Spectral properties of three-dimensional quantum billiards with a pointlike scatterer

We examine the spectral properties of three-dimensional quantum billiards with a single pointlike scatterer inside. It is found that the spectrum shows chaotic (random-matrix-like) characteristics when the inverse of the formal strength bar v-1 is within a band whose width increases parabolically as a function of the energy. This implies that the spectrum becomes random-matrix-like at very high energy irrespective to the value of the formal strength. The predictions are confirmed by numerical experiments with a rectangular box. The findings for a pointlike scatterer are applied to the case for a small but finite-size impurity. We clarify the proper procedure for its zero-size limit which involves nontrivial divergence. The previously known results in one- and two-dimensional quantum billiards with small impurities inside are also reviewed from the present perspective. copyright 1997 The American Physical Society