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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).
Figure:
Scaling eigenfunction
vanishes on a rescaled boundary
.
It is clear that when is much closer than a wavelength from ,
the wavefunction can be linearized,
giving a value on proportional to the eigenfunction normal derivative.
Also the boundary coordinate and local surface normal
are shown.

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 zerothorder 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,
where is an abbreviation for
at the location .
The final form has been written in terms of the boundary
innerproduct matrix of the
eigenstate normal derivatives,

(6.8) 
where
is the weight defining the innerproduct.
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 .
Figure:
Image of the matrix
shown as a density plot
of
,
for the case of dilation (a special deformation) of the 2D quarter
stadium billiard.
Darker pixels correspond to larger values.
The wide central white region (of width
)
corresponds to quasiorthogonality: the quantum
band profile vanishes like as
,
where
.
The diagonal is unity, and is not included in the band profile.
The matrix involves all 451 eigenstates falling in the
wavenumber range .

Next: Quasiorthogonality on the boundary
Up: The basic scaling method
Previous: The basic scaling method
Alex Barnett
20011003