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 ). 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).