and `kinks' which has high-order (preferably exponential) convergence is sought.