On the point and continuous spectra for coupled quantum waveguides and resonators
type: Item
instance of: scholarly article
title: On the point and continuous spectra for coupled quantum waveguides and resonators
zbMATH Open document ID: 0905.47060
DOI: 10.1016/S0034-4877(97)85901-0
published in: Reports on Mathematical Physics
publication date: 1999-01-10T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81T10
Mathematics Subject Classification ID: 47B25
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 81V99
Mathematics Subject Classification ID: 47A20
zbMATH DE Number: 1158138
zbMATH Keywords: resonator
zbMATH Keywords: eigenvalues imbedded in the continuous spectrum
zbMATH Keywords: trapped modes
zbMATH Keywords: eigenvalues imbedded in the waveguide
zbMATH Keywords: selfadjoint extensions of symmetric operators
zbMATH Keywords: restriction-extension procedure
zbMATH Keywords: dispersion equation
zbMATH Keywords: Neumann boundary conditions
zbMATH Keywords: Dirichlet boundary conditions
zbMATH Keywords: zero-range potential approach
author: I. Yu. Popov
reviewed by: Pavel Dyshlovenko
MaRDI profile type: Publication
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cites work: Q5752194
full work available at URL: s0034-4877(97)85901-0
OpenAlex ID: W2098961041
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description: scientific article; zbMATH DE number 1158138