Bouncing ball orbits and symmetry breaking effects in a three-dimensional chaotic billiard

We study the classical and quantum mechanics of a three-dimensional stadium billiard. It consists of two quarter cylinders that are rotated with respect to each other by 90° and it is classically chaotic. The billiard exhibits only a few families of nongeneric periodic orbits. We introduce an analytic method for their treatment. The length spectrum can be understood in terms of the nongeneric and unstable periodic orbits. For unequal radii of the quarter cylinders the level statistics agree well with predictions from random matrix theory. For equal radii the billiard exhibits an additional symmetry. We investigated the effects of symmetry breaking on spectral properties. Moreover, for equal radii, we observe a small deviation of the level statistics from random matrix theory. This led to the discovery of stable and marginally stable orbits, which are absent for unequal radii.

Figure 1

Three-dimensional stadium billiard. The plane $z=0$ is shaded.

Figure 2

(Color online) Fluctuating part $N_{\text{fluc}}$ of the staircase function for the geometry with $r_1/r_2=\sqrt{2}$ (thin line). Contributions from bouncing-ball orbits (dashed line) and from all nongeneric modes (thick line).

Figure 3

(Color online) Upper part: Length spectrum of the three-dimensional stadium billiard for $r_1/r_2=\sqrt{2}$ (full line). Contributions from unstable periodic orbits (dashed line). Lower part: Sum of contributions from nongeneric modes and unstable periodic orbits (full line). Peaks at multiples of $2r_1$ and $2r_2$ correspond to the bouncing-ball orbits in the two quarter cylinders. Diamonds: Lengths of nongeneric orbits in the $z=0$ plane.

Figure 4

(Color online) Full line: Length spectrum for $r_1=r_2$. Diamonds: Length of bouncing-ball orbits and of nongeneric orbits in the $z=0$ plane. Full arrows: Length of marginally stable orbits. Dashed arrow: length of stable periodic orbit.

Figure 5

(Color online) Upper panel: Nearest-neighbor spacing distribution (histogram) of the billiard for $r_1=\sqrt{2}{r}_2$ compared to the GOE prediction (dashed line). Lower panel: ${\Sigma }^2$ statistics (thin line) compared to the GOE prediction (dashed line).

Figure 6

(Color online) ${\Sigma }^2$ statistics for the billiard with $r_1=r_2$ (thin line) compared to the GOE prediction (dashed line) and to a random matrix model which also includes a Poissonian sequence of random levels (dots).

Figure 7

(Color online) The fluctuating part $N_{\text{fluc}}$ of the staircase function for the billiard with $r_1=r_2$ (thin line) compared to the contributions from the nongeneric modes (thick line).

Figure 8

(Color online) ${\Sigma }^2$ statistics for various ratios $r_1/r_2$ of the billiard (thin line) compared to that of two GOEs (dotted line), one GOE (dashed line), and for the random matrix model in Eq. (10) (thick line).

Figure 9

(Color online) ${\Sigma }^2$ statistics of the high-lying stretch of levels around $k{r}_2\approx 80$ (dashed-dotted line) compared to that of the low-lying levels (thin line) and the GOE prediction (dashed line).