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# Tension matrix expansion

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 off-diagonal 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 quasi-diagonal. The diagonal is , and we have from the considerations of Section 6.1.2 small off-diagonal 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 quasi-orthogonal property, so can be estimated using random waves. The estimate gives for this integral, and shows that this term dominates over any off-diagonal contribution from the first term (involving ). The factor of in this term means that there is a weak form of quasi-diagonality at this order. Importantly, for the off-diagonal error renders the error insignificant. Hence we expect the quasi-diagonality 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 random-wave estimates. A random-wave estimate of the integrals then gives on the diagonal and off-diagonal. Note that the off-diagonal has no quasi-diagonality property at this or higher orders. For higher orders for even, we expect on the diagonal and off-diagonal. 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 off-diagonal 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 quasi-diagonality, 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 off-diagonal strength of ) is negligible. It is clear that more research is needed on the properties of the higher-order terms, especially if an explanation of the tension error power-law growth (Section 6.3.1) is sought.

Next: Useful geometric boundary integrals Up: Appendix I: Scaling expansion Previous: Applying boundary conditions and
Alex Barnett 2001-10-03