Fractal Weyl law for three-dimensional chaotic hard-sphere scattering systems

The fractal Weyl law connects the asymptotic level number with the fractal dimension of the chaotic repeller. We provide the first test for the fractal Weyl law for a three-dimensional open scattering system. For the four-sphere billiard, we investigate the chaotic repeller and discuss the semiclassical quantization of the system by the method of cycle expansion with symmetry decomposition. We test the fractal Weyl law for various symmetry subspaces and sphere-to-sphere separations.

Figure 1

(Color online) The four-sphere billiard. Four spheres of equal radius $R$ are located on the vertices of an equilateral tetrahedron (indicated by bars) with edge length $d$. Shown is the case of $d/R=6$.

Figure 2

(Color online) (a) Time-delay functions in the range $2\le T\le 5$ for $d/R=3$ and (b) a magnification thereof in the range $2\le T\le 7$. The time-delay functions are drawn as functions of the coordinates $x,y$ in the surface of section. The colors indicate the value of $T$ and for clear identification some regions are explicitly labeled with $T$. Only the boundaries of individual regions are drawn; initial conditions chosen within the regions experience the same number of reflections. The self-similarity suggests that the stable manifold is a fractal set.

Figure 3

(Color online) Intersected Hausdorff sums $K^{\left(i\right)}\left(s\right)$ for various reflection numbers $i=T$ calculated from (a) a polygonal chain with 101 supporting points and (b) ellipses fitted to polygonal chains for $d/R=6$. As can clearly be seen, the intersection points for all shown curves agree perfectly. For this reason, the Hausdorff dimensions can be determined to a precision of at least 4 significant digits.

Figure 4

(Color online) Hausdorff dimension $d_{\text{H}}$ of the stable manifold ${\mathcal{W}}_{\text{s}}$ as function of the ratio $d/R$. All data points have been obtained by intersecting Hausdorff sums as demonstrated in Fig. 3.

Figure 5

(Color online) (a) Spectrum and (b) counting functions $N\left(k\right)$ for $A_1$ resonances calculated from cycle expansion in order 11 for the ratio $d/R=10$. Several counting functions for different strips $C$ are shown. The curves can be used as “raw data” to fit power laws. In this way, the exponent $\alpha$ in the fractal Weyl law can be obtained and compared to the classical calculations. More than 50 000 resonances have been used in the analysis.

Figure 6

(Color online) Exponents $\alpha$ obtained from least-squares fits of a power law to measured counting functions for (a) $A_1$ resonances and (b) $A_2$ resonances calculated for $d/R=6$ in order 13 of the cycle expansion. The power law has been fitted to the interval $k∊\left[0;2000\right]$. The horizontal dotted line gives the classical exponent $\alpha =1+d_{\text{H}}=1.2354$.

Figure 7

(Color online) Exponents $\alpha$ obtained for $d/R=8$ from least-squares fits of a power law to measured counting functions for (a) $A_1$ resonances and (b) $A_2$ resonances. The power law has been fitted to several $k$-intervals. The horizontal dotted line gives the classical exponent $\alpha =1+d_{\text{H}}=1.2063$.

Figure 8

(Color online) Exponents $\alpha$ obtained for $d/R=10$ from least-squares fits of a power law to measured counting functions for (a) $A_1$ resonances and (b) $A_2$ resonances. Results for several $k$-intervals are shown. The horizontal dotted line gives the classical exponent $\alpha =1+d_{\text{H}}=1.1888$.