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Semiclassical estimate of offdiagonal strength of
This and the next section can be skipped on first reading.
We would like an estimate of the strength of the offdiagonal part of ,
which we call .
Writing the strength of in terms of the band profile, we have
which followed from (6.9), (2.48)
and the Weyl form[35,12]
for the mean level spacing

(6.13) 
where
is the surface area of the unit sphere in dimensions.
The `distance' from the diagonal is expressed either as a frequency
or as the wavenumber difference
. The two are related by
,
where the particle speed is
at energy .
The band profile has been written
in a scaled form
.
This scaled band profile
depends only on the billiard
geometry (it is independent of ), 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
.
In this intermediate region there is nonuniversal systemspecific
structuree.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
we have
(see
Section 3.2), which upon substitution into (6.11) gives

(6.14) 
This corresponds to a randomwave assumption for the eigenstates
(Section 3.2.1), giving no dependence on (flat band
profile).
Since we have
, this gives roughly
, the inverse number
of wavelengthsized patches on the boundary.
Figure 6.3:
Strength of offdiagonal elements of the quasiorthogonal matrix ,
for the quarter stadium billiard.
The vertical axis
is a quantity
dependent only on the billiard system (not on ).
The horizontal axis is the wavenumber difference
.
The solid line is the semiclassical estimate (taken from the classical
band profile). The estimation error changes at , 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 ,
for this choice of .
The dashed line is the fit to (6.15); the dotted is the random wave
result (6.14). The two lines meet at a
around the inverse system size
.

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

(6.15) 
where we seek the prefactor , which will depend only on the billiard shape.
The prefactor is proportional to the fluctuations intensity
for the signal
(see Section 3.3.1).
This signal has a variance of
about its mean.
A rough estimate is
.
The intensity is the product of the variance and the correlation time:
.
Therefore
.
Thus the prefactor is
.
The accuracy is limited because we lack knowledge of the correlation time of
the signal
.
In the 2D quarter stadium billiard (of size
, see Fig. 2.6)
the prefactor is measured to be ,
remarkably close to unity (see Fig. 6.3).
It seems that convergence to the fourthpower law is very slow in this
system, not becoming accurate until .
This is probably due to long tails in the time correlations
(Section 3.4) which makes
close to singular at .
More study is needed of this convergence to the asymptotic form.
In systems without these nongeneric effects, we expect more rapid convergence
to the fourthpower law (for instance Fig. 3.6),
and therefore better quasiorthogonality.
Both the above estimates are for
averaged over
an energy range as in Section 2.2.2, with held fixed.
The Random Matrix Theory assumption is that the elements are
independent normallydistributed 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' (, ,
)
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 arise purely from wave considerations.
Next: Relation to strength estimate
Up: Quasiorthogonality on the boundary
Previous: Quasiorthogonality on the boundary
Alex Barnett
20011003