On the point and continuous spectra for coupled quantum waveguides and resonators
The problem of eigenvalues imbedded in the continuous spectrum is studied within the framework of the following solvable model: the system of resonators coupled with a waveguide by means of small opens that are treated as point-like ones to simplify the problem. Unlike the more conventional approach using variational inequalities, the model is based on the theory of self-adjoint extensions of symmetric operators. Except the Neumann boundary conditions, it also includes the important case of Dirichlet boundary conditions. The model is analogous to the well-known zero-range potential approach, \textit{S. Albeverio, F. Gesztesy, R. Høegh-Krohn} and \textit{H. Holden} [``Solvable models in quantum mechanics'', Berlin (1988; Zbl 0679.46057)] and \textit{B. S. Pavlov} [Russ. Math. Surv. 42, No. 6, 127-168 (1987; Zbl 0665.47004)]. It allows to obtain the dispersion equation explicitly and provides an effective tool for computing eigenvalues and bands imbedded in the continuous spectrum. Two physically important situations are considered in detail in the case of Dirichlet boundary conditions: a system of resonators connected to a waveguide and a system of waveguides coupled through a small window. In both systems eigenvalues imbedded in the continuous spectrum are possible. In the latter case, when periodic set of windows is used instead of a single window, there is a band imbedded in the continuous spectrum. Explicit dispersion equations are presented in accordance with the declared capabilities of the theory. The suggested approach can be extended to other spectral problems.
eigenvalues imbedded in the continuous spectrum
selfadjoint extensions of symmetric operators
scientific article; zbMATH DE number 1158138
scientific article