We examine the quantum energy levels of rectangular billiards with a pointlike scatterer in one and two dimensions. By varying the location and the strength of the scatterer, we systematically find diabolical degeneracies among various levels. The associated Berry phase is illustrated, and the existence of localized wave functions is pointed out. In one dimension, even the ground state is shown to display the sign reversal with a mechanism to circumvent the Sturm-Liouville theorem.
