The quantum band profile case

The understanding of the special deformation class came from classical arguments. We might wonder, is this special property is preserved for the quantum band profile? The answer is yes: for instance the graphs labelled DI in Fig. 3.7 show that the special nature of dilation is preserved to the accuracy of the quantum calculation. (The corresponding $(\partial {\mathcal{H}} / \partial x)_{nm}$ matrix is shown in Fig. 6.2). Here I used the quarter stadium billiard (Fig. 2.6) for the simple reason that efficient quantization methods exist for this shape (Chapter 6), but, as yet, do not exist for the generalized Sinai billiard (Fig. 3.2). Agreement with the classical result, and hence the power law, is maintained down to the point at which errors in the quantum calculation become dominant--this is visible as bottoming-out in the leftmost point of the log-log inset plot, at $\sim 10^{-10}$. Therefore the QCC demonstrated in Section 2.3 also holds for special deformations. This will have profound consequences for the numerical method presented in Chapter 6.

Is there a simpler direct route to the special power-law band profile behavior involving quantum-mechanical (wave) considerations alone? Koonin et al. [118] have derived the vanishing of the quantum $\nu_{{\mbox{\tiny E}}}$ for translations and rotations. Berry and Wilkinson [28] have shown that off-diagonal matrix elements vanish for translations, rotations and dilations, in the case of exact degeneracy (i.e. $\omega_{nm} = 0$). However, neither of these results addresses the finite $\omega$ dependence. It is clear that direct application of a random wave assumption, which leads to (3.12), completely fails to predict the band profile for special deformations (agreement is only reached when $\omega \gg \tau_{{\mbox{\tiny bl}}}^{-1}$). More generally, it can be said that the random wave approximation fails whenever $D{({\mathbf s})}$ is significant on a large fraction of the boundary.

It might be that there exists some transformation from the boundary overlap form Eq.(C.2) to another overlap integral which can be estimated well by a random wave approximation. For instance, (H.23) gives $(\partial {\mathcal{H}} / \partial x)_{nm}$ for translation in terms of the dipole matrix element (a weighted overlap of eigenstates in the billiard interior). A random wave argument for this overlap would predict the correct power law $\gamma = 4$ for the band profile. A similar effort has been made in the dilation case [195], however, as discussed in Section 6.1.2 this leads to the wrong power law. Generally the use of random waves for such overlap estimates is dangerous. No relation has been found relating to rotations. The prediction of special deformation band profiles using wave manipulations alone (e.g. of the type in Appendix H) is an area for research.