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## Tension matrix in a scaling eigenfunction basis

It is important to define a spatial rescaling of a wavefunction, corresponding to dilation (or contraction) about the (arbitrary) origin. If the wavefunction is a Helmholtz solution at wavenumber , then the rescaled version of the same wavefunction obeys the Helmholtz equation at a new wavenumber :

 (6.1)

The rescaled function value at a given point is found by looking at the original function at a point a fractional distance further from the origin. The derivative with respect to of the wavefunction value at a point is
 (6.2)

This means that for with small ( ), this can be expanded in powers of as
 (6.3)

We will not need any higher terms for the basic method (they are given in Appendix I).

We now choose a basis of the exact billiard eigenfunctions scaled as above to have a single wavenumber ; these are called scaling eigenfunctions [195]. The unscaled eigenfunctions are taken to have unit normalization in the billiard. The index labels the eigenfunctions in ascending order of their unscaled wavenumber . The scaling eigenfunctions no longer go to zero on the original billiard boundary , however they do go to zero on rescaled (dilated or contracted) versions of the boundary (see Fig. 6.1). Imagine that a state has a wavenumber shift which is very small ( , the inverse system size). Then will be very close to , and the value of for will be close to zero. The boundary coordinate measures location on . Applying (6.3), the difference from zero is given by

 (6.4)

where is the (unscaled) eigenstate normal derivative at location . Notice that the zeroth-order term vanished because of the BCs. (This expansion is continued in Appendix I). A general sum of scaling eigenfunctions
 (6.5)

is also a scaling wavefunction.

The tension (as defined in the previous chapter for Dirichlet BCs) of a general scaling wavefunction is

 (6.6)

where is a boundary weighting function. If is now written in the form (6.5) then the tension is . The matrix is the scaling eigenfunction representation of the tension quadratic form, and we will see that it has wonderful properties. For we can substitute (6.4) as follows,
 (6.7)

where is an abbreviation for at the location . The final form has been written in terms of the boundary inner-product matrix of the eigenstate normal derivatives,
 (6.8)

where is the weight defining the inner-product. The approximation has been used, because has only very slow dependence on . Note that all the fast dependence of on appears in the wavenumber shifts and .

Next: Quasi-orthogonality on the boundary Up: The basic scaling method Previous: The basic scaling method
Alex Barnett 2001-10-03