Relation to strength estimate of Vergini and Saraceno

I have not yet been able to find a direct wave-mechanical explanation (which would bypass the above semiclassical estimate) for the quasi-orthogonality of dilation. However a clue is given by Berry and Wilkinson's proof [28] that degenerate ($k_\mu = k_\nu$) off-diagonal elements are exactly zero. An explanation for $k_\mu \neq k_\nu$ was first attempted by VS [195,194]. They tried to establish quasi-orthogonality using the identity $V_{\mu\nu} = [(k_\nu^2 - k_\mu^2)/2k^2] B_{\mu\nu}$, with $B_{\mu\nu} \equiv \langle\psi_\mu\vert \, {\mathbf r}\,{\cdot}\nabla \, \vert\psi_\nu\rangle_{\mathcal{D}}$. This identity can be proved in a simple fashion (Eq.(H.25)). However, they then made the assumption $\vert B_{\mu\nu}\vert \sim O(1)$ (for $d=2$) by claiming that eigenstates are uncorrelated across the volume of $\mathcal{D}$ [194]. (In $d=2$ this would give $\sim ({\mathsf{L}}/\lambda_{{\mbox{\tiny B}}})^2$ variables whose variances add linearly). This would give a power law $\gamma = 2$ for the off-diagonal growth of $\vert V_{\mu \nu}\vert^2$.

This is in error for two reasons. Firstly, we know that the number of degrees of freedom in a constant-wavenumber function is actually $N_{sc} \sim ({\mathsf{L}}/\lambda_{{\mbox{\tiny B}}})^{d-1}$ scaling like the boundary (see Section 5.3.1), which would imply the naive random-wave estimate $\vert B_{\mu\nu}\vert \sim O({\mathsf{L}}/\lambda_{{\mbox{\tiny B}}})^{(3-d)/2}$. Secondly, a random wave estimate of an overlap over a large fraction of the region is generally very bad (Section 3.3.2). In fact, comparison to our band profile results (analytical and numerical) shows that $\vert B_{\mu\nu}\vert \sim \vert k_\mu - k_\nu\vert$, a result which cannot be guessed by random wave assumptions. This result has also since been verified by Vergini ^6.1.

The conclusion is that the original authors' quasi-orthogonality estimate (which they have used in [195,194,175,196,203]) was in fact pessimistic: the true power law $\gamma = 4$ actually gives much smaller elements close to the diagonal.

Figure 6.4: Eigenvalues as a function of $k$. Top: tension matrix $F$. Middle: derivative of tension matrix $dF/dk$. Bottom: the generalized eigenvalues of (6.25), plotted as the lowest-order predicted wavenumber shifts $-\delta_\mu = -2/\lambda_\mu$. The small rectangle indicates the enlargement (see Fig. 6.12). In the top two plots, the null-space of the matrix is visible as undesirable interfering eigenvalues climbing slowly upwards from zero. The bottom plot shows the remarkable result that (6.25) is free from such null-space solutions over a wide range. The quarter stadium with a basis of 120 symmetrized real plane waves was used.