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

Quantum Reflection and Transmission in Ring Systems with Double Y-Junctions: Occurrence of Perfect Reflection