Uzy Smilansky, Weizmann Institute
Counting Nodal Domains - from Chladni to Quantum Chaos

When a memberane of an arbitrary shape vibrates at one of its eigen-frequencies, one observes a pattern of nodal lines where the vibration amplitude vanishes. The nodal lines separate the nodal domains, where the wave function is of constant sign. The nodal structures were the first demonstrated and studied by Chladni, who also made the first quantitative connection between the nodal patterns, and the frequencies of the vibration modes. This was the starting point for a few 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 underlying 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 signature of quantum chaos.