Quasi-orthogonality on the boundary

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 .

- Semiclassical estimate of off-diagonal strength of
- Relation to strength estimate of Vergini and Saraceno