Coupling of bouncing-ball modes to the chaotic sea and their counting function

We study the coupling of bouncing-ball modes to chaotic modes in two-dimensional billiards with two parallel boundary segments. Analytically, we predict the corresponding decay rates using the fictitious integrable system approach. Agreement with numerically determined rates is found for the stadium and the cosine billiard. We use this result to predict the asymptotic behavior of the counting function $N_{\text{bb}}\left(E\right)\sim E^{\delta }$. For the stadium billiard we find agreement with the previous result $\delta =3/4$. For the cosine billiard we derive $\delta =5/8$, which is confirmed numerically and is well below the previously predicted upper bound $\delta =9/10$.

Figure 1

Schematic pictures of the desymmetrized (a) stadium and (b) cosine billiard. Each billiard has a rectangular bouncing-ball region (gray shaded) of width $w$ and height $h$. In the rectangular region so-called bouncing-ball orbits exist perpendicular to the parallel parts of the boundary (red vertical lines). In (c) and (d) bouncing-ball modes are shown for the stadium and the cosine billiard, respectively.

Figure 2

Comparison of bouncing-ball modes of (a) the cosine billiard with quantum number $(23,4)$ and (b) the stadium billiard with quantum number $(23,5)$ to the corresponding adiabatic modes [(c), (d)] and to the rectangle modes [(e), (f)]. The adiabatic modes (c) and (d) closely resemble the bouncing-ball modes (a) and (b), respectively. The rectangle mode (e) closely resembles the bouncing-ball mode (a) of the cosine billiard, while for the stadium billiard the rectangle mode (f) fails to reproduce the bouncing-ball mode (b).

Figure 3

Illustration of the discrete directions contributing to the random-wave model, Eq. (25), in $k$ space at energy $E_{\text{reg}}^{mn}$ (black quarter circle) with $m=10$ and $n=2$. While the wave numbers $k_{y,s}$ are equispaced, the wave numbers $k_{x,s}$ as well as the angles ${\alpha }_s$ are not equispaced. The angular regions around each ${\alpha }_s$ are marked by gray lines and the one for ${\alpha }_{s_{\text{max}}}$ is shaded.

Figure 4

Decay rates of bouncing-ball modes for (a) the cosine billiard ${\mathcal{B}}_c$ and (b) the stadium billiard ${\mathcal{B}}_s$. For the quantum numbers $n=1,2$ and increasing $m$ we compare numerical rates (dots) to the results of Eqs. (11) (solid lines) and (30) (dashed lines) on a double-logarithmic scale. The insets in (a) and (b) show the bouncing-ball modes of quantum number $(7,1)$ and $(39,2)$.

Figure 5

Decay rates of bouncing-ball modes for (a) the cosine billiard ${\mathcal{B}}_c$ and (b) the stadium billiard ${\mathcal{B}}_s$. For the quantum numbers $m=30$ and $n=1,2,\cdots$, we compare numerical rates (dots) to the results of Eqs. (11) (solid lines) and (30) (dashed lines) on a double-logarithmic scale. The insets show the bouncing-ball modes of quantum number (a) $(30,4)$ and (b) $(30,6)$.

Figure 6

Number of bouncing-ball modes $N_{\text{bb}}\left(E\right)$ for (a) the cosine billiard ${\mathcal{B}}_c$ and (b) the stadium billiard ${\mathcal{B}}_s$. We compare the results using the overlap criterion (black staircase functions) and the prediction using the decay rates (gray staircase functions) to power laws with exponent (a) $\delta =5/8$ and (b) $\delta =3/4$ (red dashed lines) on a double-logarithmic scale. In (a) we show in addition a power law with exponent $\delta =9/10$ (dotted line) [22]. The insets show the results using the overlap criterion on a linear scale compared to Eq. (32) fitted for $E>10000$ (red dashed lines). We find $\delta =0.64$, $\alpha =0.196$ in (a) and $\delta =0.73$, $\alpha =0.112$ in (b).