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## 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 (6.9)

Thus all the knowledge gained in Chapters 2 and 3 about the structure of can be applied. In particular, is a zero-mean random quantity whose variance an energy from the diagonal is given by the band profile' (see Eq.(2.48)). The band profile is to be evaluated at overall energy , where is the typical wavenumber of interest.

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 .

Subsections   Next: Semiclassical estimate of off-diagonal Up: The basic scaling method Previous: Tension matrix in a
Alex Barnett 2001-10-03