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##

Dynamic range and sweep method breakdown

I define the `dynamic range' as the ratio of the tension fluctuation
to the typical tension minima found over a sweep in .
The tension fluctuation is given roughly by the value reached inbetween
states separated by , the mean level spacing in wavenumber.
Using the above curvature
estimate
, and the Weyl law
,
the fluctuation is
.
An estimate for dynamic range is therefore
where
is a typical value of .
The scaling of
with is currently not known, but
with appropriate basis sets in the 2D stadium it appears that arbitrarily
small
can be created, even in the semiclassical limit.
(For a discussion of the diffractive contribution to
,
see [194]).
However, in billiards where such good basis sets are not known
(and therefore RPWs are used),
it seems empirically that
is independent of .
Therefore in , we expect to decrease like .

The method of this chapter, and sweep methods in general which rely on hunting
for minima in a single quantity, will certainly break down when approaches
1 or less.
By `break down' I mean that since
most minima are no longer distiguishable (or they will be eclipsed by the
parabolas from the states with smallest ), and
the fraction of missing or erroneous states approaches 1.
Encouragingly, we will find that the method of the next chapter can distinguish
close eigenstates which are indistinguishable using a sweep method (with the same
basis set).

** Next:** A more intelligent hunt?
** Up:** The hunt for local
** Previous:** Finding the nearest minimum
Alex Barnett
2001-10-03