Normal modes of billiards portrayed in the stellar (or nodal) representation

We display the eigenmodes of two plane billiards, the ellipse and the stadium, in a specially adapted nodal phase space parametrization, the stellar representation using the zeros of a reduced Husimi function. The zeros show very contrasting behaviours according to their location in the classically forbidden or allowed region of phase space. The zeros in the allowed region are ‘essential’ for specifying the shape of the wavefunction; their number is closely proportional to the wave number of the mode, and their distribution reflects the (integrable or chaotic) nature of the underlying classical dynamics. The zeros in the forbidden region are infinite in number and form vertical alignments irrespectively of the nature of the interior dynamics of the billiard.