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We discuss quantum dynamics in the ring systems with double Y-junctions inwhich two arms have same length. The node of a Y-junction can be parametrizedby U(3). Considering mathematically permitted junction conditions seriously, weformulate such systems by scattering matrices. We show that the symmetric ringsystems, which consist of two nodes with the same parameters under thereflection symmetry, have remarkable aspects that there exist localized statesinevitably, and resonant perfect transmission occurs when the wavenumber of anincoming wave coincides with that of the localized states, for any parametersof the nodes except for the extremal cases in which the absolute values ofcomponents of scattering matrices take $1$. We also investigate the magneticdisturbance to the symmetric ring systems.

Peculiarity of Symmetric Ring Systems with Double Y-Junctions and the magnetic effects