Next: Useful geometric boundary integrals
Up: Appendix I: Scaling expansion
Previous: Applying boundary conditions and
The tension matrix (using the `dilation' weighting function )
in the scaling eigenfunction basis has elements

(I.19) 
where the wavenumber shifts for the two states involved are
and
.
As usual I define the wavenumber difference
.
Substitution of (I.18) gives a power series in the 's,
which we will study and make estimates for
unknown boundary integrals to grasp the general structure.
We will assume for simplicity that the billiard size
, and
remain in .
We will ignore differences in and from when appropriate,
and look only for the dominant offdiagonal terms.
The term in comes
from the lowest order in (I.18),

(I.20) 
which we can write as
(see Section 6.1.1),
where is quasidiagonal.
The diagonal is
, and
we have from the considerations of Section 6.1.2
small offdiagonal elements of size
,
for .
The term in is, after some rearrangement,

(I.21) 
where an integral of
was performed by parts.
This required use of
,
a different expression than found by VS.
It can be proved easily using
and
, which follow from (I.8)
and (I.9).
Remarkably, the dependence then cancels out, giving the same
diagonal term as that of VS.
We believe that the integral in (I.21) does not have any
quasiorthogonal property, so can be estimated using random waves.
The estimate gives for this integral, and shows that this term
dominates over any offdiagonal contribution from the first term (involving ).
The factor of in this term means that there is a weak form
of quasidiagonality at this order.
Importantly, for the offdiagonal error
renders the error insignificant.
Hence we expect the quasidiagonality property of to play no role in errors
in the scaling method.
The and higher terms in become very messy.
I believe that the
dominant terms, both on and off the diagonal, are

(I.22) 
This can be seen by comparing powers of and using randomwave estimates.
A randomwave estimate of the integrals then gives
on the diagonal and
offdiagonal.
Note that the offdiagonal has no quasidiagonality property at this or higher
orders.
For higher orders for even, we expect
on the diagonal and
offdiagonal.
For odd, the leading diagonal
terms are down by a factor of which renders them
insignificant.
To summarize, the diagonal of the tension matrix has the form given in
Eq.(6.27), and for
the
dominant offdiagonal terms are

(I.23) 
Here contributions from orders 3 and 4 were included because it may be
that the 4th order (the lowest order with no quasidiagonality,
i.e. no powers of ) contributes most to errors in the scaling method.
It is important to note that the 2nd order term (due to offdiagonal strength
of ) is negligible.
It is clear that more research is needed on the properties of the higherorder
terms, especially if an explanation of the tension error powerlaw
growth (Section 6.3.1) is sought.
Next: Useful geometric boundary integrals
Up: Appendix I: Scaling expansion
Previous: Applying boundary conditions and
Alex Barnett
20011003