Breakdown for a state as $\vert\delta _\mu \vert$ vanishes

As observed by Vergini [194], if $k$ falls close to a true eigenstate $k_\mu$, then the scaling method fails to predict this state accurately. Fig. 6.12 shows this happening: the predicted shift heading towards zero suddenly diverges in a first-order pole, giving an inaccurate $k_\mu$ prediction. The range of $k$ affected corresponds to the tension rounding error $\Delta k(\epsilon_\mu)$ defined in Section 5.5.1. Therefore the problem does not occur very often, and can be almost (but never quite) eliminated by a good choice of basis. When it does occur it only affects the one state involved. It is very easy to detect because the norm of the state produced automatically (Section 6.2.2) grows to be much larger than 1.

The reason for this type of error is that the tension of the scaling eigenfunction $\mu$ has reached the tension minimum allowed by the basis. The diagonal of $\tilde{F}$ behaves like $2\delta_\mu^2$ for a true scaling eigenfunction. However the closest that can be achieved by the basis representation is $2 \delta_\mu^2 + \epsilon_\mu$ (see Section 5.5.1). Substitution of this diagonal term into (6.25) and then (6.26) or (6.28) gives the observed pole in the predicted shift.

It is possible to `cure' the problem completely by replacing $F$ by the true area norm matrix $G$ in (6.25), which has no singularity as $\delta _\mu$ passes through zero. Then the predicted wavenumber shift is $\delta_\mu = -4 \lambda_\mu$. This apparent cure is shown in Fig. 6.12. However, $G$ has no strong quasi-diagonality property when expressed in the scaling eigenfunction basis; the result is a severe loss in the quality of most of the $N/10$ usable states (corresponding to a choice $\beta \approx 0.03$ in Fig. 6.10).