Abstract
In this paper, the fluctuation properties of the number of energy levels (mode fluctuation) are studied in the mixed-type lemon billiards at high lying energies. The boundary of the lemon billiards is defined by the intersection of two circles of equal unit radius with the distance 2B between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by B=0.1953,0.083, is studied. It is shown that the fluctuation of the number of energy levels follows the Gaussian distribution quite accurately, even though the relative fraction of the chaotic part of the phase space is only 0.28 and 0.16, respectively. The theoretical description of spectral fluctuations in the Berry–Robnik picture is discussed. Also, the (golden mean) integrable rectangular billiard is studied and an almost Gaussian distribution is obtained, in contrast to theory expectations. However, the variance as a function of energy, E, behaves as E, in agreement with the theoretical prediction by Steiner.
Highlights
The purpose of this paper is to analyze the energy spectra of two characteristic complex mixed-type lemon billiards within the scope of quantum chaos
The statistical properties of the oscillations of the cumulative spectral staircase function around the corresponding mean value were studied, in order to compare the function obtained with the theoretical predictions of Steiner [11,12,13,14] for the fully chaotic and integrable systems
The mean behavior is asymptotically exactly described by the Weyl formula (5)
Summary
The purpose of this paper is to analyze the energy spectra of two characteristic complex mixed-type lemon billiards within the scope of quantum chaos. These lemon billiards are mixed-type billiards with several independent regular regions embedded in a chaotic sea with no significant stickiness regions, which serve as examples of systems with a complex divided phase space. These lemon billiards were selected by the criterion of the maximally complex chaotic component generated by a single chaotic orbit. It must be emphasized that all the lemon billiards belong to the same family of billiards as for the mathematical definition, individually, the lemon billiards
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