Eigenvalues and bands imbedded in the continuous spectrum for a system of resonators and a waveguide: solvable model

Abstract

A solvable model based on the operator extension theory is suggested for the description of trapped modes imbedded in the continuous spectrum. A system of resonators connected through small apertures with a waveguide is considered as for the case of Neumann and Dirichlet boundary conditions. That is, we study both acoustical and quantum waveguides. The existence of such modes is shown (the corresponding sufficient condition is derived). An effective and simple algorithm for its determination is suggested. A system of acoustic (quantum) waveguide and periodic set of coupled cavities 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.