Spectra of three-dimensional cruciform and lattice quantum waveguides
type: Item
instance of: scholarly article
title: Spectra of three-dimensional cruciform and lattice quantum waveguides
zbMATH Open document ID: 1334.82086
DOI: 10.1134/S1064562415040274
published in: Doklady Mathematics
publication date: 2015-11-12T00:00:00Z
review text: Complete conclusions for three-dimensional junctions of quantum waveguides are formulated. The main and technically most complicated results are that the discrete spectrum of the Dirichlet problem on the union of the two mutually perpendicular circular cylinders contains a single eigenvalue, while the homogeneous problem with a threshold value of the spectral parameter has nontrivial bounded solutions. As a result, the one-dimensional model of a square (or rectangular) lattice of thin circular quantum waveguides involves Dirichlet conditions rather than Kirchhoff transmission ones at the graph nodes. It is noted that, as in the case of the two-dimensional formulations, a detailed study of the boundary layer reveals various phenomena not typical of the Pauling model; specifically, in the low-frequency range of the spectrum, oscillatory processes localized near lattice nodes, leaving bridges weakly moving. In the mid-frequency range, on the contrary, they shift onto the bridges and the nodes are affected little in view of Dirichlet limit conditions.
reviewed by: Yulianna Perepelkina
Mathematics Subject Classification ID: 82D77
Mathematics Subject Classification ID: 82B20
zbMATH DE Number: 6507793
zbMATH Keywords: quantum wave guides
zbMATH Keywords: quantum wires
zbMATH Keywords: lattice systems
author: F. L. Bakharev
author: Sergey Matveenko
author: Sergei A. Nazarov
MaRDI profile type: Publication
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full work available at URL: s1064562415040274
OpenAlex ID: W2306802112
description: scientific article; zbMATH DE number 6507793