A quantum waveguide with a semitransparent barrier, placed across it, is considered. It is assumed that the barrier has a small window. This local perturbation of the waveguide causes the appearance of resonance states localized near the barrier with the window. The asymptotics (in small parameter -the window width) of the resonances (quasi-bound states) is obtained. The procedure of construction of full formal asymptotic expansion is described. The first two terms of the asymptotic expansion are obtained explicitly. These terms describe the shift of the resonance from the threshold and the life time of the corresponding resonance state.
A pair of coupled quantum waveguides with a common semitransparent wall is considered. The wall has a finite number of small windows. We consider resonance states localized near each window. The presence of several windows forces one to describe their common influence differently from that of the single-window case. Using the "matching of asymptotic expansions" method, we derive formulas for resonances and resonance states.
The paper explores the system of quantum waveguides and resonators. It suggests a solvable model of zero-width coupling windows based on the operator extensions theory in the Pontryagin space with indefinite metrics. The model self-adjoint operator is constructed explicitly, and it is similar (in some sense) to that of a physical system. We obtain an expression for the transmission coefficient for electrons and investigate its dependence on the electron energy which has a resonant character. It allows to control the electron transmission to different waveguides. The paper proposes a model of a three-channel quantum mesoscopic multiplexer.
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