Smoothing

As is clear from Fig. 2.7, individual matrix elements are random quantities. Recalling the definition (2.48), the average square matrix element a certain distance $\omega$ from the diagonal is proportional to the desired band profile $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$. I estimate $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ only at equally-spaced frequency points $\omega_j = (j + \mbox{\small$\frac{1}{2}$}\omega_s )$, where integer $j \ge 0$. The value at each point $j$ is found by averaging the squared elements whose frequency difference $\vert\omega_n - \omega_m\vert$ falls within the `bin' from $j\omega_s$ to $(j+1)\omega_s$. The constant bin width $\omega_s$ determines the resulting frequency resolution. This is seen to be simply smoothing via `histogram binning'. All such bin averages are multiplied by $2\pi \hbar/\Delta$, if needed to give $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ in the correct units. Since the distance from the diagonal is classically small ( $\hbar \vert\omega_n - \omega_m\vert \ll E$), linearization of the dispersion relation is a good approximation giving $k_n - k_m = (\omega_n - \omega_m)/v$. In practice, I bin in terms of the wavenumber difference $\kappa \equiv k_n - k_m$. Note that the phrase `distance from the diagonal' would strictly imply an integer $r \equiv n-m$. The difference between this interpretation ($r$) and the continuous version based on the corresponding eigenvalues ($\kappa$) is small, having jitter on the scale of a single level-spacing. This small size of jitter is due to spectral rigidity[24]. The choice of continuous over integer variable is therefore arbitrary if only features larger than the level spacing are desired. The only time my choice is important is in Section 2.3.4.

The diagonal elements $n=m$ are not treated as part of the band profile, and are removed before binning. Since the band profile is symmetric (the matrix is Hermitian), a `single-sided' band profile was taken (discarding the sign of $\kappa$). However the above can easily be extended to a two-sided version (preserving the sign of $\kappa$) if the band profile of a non-Hermitian, possibly rectangular, matrix is desired.

Statistical errors result from any estimation of average value. If the number of matrix elements collected in a bin is $\mathcal{N}_j$, then the fractional error is Gaussian with standard deviation ${\mathcal{N}}_j^{-1/2}$ (for $\mathcal{N}_j \gg 1$). This assumes that all elements were uncorrelated random variables. This was found to be a good assumption, except at certain $\kappa$ in billiards which have strong scarring. As discussed in Section 2.3.2, at these $\kappa$ the average is determined by a few very large values of $\vert(\partial {\mathcal{H}} / \partial x)_{nm}\vert^2$, giving poorer estimation errors at certain places in the band profile. For the quarter-stadium example at $k \approx 400$, I used $N=451$ states, giving an estimation error (varying with distance from the diagonal) at the 10% level.