Firstly, re-examining Fig. 2.3, it should be noted that if the smoothing energy range is sufficiently large, then a smooth results even if the off-diagonal smoothing range is smaller than the level spacing . In this case, there is meaning to structure in 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 -functions to reach (2.48). In other words, can no longer be taken to be : the exact energy levels and their spacings are now needed. In Fig. 2.3 this can be visualized by replacing the regular energy grid with the irregular grid .
This point
manifests itself most clearly at distances or less from the
diagonal.
The quantum band profile
is defined by a -function `comb'
averaged over (see Section 2.2.2).
Imagine the case where
has constant average.
The band profile will drop to zero as
because
there is level repulsion: the likelihood of a spacing
drops to zero like a power law for
.
The power is
determined by the symmetry class of the chaotic system [35].
More precisely, one can say that
would be given by the 2-level
correlation function [35] at .
If the mean square matrix element
is now given by a continuous
function
which varies with distance
from the diagonal,
one expects the band profile to be similarly modified,
For most of my work, the smoothing width is larger than , so the above will not have an effect.