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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,
![\begin{displaymath}
{\mathbf x}_\mu \ \longleftarrow \ \left(\tilde{F}_{\mu\mu}\right)^{1/2}
{\mathbf x}_\mu.
\end{displaymath}](img1037.gif) |
(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 (2-norm 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.
![\begin{figure}\centerline{\epsfig{figure=fig_vergini/err.eps,width=0.7\hsize}}\end{figure}](img1039.gif) |
Figure 6.10:
Exploration of the effect of weakening the quasi-diagonality 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 power-law and much greater random state-to-state fluctuations.
Examining the
boundary errors shows that they are dominated by the
effect of the one or two states with smallest
.
![\begin{figure}\centerline{\epsfig{figure=fig_vergini/mix.eps,width=0.7\hsize}}\end{figure}](img1042.gif) |
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
basis-related errors.
The billiard is the 2D quarter stadium at
.
The coordinate
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
corresponds to the `kink' in the
stadium boundary.
![\begin{figure}\centerline{\epsfig{figure=fig_vergini/ngrper500.eps,width=\hsize}}\end{figure}](img1044.gif) |
Next: Sources of error in
Up: Higher-order correction and normalization
Previous: Correction of the eigenwavenumbers
Alex Barnett
2001-10-03