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 from the diagonal
is proportional to the desired band profile
.
I estimate
only at equally-spaced frequency points
, where integer .
The value at each point is found by averaging the squared elements
whose frequency difference
falls within the `bin' from to .
The constant bin width determines the resulting
frequency resolution.
This is seen to be simply smoothing via `histogram binning'.
All such bin averages are multiplied by
, if needed
to give
in the correct units.
Since the distance from the diagonal
is classically small (
), linearization
of the dispersion relation is a good approximation giving
.
In practice, I bin in terms of the wavenumber difference
.
Note that the phrase `distance from the diagonal' would
strictly imply an *integer*
.
The difference between this interpretation () and the continuous version
based
on the corresponding eigenvalues () 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 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 ). However the above can easily be extended to a two-sided version (preserving the sign of ) 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 ,
then the fractional error is Gaussian with standard deviation
(for
).
This assumes that all elements were uncorrelated random variables.
This was found to be a good assumption, *except* at certain
in billiards which have strong scarring.
As discussed in Section 2.3.2, at these
the average is determined by a few
very large values of
, giving
poorer estimation errors at certain places in the band profile.
For the quarter-stadium example at ,
I used states, giving an estimation error
(varying with distance from the diagonal) at the 10% level.