The spectrum of the billiard Laplacian of a family of random billiards

Random billiards are billiard dynamical systems for which the reflection law giving the postcollision direction of a billiard particle as a function of the pre-collision direction is specified by a Markov (scattering) operator P . Billiards with microstructure are random billiards whose Markov operator is derived from a “microscopic surface structure” on the boundary of the billiard table. The microstructure in turn is defined in terms of what we call a billiard cell Q, the shape of which completely determines the operator P . This operator, defined on an appropriate Hilbert space, is bounded self-adjoint and, for the examples considered here, a Hilbert-Schmidt operator. A central problem in the statistical theory of such random billiards is to relate the geometric characteristics of Q and the spectrum of P . We show, for a particular family of billiard cell shapes parametrized by a scale invariant curvature K (figure 3), that the billiard Laplacian P − I is closely related to the ordinary spherical Laplacian, and indicate, by partly analytical and partly numerical means, how this provides asymptotic information about the spectrum of P for small values of K. It is shown, in particular, that the second moment of scattering about the incidence angle closely approximates the spectral gap of P .