Effects below the quantum level spacing

Firstly, re-examining Fig. 2.3, it should be noted that if the smoothing energy range $M\Delta$ is sufficiently large, then a smooth $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ results even if the off-diagonal smoothing range $\hbar \varepsilon$ is smaller than the level spacing $\Delta$. In this case, there is meaning to structure in $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ below the level spacing. However, the level spacing can no longer be treated as uniform, as was done in taking the continuum limit of the sum of $\delta$-functions to reach (2.48). In other words, $E_m$ can no longer be taken to be $m\Delta$: the exact energy levels and their spacings are now needed. In Fig. 2.3 this can be visualized by replacing the regular energy grid $(n\Delta,m\Delta)$ with the irregular grid $(E_n,E_m)$.

This point manifests itself most clearly at distances $\Delta$ or less from the diagonal. The quantum band profile $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ is defined by a $\delta$-function `comb' $\sum_n \vert{\mathcal{F}}_{nm}\vert^2 2\pi\delta(\omega_{nm} - \omega)$ averaged over $m$ (see Section 2.2.2). Imagine the case where $\vert{\mathcal{F}}_{nm}\vert^2$ has constant average. The band profile will drop to zero as $\omega\rightarrow0$ because there is level repulsion: the likelihood of a spacing $s \equiv \hbar\omega_{nm}$ drops to zero like a power law $s^\beta$ for $s\ll \Delta$. The power $\beta = 1,2,4$ is determined by the symmetry class of the chaotic system [35]. More precisely, one can say that $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$ would be given by the 2-level correlation function $R_2(E)$ [35] at $E=\hbar\omega$. If the mean square matrix element is now given by a continuous function $\sigma^2(\omega)$ which varies with distance $\hbar \omega$ from the diagonal, one expects the band profile to be similarly modified,

$$\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)\; ... ...rac{2\pi\hbar}{\Delta} R_2(\hbar\omega) \, \sigma^2(\omega) .$$ (2.59)

An important point is that it is $\sigma^2(\omega)$ (rather than $\tilde{C}^{{\mbox{\tiny qm}}}_{{\mbox{\tiny E}}}(\omega)$) which has correspondence with the classical $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$. This is verified numerically in Section 3.3.2. So in effect, the quantum band profile is reached by `punching a hole' of width $\sim \Delta$ at the origin in the classical band profile. Further than $\Delta$ from the diagonal, $R_2\rightarrow1$ so the two become equal. Level-spacing effects on the quantum band profile are also discussed in [11].

For most of my work, the smoothing width $\hbar \varepsilon$ is larger than $\Delta$, so the above will not have an effect.