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In Fig. 6.4 (bottom) the consistent curvature in the predicted
shifts implies that higherorder effects could be important
in predicting the values.
There are many levels of higherorder correction possible
once the basic diagonalization of (6.25) has been performed:
1) identification of the higherorder diagonal terms in
and for an average state, then correcting all the
according to the same resulting formula,
2) using
for each state on the boundary to
compute a correction for each corresponding ,
3) using
for each state on the boundary and knowledge of
offdiagonal terms in and to compute
corrections to the eigenvectors
as well as the .
I will only report on the first of these ideas, since it is easiest to
routinely implement.
The other two levels of correction are areas for investigation, however
it is not known whether they will prove even worthwhile (there
is a tradeoff betweeen correcting the results of one diagonalization and
simply performing more diagonalizations).
My inkling is that in practice
the corrections presented here will be the only worthwhile ones.
Appendix I shows that the expansion of the diagonal of the
tension matrix is

(6.27) 
where the coefficient is the sum of many complicated boundary integrals
of
,
, etc, and
the local curvature and its derivative.
It is possible in theory to perform a randomwave estimate for these integrals
to reach an expression in terms of the billiard shape.
However in practice it is much easier to fit for by measuring the
average value for
a few states.
For instance, in the quarter stadium of length 2 by height 1,
.
However, it varies from state to state
^{6.3},
taking smaller values for states with higher average transverse momentum on the
boundary, or for states which are scarred so as to avoid regions of high
.
Note that has units of length squared, so that if the billiard changes
size, then will need to be scaled accordingly.
The generalized eigenvalue is
which can be
found from (6.27). Inversion of the series expansion
is needed to get the series for our estimate for , giving

(6.28) 
where corrections of
have been ignored. This is our
improved version of (6.26).
In tests, use of this formula had little effect on the
most accurate states, but the resulting wavenumber shift reduced the
tension of the remainder of the
useful states by a factor of (see Fig. 6.9).
Also, the
term had very little effect (much less
than the
) because
of the factor.
The utility of higherorder expansion is minimal (the next significant, the
sixthorder, has little effect).
Figure 6.8:
Automatic normalization of states returned by the scaling method at
, using (6.29). The error from 1 is shown.
The growth of random norm errors roughly follows a second power law
(shown by the staight line).

Next: `Automatic' normalisation of states
Up: Higherorder correction and normalization
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Alex Barnett
20011003