Dynamic range and sweep method breakdown

I define the `dynamic range' $r_d$ as the ratio of the tension fluctuation to the typical tension minima found over a sweep in $k$. The tension fluctuation is given roughly by the value reached inbetween states separated by $\Delta_k$, the mean level spacing in wavenumber. Using the above curvature estimate $c_\mu \sim {\mathsf{L}}$, and the Weyl law $\Delta_k \sim k^{1-d}/{\mathsf{V}}$, the fluctuation is $\sim k^{2-2d} {\mathsf{L}}^{1-2d}$. An estimate for dynamic range is therefore $r_d \sim (\epsilon_{{\mbox{\tiny typ}}} k^{2d-2} {\mathsf{L}}^{2d-1})^{-1}$ where $\epsilon_{{\mbox{\tiny typ}}}$ is a typical value of $\epsilon_\mu$. The scaling of $\epsilon_{{\mbox{\tiny typ}}}$ with $k$ is currently not known, but with appropriate basis sets in the 2D stadium it appears that arbitrarily small $\epsilon_{{\mbox{\tiny typ}}}$ can be created, even in the semiclassical limit. (For a discussion of the diffractive contribution to $\epsilon_{{\mbox{\tiny typ}}}$, see [194]). However, in billiards where such good basis sets are not known (and therefore RPWs are used), it seems empirically that $\epsilon_{{\mbox{\tiny typ}}}$ is independent of $k$. Therefore in $d=2$, we expect $r_d$ to decrease like $k^{-2}$.

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 $r_d$ 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 $\epsilon_\mu$), 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).