`Automatic' normalisation of states

A feature of the scaling method, as VS realised, is that the eigenstates found are already normalized in a known fashion. (This contrasts the BIM, Heller's PWDM for which explicit normalization is required). The numerical diagonalization of (6.25) usually returns eigenvectors normalized so ${\mathbf x}^{{\mbox{\tiny T}}} F {\mathbf x} = 1$, corresponding to scaling eigenfunctions with unit tension. However, we already know the tension of such functions when they are normalized to unity in the domain $\mathcal{D}$: this is given by (6.27). Therefore by taking the square root we obtain the amplitude correction factor,

$${\mathbf x}_\mu \ \longleftarrow \ \left(\tilde{F}_{\mu\mu}\right)^{1/2} {\mathbf x}_\mu.$$ (6.29)

Of course, with (H.9) and the method of Appendix G we are armed with a rapid tool for checking the normalization of states. This is done in Fig. 6.8, showing that for high $k$, the normalization is correct to 1% for the $N/10$ useful states, and correct to 0.01% for the $N/50$ highest accuracy ones. The growth of norm errors is random from state to state, and follows a second power law. In practice, since the boundary derivatives of the eigenstates are needed anyway, a final normalization using (H.9) was performed. If the norm deviates much from 1, it is a very useful indicator that something is wrong (e.g. a spurious state has been found). This is probably the single most important use of automatic normalisation. Such errors are now analysed in the next section.

Figure 6.9: Growth of tension (2-norm of error from obeying the boundary conditions) with $\delta _\mu$ of eigenstates returned from a single scaling diagonalization at $k\approx 10^3$. The basic method using (6.26) (crosses) and corrected wavenumbers (6.28) (dots) are shown. Both show a sixth power law (straight line), with small deviations from state to state. Truncation at $t\sim10^{-11}$ is due to limitations of the basis set.

Figure 6.10: Exploration of the effect of weakening the quasi-diagonality of $M_{\mu\nu}$. The dots show the same as Fig. 6.9. The other sets of points show the tension errors for $w{({\mathbf s})}= (1-\beta) r_n^{-1} + \beta r_n^{-2}$, with the different choices of $\beta$ labelled in the upper left corner. This corresponds to $D{({\mathbf s})}$ being the dilation deformation with some `constant' (CO) deformation mixed in (see Table 3.1). As $\beta$ increases a new type of error emerges with a different power-law and much greater random state-to-state fluctuations. Examining the $\psi$ boundary errors shows that they are dominated by the effect of the one or two states with smallest $\vert\delta _\mu \vert$.

Figure 6.11: Normal gradients $\partial_n \psi_\mu$ (upper) and values $\psi_\mu$ (lower) as a function of the boundary coordinate $s$. The plots have been displaced vertically by state (with increasing $k$). The states correspond to those shown in Fig. 6.5, except that a better RPW and EPW basis set has been used to reduce basis-related errors. The billiard is the 2D quarter stadium at $k=500$. The coordinate $s$ is measured from the upper-left to the lower-right `corner' of the quarter stadium, on the outer edge (which is also present in the full stadium). The small `blip' at $s=1$ corresponds to the `kink' in the stadium boundary.