Structure of the spectrum of a periodic family of identical cells connected by converging apertures
type: Item
instance of: scholarly article
title: Structure of the spectrum of a periodic family of identical cells connected by converging apertures
zbMATH Open document ID: 1304.35458
author: Sergei A. Nazarov
author: J. Taskinen
published in: Journal of Mathematical Sciences (New York)
publication date: 2014-09-25T00:00:00Z
review text: For \(d\geq 2\), the authors define \(\Pi\), the \(d\)-dimensional waveguide comprised by the union of identical cells (its elements belong to \(\Omega\), a bounded subset of an explicit strip in \(\mathbb R^{d}\)) and apertures (their sizes decay). The authors show that the spectra of \(DP_\Pi\) and \(DP_\Omega\), the Dirichlet problem on \(\Pi\) and on \(\Omega\), respectively, are closely related. Precisely, they show that the essential spectrum of \(DP_\Pi\) is the family of the eigenvalues of \(DP_\Omega\) and \(DP_\Pi\) has an increasing sequence of eigenvalues converging to the first eigenvalue of \(DP_\Omega\) (Theorem, section 2). Among the apparatus used on the existence of the essential spectrum, they apply the Weyl criteria, and in the rest of the proof, they use the Hardy inequality -- the logarithm Hardy inequality for the planar waveguide -- and the Friedrichs inequality on the semi-hypersphere of the hypersphere \(\mathbb S^{d-1}\).
reviewed by: Mohammed El Aïdi
Mathematics Subject Classification ID: 35P20
zbMATH DE Number: 6347702
zbMATH Keywords: \(d\)-dimensional waveguide
zbMATH Keywords: apertures
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: eigenvalues
MaRDI profile type: Publication
full work available at URL: s10958-013-1508-1
OpenAlex ID: W1982530652
cites work: Q3737900
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cites work: Q3005477
DOI: 10.1007/S10958-013-1508-1
description: scientific article; zbMATH DE number 6347702