Statistical properties of high-lying chaotic eigenstates

We study the statistical properties of the high-lying chaotic eigenstates (200000 and above) which are deep within the semiclassical regime. The system we are analysing 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: (i) 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; (ii) the autocorrelation function is found to be strongly direction dependent and only after averaging over all directions agrees well with Berry's (1977) prediction; (iii) although the scars of unstable classical periodic orbits (in such an ergodic regime) are expected to exist, so far we have not found any (around the 200000th state) other than a scar-like feature resembling the whispering-gallery modes. (author)