Point perturbations of circle billiards

The spectral statistics of the circular billiard with a point-scatterer is investigated. In the semiclassical limit, the spectrum is demonstrated to be composed of two uncorrelated level sequences. The first corresponds to states for which the scatterer is located in the classically forbidden region and its energy levels are not affected by the scatterer in the semiclassical limit while the second sequence contains the levels which are affected by the point-scatterer. The nearest neighbor spacing distribution which results from the superposition of these sequences is calculated analytically within some approximation and good agreement with the distribution that was computed numerically is found. Classical dynamics may be illuminating for the understanding of the corresponding quantum mechanical systems. One of the most studied aspects of the relation between classical and quantum mechanics is the connection between the spectral statistics of the quantum system and the dynamical properties of its classical counterpart. Classically integrable systems typically exhibit Poisson-like spectral statistics [1] while classically chaotic systems exhibit spectral statistics of random matrix ensembles [2, 3, 4, 5]. The spectral statistics of integrable and chaotic systems are universal, that is, they do not depend on specific details of the system but rather on the type of motion and its symmetries. There are systems which are intermediate between integrable and chaotic ones and their spectral properties are not known to be universal. Such systems are of experimental relevance. The spectral statistics of mixed systems, for which the phase space is composed of both integrable and chaotic regions, were studied by Berry and Robnik [6]. The spectrum can be viewed as a superposition of uncorrelated level sequences, corresponding to the various regions, which are either chaotic or integrable. The nearest neighbor spacing distribution (NNSD) of such a superposition of sequences was calculated in [6]. The resulting statistics are, in some sense, intermediate between those of integrable and chaotic systems. Other types of systems with intermediate statistics include pseudointegrable systems and integrable systems with flux lines or