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.

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The electron transmission properties in a model of two chained orthogonal quantum rings with input and output wires were investigated by using the quantum graph theory and quantum waveguide theory. The model was obtained for three-dimensional space. It also was shown that changing of the orientation of second ring in the respect to the field is a way to control the electron transmission and reflection.

We consider the Laplace operator with the Neumann boundary condition in a two-dimensional domain divided by a barrier composed of many small Helmholtz resonators coupled with the both parts of the domain through small windows of diameter 2a. The main terms of the asymptotic expansions in a of the eigenvalues and eigenfunctions are considered in the case when the number of the Helmholtz resonators tends to infinity. It is shown that such a homogenization procedure leads to some energy-dependent boundary condition in the limit. We use the method of matching the asymptotic expansions of the boundary value problem solutions.