Resonances and tunneling in a quantum wire

We consider a problem in mathematical scattering theory related to the ballistic conductance model. The model under investigation describes the charge propagation in a quantum wire. We assume that the charge carrier has a spin and take the Rashba spin-orbital interaction into account. We study the conductance resonances generated by the weak quantum-wire interaction with the quasistationary state of a parallel-connected quantum dot or with the tunneling through a series-connected quantum dot. Such a quantum dot is usually the control element. We present sufficient conditions for the spatial symmetry of the system to ensure that the quasistationary state of the quantum dot generates a conductance resonance. We assume that the conductance is related to the scattering matrix by the Landauer formula.