Semiclassical estimate of off-diagonal strength of $M$

This and the next section can be skipped on first reading. We would like an estimate of the strength of the off-diagonal part of $M$, which we call $V$. Writing the strength of $V$ in terms of the band profile, we have

$$M_{\mu\nu} \ = \ \delta_{\mu\nu} + V_{\mu\nu},$$ (6.10)

$$\langle \vert V_{\mu \nu}\vert^2 \rangle \approx \frac{(2\pi)^d}{\Omega_d} \frac{1}{{\mathsf{V}}} \frac{m}{2\pi\hbar^3 k^{2+d}} \tilde{C}_{{\mbox{\tiny E}}}(\omega)$$ (6.11)

$$\approx \frac{1}{\Omega_d} \frac{1}{{\mathsf{V}}} \left(\frac{2\pi}{k}\right)^{d-1} \tilde{C}(\kappa) ,$$ (6.12)

which followed from (6.9), (2.48) and the Weyl form[35,12] for the mean level spacing

$$\Delta \ = \ \frac{2}{\Omega_d{\mathsf{V}}} \left(\frac{2\pi^2\hbar^2}{m}\right)^{d/2} E^{1-d/2},$$ (6.13)

where $\Omega_d \equiv 2\pi^{d/2}/\Gamma(d/2) = 2\pi,4\pi \cdots$ is the surface area of the unit sphere in $d = 2,3\cdots$ dimensions. The `distance' from the diagonal is expressed either as a frequency $\omega = (E_\mu - E_\nu)/\hbar$ or as the wavenumber difference $\kappa \equiv k_\mu - k_\nu$. The two are related by $\omega = v_{{\mbox{\tiny E}}} \kappa$, where the particle speed is $v_{{\mbox{\tiny E}}}$ at energy $E$. The band profile has been written in a scaled form $\tilde{C}_{{\mbox{\tiny E}}}(\omega)= m^2 v_{{\mbox{\tiny E}}}^3 \tilde{C}(\kappa)$. This scaled band profile $\tilde{C}(\kappa)$ depends only on the billiard geometry (it is independent of $k$), because the system has hard walls. It has universal forms at both large and small `distances' from the diagonal. The crossover happens at typical correlation scales of the system, given by the system size, namely at $\kappa \sim 1/{\mathsf{L}}$. In this intermediate region there is non-universal system-specific structure--e.g. see the band profile in Fig. 3.7 (graph DI). The equivalence of classical and quantum band profiles was demonstrated in Section 2.3.

For $\kappa \gg 1/{\mathsf{L}}$ we have $\tilde{C}_{{\mbox{\tiny E}}}(\omega)\approx \nu_{{\mbox{\tiny E}}}^{{\mbox{\tiny WNA}}}$ (see Section 3.2), which upon substitution into (6.11) gives

$$\langle \vert V_{\mu \nu}\vert^2 \rangle \ \approx \ \frac... ...,r_n^2 \hspace{0.5in} \mbox{for $\kappa \gg 1/{\mathsf{L}}$.}$$ (6.14)

This corresponds to a random-wave assumption for the eigenstates (Section 3.2.1), giving no dependence on $\kappa$ (flat band profile). Since we have $r_n \sim {\mathsf{L}}$, this gives roughly $\langle \vert V_{\mu \nu}\vert^2 \rangle \sim (\lambda_{{\mbox{\tiny B}}}/{\mathsf{L}})^{d-1}$, the inverse number of wavelength-sized patches on the boundary.

Figure 6.3: Strength of off-diagonal elements $V$ of the quasi-orthogonal matrix $M$, for the $d=2$ quarter stadium billiard. The vertical axis $k^{d-1}\langle \vert V_{\mu\nu}\vert^2 \rangle$ is a quantity dependent only on the billiard system (not on $k$). The horizontal axis is the wavenumber difference $\kappa \equiv k_\mu - k_\nu$. The solid line is the semiclassical estimate (taken from the classical band profile). The estimation error changes at $\kappa = 0.01$, allowing a dynamic range of 16 orders of magnitude. The quantum band profile (taken from the matrix of Fig. 6.2) is shown as dots, and it becomes unreliable at the level $10^{-8}$, for this choice of $k$. The dashed line is the fit to (6.15); the dotted is the random wave result (6.14). The two lines meet at a $\kappa$ around the inverse system size $1/{\mathsf{L}}\sim 1$.

Quasi-orthogonality (and the resulting success of the VS method) is determined by very small elements close to the diagonal. Here we have

$$\langle \vert V_{\mu \nu}\vert^2 \rangle \ \approx \ \frac{... ...\gamma \hspace{0.5in} \mbox{for $\kappa \ll 1/{\mathsf{L}}$},$$ (6.15)

where we seek the prefactor $a$, which will depend only on the billiard shape. The prefactor is proportional to the fluctuations intensity $\nu_{{\mbox{\tiny E}}}^{(\mathcal{G})} = \tilde{C}_{\mathcal{G}}(\omega\rightarrow0)$ for the signal ${\mathcal{G}}(t) = -\mbox{\small$\frac{1}{2}$}m {\mathbf r}^2(t)$ (see Section 3.3.1). This signal has a variance of $\sigma_{\mathcal{G}}^2 \equiv {\textstyle\frac{1}{4}} m^2 (\langle r^4 \rangle - \langle r^2 \rangle^2)$ about its mean. A rough estimate is $\sigma_{\mathcal{G}}^2 \sim m^2{\mathsf{L}}^4$. The intensity is the product of the variance and the correlation time: $\nu_{{\mbox{\tiny E}}}^{(\mathcal{G})} \sim m^2{\mathsf{L}}^4\tau_{{\mbox{\tiny bl}}}$. Therefore $\tilde{C}_{{\mbox{\tiny E}}}(\omega)= \omega^4 \tilde{C}_{\mathcal{G}}(\omega) \sim (\kappa {\mathsf{L}})^4 \nu_{{\mbox{\tiny E}}}^{{\mbox{\tiny WNA}}}$. Thus the prefactor is $a \sim {\mathsf{L}}^{5-d}$. The accuracy is limited because we lack knowledge of the correlation time of the signal ${\mathcal{G}}(t)$. In the 2D quarter stadium billiard (of size ${\mathsf{L}}\sim 1$, see Fig. 2.6) the prefactor is measured to be $a \approx 1.2$, remarkably close to unity (see Fig. 6.3). It seems that convergence to the fourth-power law is very slow in this system, not becoming accurate until $\kappa < 0.01$. This is probably due to long tails in the time correlations (Section 3.4) which makes $\tilde{C}_{\mathcal{G}}(\omega)$ close to singular at $\omega=0$. More study is needed of this convergence to the asymptotic form. In systems without these non-generic effects, we expect more rapid convergence to the fourth-power law (for instance Fig. 3.6), and therefore better quasi-orthogonality.

Both the above estimates are for $\vert V_{\mu \nu}\vert^2$ averaged over an energy range as in Section 2.2.2, with $\kappa$ held fixed. The Random Matrix Theory assumption is that the elements $V_{\mu \nu}$ are independent normally-distributed random quantities. However, deviations from strong chaos (due to periodic orbits and scars; see Section 2.3.3) will affect the distribution and correlation of the elements.

Thus we have estimated a wave overlap quantity semiclassically. In both the above expressions all the dependence on `particle quantities' ($\hbar$, $m$, $v_{{\mbox{\tiny E}}}$) has cancelled, which is necessary since the particle picture is merely a construction allowing semiclassical estimates. (For instance in an acoustic Helmholtz application the particle model is completely fictitious). Rather, everything is in terms of lengths, as expected since the matrix elements of $V$ arise purely from wave considerations.