Quantum structure beyond the band profile

For the purposes of establishing QCC for the heating rate, demonstration of band profile agreement is sufficient. In this section I discuss further structure in the matrix $(\partial {\mathcal{H}} / \partial x)_{nm}$, which is irrelevant for heating rate, but is interesting in itself.

The classical band profile $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ is a smooth function of $E$, changing only on classically-significant scales ($\hbar^0$). To claim this, I must assume that there are no discontinuities in the phase-space energy-shell as a function of $E$; such pathologies would be a new area of study.

This constrasts the quantum case, where there is much $E$-dependent structure before smearing is performed: Figs. 2.7 and 2.8 show images of $\vert(\partial {\mathcal{H}} / \partial x)_{nm}\vert^2$ for parametric changes (deformations) of the billiard of Fig. 2.6. Features to notice are,

1) The overall `banded' shape--this is what the smeared $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ measures.

2) Generally matrix elements are random and uncorrelated. It is the local mean that follows the banded shape.

3) There is a slight wobble (fluctuation in band profile) along the diagonal direction, best viewed by looking down the diagonal.

4) Some matrix elements (or blocks of elements) are anomalously large (especially see Fig. 2.8), and these large elements form periodic structures along the diagonal direction. When the corresponding eigenfunctions are examined, they are often found to be scarred states [90].

5) Whenever there is such a large element, the remaining row and column are very small, giving a vertical and horizontal streaking effect. This is an example of anti-scarring [111].

I now discuss these effects further. Wilkinson [199] has extended the sum rule for the band profile to include periodic orbit effects which manifest themselves once the smoothing width $M\Delta$ is less than $\hbar / \tau_1$, where $\tau_1$ is the period of the shortest periodic orbit. The contribution from a single orbit of period $\tau$ is structure in the matrix at a `distance' from the diagonal (in energy units) of $2\pi\hbar/\tau$, and its integer multiples. This structure is periodic in $E$ (i.e. along the diagonal direction), with period given by the action integral of the orbit. Therefore $E$-dependent structure up to energy scales $O(\hbar)$ exists, on top of the $O(\hbar^0)$ smooth classical changes. This oscillating correction to the off-diagonal mean value is analogous to Berry's periodic orbit correction to the mean level density [25]. The $E$-oscillatory peaks (effects 3 and 4 above) can be seen as an example of these oscillations. The clear period of $\pi$ in the figures corresponds to the shortest periodic orbit (the `bouncing ball' orbit of length $\tau_1 = 2$).

Scarring [90,91,111], a result of the classical dynamics, occurs with much more strength in the stadium than in a generic `hard chaos' system, because of the marginally-stable bouncing ball orbit family (such non-generic effects are discussed in Section 3.4). However, it is always present to some degree even in hard chaos systems. Its effects (4 and 5 above), that of introducing matrix elements which are anomalously large compared to a Random Matrix Theory prediction, can be viewed in another way. When an integrable system is perturbed (to give a KAM system), the corresponding off-diagonal matrix elements are sparse [164]. Scarred states are due to the closest-to-stable classical structures (in this stadium example the main scars are due to marginally stable orbits [91]), so one might expect more sparse rows and columns at these states. Fig. 2.7 is for a deformation which is zero where the bouncing ball orbits hit the wall, and it shows little sparsity. In contrast, Fig. 2.8 deforms this part of the wall, and has much more sparsity and structure.

Why is there `anti-scarring' (effect 5 above)? To answer this I use a simple formula for the sum of any row or column of the squared matrix elements,

$$\sum_m \left\vert \left( \frac{\partial \mathcal{H}}{\parti... ...{\partial x}{({\mathbf r})}\right]^2 \psi_n^2{({\mathbf r})}.$$ (2.58)

I will be interested in how this quantity varies with $n$. Clearly if $\frac{\partial \mathcal{H}}{\partial x}{({\mathbf r})}^2 = 1$ over all space, then the required sum is unity, independent of $n$. However, if the positive weighting function $\frac{\partial \mathcal{H}}{\partial x}{({\mathbf r})}^2$ covers any classically-large region of position space then the required sum will also vary relatively little with $n$, because of ergodicity of almost all quantum states [91]. The result is a sum rule (note that it does include the diagonal). Unfortunately, the sum diverges in the case of hard-walled systems, because the matrix bandwidth is then infinite [46], but the idea is clear: if one matrix element has a anomalously large value then its row and column must be correspondingly small. The states that are likely to disobey this sum rule are the strongly scarred ones themselves, in the case that the choice of $\frac{\partial \mathcal{H}}{\partial x}{({\mathbf r})}$ is sensitive to certain orbits and not others. Further study is needed in this area, in particular quantitative comparison to Wilkinson's predictions.