On the point and continuous spectra for coupled quantum waveguides and resonators

A system of quantum resonators connected through small apertures with a quantum waveguide and a system of waveguides coupled through small windows are considered in the case of Dirichlet boundary condition. Solvable model based on the operator extension theory in Pontryagin space is suggested for the description of trapped modes imbedded in the continuous spectrum for the systems. The model allows one not only to prove the existence of such modes but also to suggest an effective and simple algorithm for its determination. A system of quantum waveguides coupled laterally through small windows is considered. The existence of eigenvalue imbedded in the continuous spectrum is shown. The case of periodic set of windows is studied in the framework of the model. The dispersion equation is obtained in an explicit form. The existence of bands imbedded in the continuous spectrum is proved, and an algorithm for its determination is described.