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.