Statistics of Wave-Function Scars

The properties of ‘‘scars’’ on eigenfunctions (i.e., enhancements along unstable classical periodic orbits) of a two-dimensional, classically chaotic billiard are studied. It is shown that the tendency for a scar to form is controlled by both the stability of the periodic orbit and the statistical fluctuations in the time for wave density to return to the unstable orbit once having left. Both scars and ‘‘antiscars’’ are predicted to occur depending on the nearness of the eigenvalue of the chaotic eigenfunction in question to a value that quantizes the periodic orbit. The theoretical predictions are compared with direct numerical solutions for a bowtie shaped billiard.