The aim of this section is to show that can be made
very close to the identity matrix
via a certain choice
of weighting function
.
The matrix
can be related to the rate of change of the Hamiltonian
under deformation of
, written in the energy basis.
is then interpreted as the
deformation function
associated with a deformation `field'
(see
Section 3.1).
Looking at Appendix C we have
In order to allow the scaling method to extract the eigenstates ,
we seek a choice of
(and therefore
) which makes
as
close as possible to the identity matrix
.
This is the issue of orthogonality on the boundary.
Note that in the domain
there is an exact orthogonality relation
for the eigenstates:
.
There is no such exact relation on the boundary.
One cannot define a Hilbert space which consists of the boundary functions
(eigenstate normal derivatives)
at a single wavenumber
.
Still, the boundary functions `live' in
an effective Hilbert space of dimension
.
This dimension is just the semiclassical basis size
of
Section 5.3.1.
On the boundary the best that can be done is a quasi-orthogonality
relation--approximate orthogonality of
adjacent eigenstate
boundary-derivative functions, which span a range
in wavenumber.
The smallest off-diagonal elements of close to the diagonal are found
when
corresponds to a `special deformation'. As discussed in
Section 3.3, the band profile
then vanishes
as
like a power law
.
On the other hand, the diagonal elements of
are given by the average
generalized force
on the deformation (see Section 3.1).
So far the only special deformation
known to the author (see Appendix D)
which has a non-zero
is dilation
.
(Translations and rotations do not change the billiard volume so have
).
For dilation, the diagonal elements of
are all
unity (proofs of this for
are known [28,33];
I have proved it for arbitrary
with
Eq.(H.9) of Appendix H--also see our work[14]).
The power law is
, a result which has a classical
origin (Section 3.3.1).
The quasi-orthogonality of
for dilation
is illustrated by Fig. 6.2.
The weighting function which gives
is the one found by
VS [195], namely
.
Therefore from now on I will assume this weighting, unless stated otherwise.
It seems that in order to prevent
from diverging, this introduces the
restriction that there must exist an origin from which all parts of the billiard
wall are directly `visible' (connected by a chord which always remains inside
), so that
never passes through zero.
Such billiards have been called `star-shaped'[195].
However (despite protest to the contrary[39]),
it seems that a slight generalization can be made to include
non-simply-connected billiards composed of a star-shaped exterior surface and
a star-shaped (concentric) interior surface, without breaking the
above restriction on
.