Spectral statistics in the quantized cardioid billiard

The spectral statistics of the strongly chaotic cardioid billiard is studied. The analysis is based on the first 11 000 quantal energy levels for both odd and even symmetry. It is found that the level-spacing distribution is in good agreement with the Gaussian-orthogonal-ensemble distribution of random-matrix theory. In the cases of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode fluctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosine-modulated heat kernel is studied. Since the billiard boundary is focusing, there are conjugate points that give rise to zeros instead of exclusively at Gaussian peaks at the locations of the periodic orbits.