Conclusion and discussion

In this chapter I have presented the main components of an improved `sweep' method (so named because a sweep in $k$ is required to locate eigenstates) for billiard diagonalization. It falls squarely in Class b) of Table 5.1, although the zero-determinant condition has been replaced by a maximum-eigenvalue one. The resulting eigenstates once found can be normalized accurately using the boundary formula for $\left\langle \psi \left\vert\psi\right.\right\rangle _{cal D}$ presented in Appendix H. The method is a simplified but more rigorous version of Heller's original PWDM, and reduces to the key component of Heller's method in the case of a particular choice of $G$.

Several problems in Heller's method have been solved. The choice of basis size $N$ in Heller's method requires careful adjustment ($\pm 10\%$ or so), however the generalized eigenproblem approach of this chapter truncates $N$ automatically. The method has been reduced to a simpler algorithm which is entirely expressible on the boundary, and therefore scales with the area of the boundary. By reducing to an order-$N$ symmetric eigenproblem, the choice $M \gg N$ can be made without the disadvantage of slowing down any determinant (SVD) evaluations. This enables very accurate boundary sampling without slowing down the $O(N^3)$ matrix operations. $N$ and $M$ are no longer parameters whose choice has a sensitive effect on the success of the method; they simply need to be sufficiently large. The accurate norm matrix approaches of Section 5.4 gives clean `switching' between states corresponding to tension minima--this is an improvement over Heller's formulation, and would allow such minima to be distiguished up to a higher $k$.

The scaling of the numerical effort required per found eigenstate is not hard to compute. The basis size $N$ is of order the semiclassical basis size ( $N_{sc} = kL/\pi$ in $d=2$, $N_{sc} \sim k^{d-1}{\mathsf{A}}$ in higher $d$). The matrices $F$ and $G$ are dense, requiring $O(N^3)$ effort to diagonalize (5.14) at a single $k$. A few of these are required per state found. Therefore the effort per state found scales like $k^{3(d-1)}$. The accuracy in $k$ found is determined entirely by the basis set limitations: it appears to be an order of magnitude better than the `tension rounding error' $\Delta k(\epsilon_\mu)$.

The method of this chapter breaks down when the maxima of tension reached between $k_\mu$ values becomes about equal to the typical tension minima $\epsilon_\mu$, in which case the fraction of `missing states' becomes large. Even long before this is reached, close eigenvalues become indistiguishable (see Fig. 6.5). Improved basis sets which can reduce $\epsilon_\mu$ at a given $k$ require the use of evanescent waves: the reduction can then be by orders of magnitude in certain billiard shapes (see Fig. 6.6).

Finally, it is clear that there is much information that is being wasted by the whole `tension minima hunt' approach described here and very common elsewhere (including in the vast array of engineering applications of the BIM). Fig. 5.6 shows the tension eigenvalues (inverse generalized $\lambda_n$ of (5.14)). The square root is plotted so that parabolas become straight lines, for clarity. (The finite $\epsilon_\mu$ actually round the bottoms to give hyperbolae). It is clear that the eigenstates associated with each minimum are in fact long-lived as $k$ changes over many level-spacings, and therefore prediction of a large number of eigenstates from a single diagonalization is possible. This is the essence of the method of the next chapter.