Test of a numerical approach to the quantization of billiards

A method for computing large numbers of eigenvalues of self-adjoint elliptic operators [J. Comput. Phys. 184, 321 (2003)] is tested in numerical studies of the spectral properties of quantum billiards. To this extent, we study a time-reversal invariant quantum billiard of threefold symmetry, that undergoes a transformation in its symmetry properties from $C_{3v}$ to $C_3$. Thereby a transition from Gaussian orthogonal to Gaussian unitary ensemble statistics is observed, verifying earlier experimental indications and theoretical predictions. At the same time our numerical ansatz is shown to be applicable to arbitrary billiard shapes.

Figure 1

Shape of the billiard. When rotating the inner boundary from $\alpha =0$ to $\alpha \ne 0$, a transition of the symmetry properties from $C_{3v}$ to $C_3$ is induced. The dashed line shows a periodic classical orbit that is nonchaotic and that influences the spectral properties of the quantum system.

Figure 2

Percentage of doublets as a function of the assumed maximum doublet splitting, shown for the $\alpha =0$ case. As one can see it is possible to vary the parameter taken for the maximum splitting from ${10}^{-10}$ up to ${10}^{-5}$, i.e., by five orders of magnitude, without changing the singlet and doublet subspectra. This indicates that the numerical lifting of the degeneracies is extremely small compared to the smallest eigenvalue spacings.

Figure 3

Nearest-neighbor-spacings distributions (NNSDs) of the spectra calculated for the three different configurations of the billiard (histograms). The full lines and the dashed lines mark the theoretical predictions for $C_{3v}$ and $C_3$ symmetry, respectively. For all cases the results coincide with the predictions: for $\alpha =0°$ one has two superimposed GOE spectra for the singlets and a GOE spectrum for the doublets. The other configurations show GOE statistics for the singlets and GUE behavior for the doublets.

Figure 4

Long-range correlations of the calculated spectra for three different configurations of the billiard (thick lines). The thin, full, and dashed lines mark the theoretical predictions for $C_{3v}$ and $C_3$ symmetry, respectively. For the limiting cases of $\alpha =0°$ the ${\Delta }_3$ statistics coincides with the predictions for the $C_{3v}$ case. In the $\alpha =1.7°$ case and also for $\alpha ={10}^{\circ }$ the statistics deviates from the theoretical predictions for $C_3$ symmetry.

Figure 5

Length spectrum, i.e., the Fourier transform of the eigenvalue spectrum according to Eq. (10), for the singlet spectrum of the $\alpha =0°$ configuration. The prominent peaks at $x\approx 0.59\mathrm{m}$ and at $x\approx 1.18\mathrm{m}$ are due to the orbit shown in Fig. 1 and have been extracted from the data discussed in Fig. 6. Note that we do not simply remove the strongest peaks, but those peaks that appear in all spectra and that show at least one or two repetitions.

Figure 6

Long-range correlations of the calculated spectra after extraction of the influence of nonchaotic periodic orbits (Fig. 1). The resulting long-range correlations are closer to the theoretical predictions as in Fig. 4, except for the case $\alpha =0°$. Most importantly, the doublets show GOE statistics for $\alpha =0°$, while different angles $\alpha$ lead to GUE-like spectral correlations.

Figure 7

Nearest-neighbor-spacings distributions for the experimental single and double eigenfrequencies of the $C_{3v}$ billiard (histograms), to be compared with the uppermost part of Fig. 3. Degenerate modes (doublets) and nondegenerate modes (singlets) both obey GOE statistics. However, the singlet modes have odd or even parity with respect to the symmetry axes and thus one sees a superposition of two statistically independent GOEs.