Deformations and dilations of chaotic billiards: Dissipation rate, and quasiorthogonality of the boundary wave functions

We consider chaotic billiards in d dimensions, and study the matrix elements Mnm corresponding to general deformations of the boundary. We analyze the dependence of |Mnm|2 on ω=(En−Em)/ħ using semiclassical considerations. This relates to an estimate of the energy dissipation rate when the deformation is periodic at frequency ω. We show that, for dilations and translations of the boundary, |Mnm|2 vanishes like ω4 as ω→0, for rotations such as ω2, whereas for generic deformations it goes to a constant. Such special cases lead to quasiorthogonality of the eigenstates on the boundary.