  We study the statistical properties of the high-lying chaotic eigenstates
(200,000 and above) which are deep in the semiclassical regime. The system we
are analyzing is the billiard system inside the region defined by the quadratic
(complex) conformal map of the unit disk as introduced by Robnik (1983). We are
using Heller's method of plane wave decomposition of the numerical
eigenfunctions, and perform extensive statistical analysis with the following
conclusions: (1) The local average probability density is in excellent
agreement with the microcanonical assumption and all statistical properties are
also in excellent agreement with the Gaussian random model; \qquad (2) The
autocorrelation function is found to be strongly direction dependent and only
after averaging over all directions agrees well with Berry's (1977) prediction;
\qquad (3) Although the scars of unstable classical periodic orbits (in such
ergodic regime) are expected to exist, so far we have not found any (around
200,000th state) but a scar-like feature resembling the whispering gallery
modes.

Chaotic Dynamics;Chaotic Dynamics