Weyl asymptotics: From closed to open systems

We present microwave experiments on the symmetry reduced five-disk billiard studying the transition from a closed to an open system. The measured microwave reflection signal is analyzed by means of the harmonic inversion and the counting function of the resulting resonances is studied. For the closed system this counting function shows the Weyl asymptotic with a leading exponent equal to 2. By opening the system successively this exponent decreases smoothly to a noninteger value. For the open systems the extraction of resonances by the harmonic inversion becomes more challenging and the arising difficulties are discussed. The results can be interpreted as a first experimental indication for the fractal Weyl conjecture for resonances.

Figure 1

(a) A sketch of the five-disk scattering system is given. The gray shaded part shows the fundamental domain of the symmetry reduction. The radius of the disks is denoted by $a$ and their distance by $R$. (b) A photograph of the experimental setup is presented.

Figure 2

The plot shows measured reflection spectra for the closed system $R/a=2$ (black solid line), a system in the transition region with $R/a=2.25$ (red dotted line), and an open system with $R/a=3.83$ (blue dashed line).

Figure 3

Fast Fourier transform of the reflection signal of the frequency window from 20 to 23 GHz for the closed system ($R/a=2$, upper black curve) and an open system ($R/a=3.83$, lower orange curve). The unit of the $x$ axis is the number of data points of this discrete time signal. The inset shows the decay for the closed system in a double logarithmic plot. The red straight line correspond to a $1/n$-decay. The light blue shaded region indicates the variation of the cutoff parameter between 80 and 120 data points, which is used in the data analysis described in Sec. 5.

Figure 4

This figure shows the resonances of the five-disk system with $R/a=3.83$ in the complex plane obtained via the HI for one set of parameters. The light blue circles are resonances that are removed by the filter, whereas the black dots correspond to resonances that are assumed to be true.

Figure 5

The top plot shows the measured microwave spectrum ($R/a=3.83$) (black solid line) as well as two reconstructions obtained by the HI results for two different parameter sets (orange dashed and red dotted lines). The bottom plot shows the corresponding complex resonance positions. The red circles correspond to the resonances of the reconstructed signal shown above as red dotted line and the blue triangles to the orange dashed line. Light blue dots represent the resonance positions of the HI results for all parameter sets. The two vertical dashed lines mark the valid range of the analyzed window.

Figure 6

The counting functions for $R/a=2$, 2.25, and 3.9 (in this order from bottom to top) are plotted in black (histograms). Fits of their slope in the frequency range 15–24 GHz, corresponding to a $k$ range of 315–500 mm${}^{-1}$ (dotted vertical line) are shown in blue (straight lines). The light orange curve over the lower histogram corresponds to the Weyl formula with 12$\%$ loss for the closed system. Plotted in the inset is the difference between the Weyl formula with 12$\%$ loss and the experimental counting function for the closed system ($R/a=2$).

Figure 7

The data points correspond to the fitted exponent of the counting function in dependence of the $R/a$ parameter. The solid line shows the asymptotic exponent $1+d_H$ predicted by the fractal Weyl law and the dashed curve is the same shifted by 0.4 as a guide for the eye. The three squares mark the examples which have already been presented in the previous figures. The darker shaded blue region indicates the $R/a$ values without open channels, whereas in the range marked by the lighter shaded blue region only a few open channels (1, … , 8) exist.