  Defects or junctions in materials serve as a source of interactions for
particles, and in idealized limits they may be treated as singular points
yielding contact interactions. In quantum mechanics, these singularities
accommodate an unexpectedly rich structure and thereby provide a variety of
physical phenomena, especially if their properties are controlled properly.
Based on our recent studies, we present a brief review on the physical aspects
of such quantum singularities in one dimension. Among the intriguing phenomena
that the singularities admit, we mention strong vs weak duality, supersymmetry,
quantum anholonomy (Berry phase), and a copying process by anomalous caustics.
We also show that a partition wall as a singularity in a potential well can
give rise to a quantum force which exhibits an interesting temperature behavior
characteristic to the particle statistics.
