Fluctuating Number of Energy Levels in Mixed-Type Lemon Billiards - Črt Lozej; Dragan Lukman; Marko Robnik (Physics, 2021)
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 <semantics>2B</semantics> between the centers, as introduced by Heller and Tomsovic. In this paper, the case of two billiards, defined by <semantics>B=0.1953,<mspace width="0.277778em"></mspace>0.083</semantics>, 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 <semantics><msqrt>E</msqrt></semantics>, in agreement with the theoretical prediction by Steiner.