# Deformations and dilations of chaotic billiards: dissipation rate, and quasiorthogonality of the boundary wave functions Academic Article

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• We consider chaotic billiards in $d$ dimensions, and study the matrix elements ${M}_{\mathrm{nm}}$ corresponding to general deformations of the boundary. We analyze the dependence of $|{M}_{\mathrm{nm}}{|}^{2}$ on $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}({E}_{n}{\ensuremath{-}E}_{m})/\ensuremath{\Elzxh}$ using semiclassical considerations. This relates to an estimate of the energy dissipation rate when the deformation is periodic at frequency $\ensuremath{\omega}$. We show that, for dilations and translations of the boundary, $|{M}_{\mathrm{nm}}{|}^{2}$ vanishes like ${\ensuremath{\omega}}^{4}$ as $\ensuremath{\omega}\ensuremath{\rightarrow}0$, for rotations such as ${\ensuremath{\omega}}^{2}$, whereas for generic deformations it goes to a constant. Such special cases lead to quasiorthogonality of the eigenstates on the boundary.