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`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
,
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 : this is given by (6.27).
Therefore by taking the square root we obtain the
amplitude correction factor,

(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 , the
normalization is correct to 1% for the useful states, and
correct to 0.01% for the 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 (2norm of error from obeying the boundary conditions)
with
of eigenstates returned from a single scaling diagonalization at
.
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 is due to limitations of the basis set.

Figure 6.10:
Exploration of the effect of weakening the quasidiagonality of
.
The dots show the same as Fig. 6.9.
The other sets of points show the tension errors for
, with the
different choices of labelled in the upper left corner.
This corresponds to
being the dilation deformation with some
`constant' (CO) deformation mixed in (see Table 3.1).
As increases a new type of error emerges
with a different powerlaw and much greater random statetostate fluctuations.
Examining the boundary errors shows that they are dominated by the
effect of the one or two states with smallest .

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

Next: Sources of error in
Up: Higherorder correction and normalization
Previous: Correction of the eigenwavenumbers
Alex Barnett
20011003