  The spectra of a microwave cylindrical resonator with the embedded thin metal
rod playing the role of a singular perturbation are studied both theoretically
and experimentally. The intra- and inter-mode scattering caused by the
perturbation are clearly distinguished and recognized to play essentially
different parts in the appearance of spectrum chaotic properties. The analysis
based on the mode-mixing operator norm shows that the inter-mode scattering
dominates over the intra-mode scattering and basically determines statistical
properties of the resonator spectrum. The results we have obtained in the
experiment are in good conformity with our theory. Clear manifestations of
quantum chaos are revealed for the resonator with the asymmetrically inserted
rod, namely, the Wigner-type distribution of the inter-frequency intervals, the
apparent correlation between spectral lines, and the characteristic curve of
the spectral rigidity. By comparing the theory and the experiment we succeeded
in establishing for the first time that it is just the inter-mode scattering
that is responsible for the quantum chaos manifestations in the singularly
perturbed integrable wave-billiard system.
