Localization in a rough billiard: A sigma model formulation

We consider the quantum dynamics of a particle in a weakly rough billiard. The Floquet operator for reflection at the boundary is obtained as a unitary band matrix. The resulting dynamics in angular momentum space can be treated in the framework of the one-dimensional supersymmetric nonlinear sigma model. We find analytically localization and the corresponding localization length $\xi=D_{cl}$ where $D_{cl}$ is the classical diffusion constant due to boundary scattering.