Electron transport in a two-terminal Aharonov-Bohm ring with a few shortrange scatterers is investigated. An analytical expression for the conductance as a function of the electron Fermi energy and magnetic flux is obtained using the zero-range potential theory. The dependence of the conductance on positions of scatterers is studied. We have found that the conductance exhibits asymmetric Fano resonances at certain energies. The dependence of the Fano resonances on magnetic field and positions of impurities is investigated. It is found that collapse of the Fano resonances occurs and discrete energy levels in the continuous spectrum appear at certain conditions. An explicit form for the wave function corresponding to the discrete level is obtained.

Spectral properties of periodic one-dimensional array of nanorings in a magnetic field are investigated. Two types of the superlattice are considered. In the first one, rings are connected by short one-dimensional wires while in the second one rings have immediate contacts between each other. The dependence of the electron energy on the quasimomentum is obtained from the Schrödinger equation for the Bloch wave function. We have found an interesting feature of the system, namely, presence of discrete energy levels in the spectrum. The levels can be located in the gaps or in the bands depending on parameters of the system. The levels correspond to bound states and electrons occupying these levels are located on individual rings or couples of neighboring rings and do not contribute to the charge transport. The wave function for the bound states corresponding to the discrete levels is obtained. Modification of electron energy spectrum with variation of system parameters is discussed.

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