  We study the low energy quantum spectra of two-dimensional rectangular
billiards with a small but finite-size scatterer inside. We start by examining
the spectral properties of billiards with a single pointlike scatterer. The
problem is formulated in terms of self-adjoint extension theory of functional
analysis. The condition for the appearance of so-called wave chaos is
clarified. We then relate the pointlike scatterer to a finite-size scatterer
through the appropriate truncation of basis. We show that the signature of wave
chaos in low energy states is most prominent when the scatterer is weakly
attractive. As an illustration, numerical results of a rectangular billiard
with a small rectangular scatterer inside are exhibited.
