Nodal structures in billiard wave functions

We consider real wave functions of quantum billiards, and study their nodal lines - where the wave-functions vanish, and the nodal domains - where the wave-function is of constant sign. The nodal structures were first demonstrated and studied by Chladni, whose pioneering work led to many classical studies (Rayleigh, Courant, Pleijel) which form the basis for the results to be described in the present talk. We have recently developed a statistical approach to the problem of nodal domains counting. The resulting distribution functions depend crucially on the undelying classical flow (billiards) - whether it is chaotic or integrable. Within each class, the distribution functions have universal (system independent) features, with scaling parameters which depend on a few parameters (area, circumference) of the vibrating membrane. Thus, counting nodal domains offers a new criterion for quantum chaos.