  We consider the scattering problems of a quantum particle in a system with a
single Y-junction and in ring systems with double Y-junctions. We provide new
formalism for such quantum mechanical problems. Based on a path integral
approach, we find compact formulas for probability amplitudes in the ring
systems. We also discuss quantum reflection and transmission in the ring
systems under scale-invariant junction conditions. It is remarkable that
perfect reflection can occur in an anti-symmetric ring system, in contrast with
the one-dimensional quantum systems having singular nodes of degree 2.
