Discussion

I have explained and demonstrated the workings of what is probably the most powerful tool for billiard eigenproblems in existence today: the scaling method of Vergini and Saraceno. Certain errors in the original authors' work have been corrected, most notably the understanding of quasi-orthogonality (which has been deepened through a semiclassical estimate), and the derivation of higher-order terms (Appendix I). A comparison between the results of the scaling method and of the sweep method of the previous chapter has also been made. An analysis of the different types of error in the scaling method has been given. Most surprising is the fact that the deviation of $M$ (equivalently the tension matrix) from diagonality seems not to be responsible for the dominant error, namely the sixth-power-law growth of tension error with the wavenumber shift $\delta$. The understanding of this type of error is a most pressing area of research in the scaling method.

However there are many more directions to investigate. Work on the deformed stadium shows that much more research is sorely needed in the area of basis set choice, in general billiard shapes. It suggests that Vergini's EPW basis for the stadium is in fact very special. Vergini has noted that for the Sinai billiard no such high-quality basis has yet been found[194]. I have found similarly-disappointing results in the generalised Sinai billiard of Fig. 3.2a (the tension minima barely reach $10^{-3}$). Currently investigations are underway of the use of a BIM-type basis set of singular angular-momentum states (the Green's function, or $Y_0(kr)$ Neumann functions or their derivatives).

How could the scaling method be applied elsewhere? The BIM for Helholtz eigenvalue problems is very widely used in the physics, engineering, and acoustics communities [36,121] (in contrast the PWDM has not been used outside the quantum chaos sub-field to my knowledge). It has an basis set which is `pinned' to the boundary (the effective finite basis is due to discretization of the integral equation). Therefore this basis, which has been immensely successful and seems not to suffer from the sensitivity to evanescent waves that a plane-wave basis set does, cannot be used with the scaling method. Either a quasi-orthogonality relation could be found for such a basis, or more likely, a BIM-like basis which is a scaling basis could be created for use with the scaling method. Work is in progress on the latter idea. This way, the very expensive `hunt' for zeros of the BIM determinant could be replaced by a diagonalization returning a large number of states at once. The resulting impact on any research field where Helmholtz eigenstates are needed could be huge.

Other generalizations of the scaling method may also exist. One possibility is that a quasi-orthogonality relation exists for Neumann (or even mixed) boundary conditions. This would allow solution of for example acoustic problems where Neumann BCs are appropriate. Another is the generalization to other scaling systems, for instance those given by homogeneous potentials of the form (D.11).