Appendix I: Scaling expansion of eigenfunctions and tension matrix

Here I derive the expansion of the value of a scaling eigenfunction on the unscaled boundary $\Gamma$, in powers of the wavenumber shift $\delta$. Section 6.1.1 is a necessary prerequisite. The expansion will be in terms of normal and tangential derivatives of the unscaled eigenfunction on $\Gamma$. Dirichlet boundary conditions will be assumed for this eigenfunction. From this will follow the expansion of the tension on $\Gamma$, in the scaling eigenfunction basis. This is a vital part of the understanding of the scaling method of Vergini and Saraceno (VS).

It is my belief that although the lowest-order term found by those authors [195,194] is correct (giving the basic explanation of the scaling method), the higher-order terms are not, because the curvature (metric) of the boundary was not taken into account. My aim here is to correct this oversight. This will affect the understanding of the growth of errors in the method. I will stick to $d=2$, although the generalization to higher dimension is believed to be simply a matter of introducing vector notation for the tangential coordinate.