Deformation matrix elements

When a billiard is deformed, the potential $U{({\mathbf r})}$ changes by an infinite amount (in the hard-wall limit) in narrow regions of space on the boundary. These regions are simply those excluded from or newly-included into the cavity interior by the deformation. The operator $\partial {\mathcal{H}}/\partial x$ is therefore either infinite or zero in a position representation. Therefore it might be a surprise that the matrix elements $(\partial {\mathcal{H}} / \partial x)_{nm}$ in the energy representation are in fact finite. The following derivation has been known for some time [28,118,46].

The position of a particle in the vicinity of a wall element is conveniently described by ${\mathbf r}=({\mathbf s},z)$, where ${\mathbf s}$ is a $d{-}1$ dimensional surface coordinate and $z$ is a perpendicular `radial' coordinate. I set $U{({\mathbf r})}= U_0$ outside the undeformed billiard; this will relax the usual Dirichlet condition $\psi = 0$ on the boundary (at $z=0$). Later I will take the limit $U_0 \rightarrow \infty$. I have

$$\frac{\partial \mathcal{H}}{\partial x}\; = \; - [{\mathb... ...}) {\cdot} \hat{{\mathbf D}}({\mathbf s})] \, U_0 \delta(z) .$$ (C.1)

The normal derivative (derivative with respect to $z$) of an eigenfunction $\psi{({\mathbf r})}$ measured on the boundary is $\varphi({\mathbf s}) \equiv {\mathbf n}{\cdot}\nabla\psi$. The `logarithmic derivative' is defined as the ratio of gradient to value, $\varphi({\mathbf s})/\psi({\mathbf s})$. For $z>0$ the wavefunction $\psi{({\mathbf r})}$ is a decaying exponential in $z$, using simple one-dimensional considerations which hold true if the decay length is much smaller than any radius of curvature. Hence the logarithmic derivative of the wavefunction on the boundary should be equal to $-\sqrt{2mU_0}/\hbar$. Consequently one obtains

$$\left( \frac{\partial \mathcal{H}}{\partial x} \right)_{\!n... ...bf s}) \, ({\mathbf n}{\cdot}\hat{{\mathbf D}}) d{\mathbf s} .$$ (C.2)

The weighting $({\mathbf n}{\cdot}\hat{{\mathbf D}})$ is simply the deformation function $D{({\mathbf s})}$. Note that for any finite deformation $x$, the matrix elements of ${\mathcal{H}}(x)$ in the original energy basis become infinite--this problem is avoided by always remaining in the local adiabatic energy basis [46,48].

Therefore, given a subset of $N$ adjacent eigenstates, an on-diagonal block of the matrix $(\partial {\mathcal{H}} / \partial x)_{nm}$ can be found. A resulting example is shown in Fig. 2.7. The integrals are evaluated using the techniques of Appendix G.