Symmetry reduced three-disk and five-disk systems are studied in a microwave setup. Using harmonic inversion the distribution of the imaginary parts of the resonances is determined. With increasing opening of the systems, a spectral gap is observed for thick as well as for thin repellers and for the latter case it is compared with the known topological pressure bounds. The maxima of the distributions are found to coincide for a large range of the distance to radius parameter with half of the classical escape rate. This confirms theoretical predictions based on rigorous mathematical analysis for the spectral gap and on numerical experiments for the maxima of the distributions. A paradigm for systems with fractal repellers is the ndisk scattering system (see Fig. 1). It was introduced in the 1980s by Ikawa in mathematics [16] and by Gaspard and Rice [17][18][19] and Cvitanović and Eckhardt [20] in physics. It is given by n hard disks with centers forming a regular polygon. The distance between the centers is denoted by R and the disk radius by a; R/a determines the system up to scaling (see Fig. 1).The quantum system is described by the Helmholtz equationThe quantum resonances k n = Re k n + iIm k n , are the complex poles of the scattering matrix. For the three-disk system this scattering matrix is expressed using Bessel functions and that allowed Gaspard and Rice [19] to calculate the quantum resonances numerically. Classically, particle trajectories are given by straight lines reflected by the disks. From periodic trajectories a wide range of classical quantities such as the classical escape rate, the fractal dimension of the repeller and the topological pressure can be calculated using the Ruelle zeta function [18],where the product runs over the primitive periodic orbits, T p are the corresponding period lengths, and Λ p are the stabilities. The topological pressure P (β) is then defined as the largest real pole of ζ β (z). An effective method for its calculation is the cycle expansion [20,21]. The classical escape rate is given by γ cl = −P (1) and the FIG. 1. (color online)A sketch of a three-disk system is shown on the left side, where one fundamental domain is shaded.On the right side a photograph of the experimental cavity without top plate supporting the disk inset and the absorber is presented.

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.