Effects of symmetries on single-channel systems: Perfect transmission and reflection

We study the effects of symmetries on resonant transport in single-channel systems with an emphasis on their implications for atomic wires. Employing an expression of the scattering matrix in terms of its poles, it is shown that the mirror reflection and time-reversal symmetries lead generically to transmission and reflection peaks of magnitude I, resulting in perfect transmission and perfect reflection, respectively. In particular, it is demonstrated that even when multiple resonances overlap and resonance shapes are distorted considerably, the perfect transmission (perfect reflection) can still occur provided that a system is free from ''pair annihilation" events of two perfect transmissions (two perfect reflections). It is argued that these results can remain valid even when the employed expression of the scattering matrix becomes inapplicable, because the symmetries create certain topological structures of transmission and reflection amplitudes that stabilize the perfect transmission and reflection. Crossover from resonant transport to ballistic transport is also discussed. Results are discussed in connection with recent works on atomic wires.