Quasi-orthogonality on the boundary

The aim of this section is to show that $M_{\mu\nu}$ can be made very close to the identity matrix $\delta_{\mu\nu}$ via a certain choice of weighting function $w{({\mathbf s})}$. The matrix $M$ can be related to the rate of change of the Hamiltonian under deformation of $\Gamma$, written in the energy basis. $D{({\mathbf s})}= {\mathbf n}{({\mathbf s})}\cdot {\mathbf D}{({\mathbf s})}$ is then interpreted as the deformation function associated with a deformation `field' ${\mathbf D}{({\mathbf r})}$ (see Section 3.1). Looking at Appendix C we have

$$M_{\mu\nu} \ = \ -\frac{m}{(\hbar k)^2} \left(\frac{\partial \mathcal{H}}{\partial x}\right)_{\!\mu\nu}.$$ (6.9)

Thus all the knowledge gained in Chapters 2 and 3 about the structure of $\partial {\mathcal{H}}/\partial x$ can be applied. In particular, $\partial {\mathcal{H}}/\partial x$ is a zero-mean random quantity whose variance an energy $\hbar \omega$ from the diagonal is given by the `band profile' $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ (see Eq.(2.48)). The band profile is to be evaluated at overall energy $E = (\hbar k)^2/2m$, where $k$ is the typical wavenumber of interest.

In order to allow the scaling method to extract the eigenstates $\psi_\mu$, we seek a choice of $D{({\mathbf s})}$ (and therefore $w{({\mathbf s})}$) which makes $M_{\mu\nu}$ as close as possible to the identity matrix $\delta_{\mu\nu}$. This is the issue of orthogonality on the boundary. Note that in the domain $\mathcal{D}$ there is an exact orthogonality relation for the eigenstates: $\langle\psi_\mu\vert\psi_\nu\rangle_{\mathcal{D}} = \delta_{\mu\nu}$. 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 $k$. Still, the boundary functions `live' in an effective Hilbert space of dimension $\sim (k{\mathsf{L}})^{d-1}$. This dimension is just the semiclassical basis size $N_{sc}$ of Section 5.3.1. On the boundary the best that can be done is a quasi-orthogonality relation--approximate orthogonality of $N_{sc}$ adjacent eigenstate boundary-derivative functions, which span a range $\sim 1/{\mathsf{L}}$ in wavenumber.

The smallest off-diagonal elements of $M$ close to the diagonal are found when $D{({\mathbf s})}$ corresponds to a `special deformation'. As discussed in Section 3.3, the band profile $\tilde{C}_{{\mbox{\tiny E}}}(\omega)$ then vanishes as $\omega\rightarrow0$ like a power law $\omega^\gamma$. On the other hand, the diagonal elements of $M$ are given by the average generalized force $F(x)$ 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 $F(x)$ is dilation ${\mathbf D}{({\mathbf s})}= {\mathbf r}$. (Translations and rotations do not change the billiard volume so have $F(x) = 0$). For dilation, the diagonal elements of $M$ are all unity (proofs of this for $d=2$ are known [28,33]; I have proved it for arbitrary $d$ with Eq.(H.9) of Appendix H--also see our work[14]). The power law is $\gamma = 4$, a result which has a classical origin (Section 3.3.1). The quasi-orthogonality of $M$ for dilation is illustrated by Fig. 6.2.

The weighting function which gives $D{({\mathbf s})}= r_n$ is the one found by VS [195], namely $w{({\mathbf s})}= 1/r_n$. Therefore from now on I will assume this weighting, unless stated otherwise. It seems that in order to prevent $w$ 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 $\mathcal{D}$), so that $r_n$ 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 $r_n$.