  The existence of bound states in quantum mechanics with no classical
counterpart has been a subject of interest for a long time. Cross-wires and
cavities connected to infinite leads are typical examples in which open
geometries with bulges support bound solutions, in two or more dimensions. Here
we find that the role of topology can be even more important than space
availability, by showing the existence of bound solutions in one-dimensional
systems such as quantum cross-chains and, in general, chains tied in geometries
without loops. It is shown that these examples of unusual binding can be solved
analitically for energies and eigenvectors. An experimental proposal is given
in the form of tight-binding arrays of electromagnetic resonators, as the
effects in question are of a wave-like nature.
