Deterioration of the eigenstates with $\vert\delta _\mu \vert$

For a given state $\mu$ returned by the generalized eigenproblem, there will be an optimal adjustment to its $k_\mu$ that can be made to reduce the tension to a minimum. (Generally this will be close to that returned by the state-independent higher-order correction formula (6.28)). This adjustment has the effect of rescaling the eigenfunction so that its node best approximates the Dirichlet boundary $\Gamma$. However the tension minimum $t_\mu$ thus achievable is much higher than the intrinsic basis limitation $\epsilon_\mu$, for most of the $N/10$ useful states, so is the dominant error in practical applications. Therefore it would be good to understand its origin.

In Fig. 6.9 I show that the growth of tension error is $t_\mu \approx A \delta_\mu^6$ with high accuracy. There is relatively little state-to-state fluctuation, that is, $t$ is not simply a random quantity with growing mean (as would be expected if it were the square of a zero-mean gaussian variable whose variance is given). In the same system at lower $k$ I show the normal gradient, and wavefunction error on the boundary for a set of states, in Fig. 6.11. It is interesting to note that the error arises roughly in proportion to $r_n$ (or some power) on the boundary. Also, the spatial frequency of the error at surprisingly low at large $r_n$. The constant $A$ seems relatively independent of $k$ (it certainly does not change with any higher power than $k^{1/2}$).

This sixth power law is not affected by changing the basis, or by whether $k$ is near or far from the closest $k_\mu$. The cause, as one would expect, seems to be the non-zero off-diagonal elements of $\tilde{F}$ and $d\tilde{F}/dk$, which causes the eigenvectors $\tilde{{\mathbf x}}_\mu$ (in the scaling eigenfunction representation) to be rotated slightly away from the unit vectors. At first sight it might be thought that the quasi-diagonality of $M$ is to blame, because this appears with the lowest power of $\delta$, namely $\delta^2$. However, Appendix I shows that in fact the 3rd and 4th order terms in $\delta$ contain much more off-diagonal strength for $\delta \ll {\mathsf{L}}^{-1}$. This is evidenced by Fig. 6.10, which shows that by increasing the off-diagonal strength of $M$ by controlled amounts, a new type of error results. This shows that the off-diagonal strength of $M$ in the standard case of $w{({\mathbf s})}= 1/r_n$ does not seem to cause the sixth power law error growth.

I have attempted to model the effect of the 3rd- and 4th-order in $\delta$ off-diagonal terms by applying them in first-order perturbation theory to the generalized eigenproblem (6.25) (in the scaling eigenfunction representation). However, it seems that the perturbing effect of scaling eigenfunctions with distant wavenumbers is divergent, and grows like $\delta_\mu^2$ not the observed $\delta_\mu^6$. The role of these distant states is dubious because they fall outside the wavenumber range $O({\mathsf{L}}^{-1})$ encompassing $N_{sc}$ adjacent states. It is unknown whether near or distant states are the cause of this error. Clearly this is an area for more study.

Figure 6.12: Breakdown of the prediction of wavenumber shift $\delta_\mu = -2/\lambda_\mu$ within $\Delta k(\epsilon_\mu)$ of a shift of zero. The crosses are with the generalized problem of (6.25), and the smaller dots using the modified version with $F$ replaced by $G$ as described in the text.