  We report on the experimental investigation of the properties of the
eigenvalues and wavefunctions and the fluctuation properties of the scattering
matrix of closed and open billiards, respectively, of which the classical
dynamics undergoes a transition from integrable via almost integrable to fully
chaotic. To realize such a system we chose a billiard with a 60 degree sector
shape of which the classical dynamics is integrable, and introduced circular
scatterers of varying number, size and position. The spectral properties of
generic quantum systems of which the classical counterpart is either integrable
or chaotic are universal and well understood. If, however, the classical
dynamics is pseudo-integrable or almost-integrable, they exhibit a
non-universal intermediate statistics, for which analytical results are known
only in a few cases, like, e.g., if it corresponds to semi-Poisson statistics.
Since the latter is, above all, clearly distinguishable from those of
integrable and chaotic systems our aim was to design a billiard with these
features which indeed is achievable by adding just one scatterer of appropriate
size and position to the sector billiard. We demonstrate that, while the
spectral properties of almost-integrable billiards are sensitive to the
classical dynamics, this is not the case for the distribution of the
wavefunction components, which was analysed in terms of the strength
distribution, and the fluctuation properties of the scattering matrix which
coincide with those of typical, fully chaotic systems.
