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
associated with a deformation `field'
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). 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 , 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'. However (despite protest to the contrary), 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 .