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Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki, 2014, Volume 54, Number 8, Pages 1299–1318
DOI: https://doi.org/10.7868/S0044466914080110 (Mi zvmmf10077) 		 
This article is cited in 28 scientific papers (total in 28 papers)

Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder

S. A. Nazarov

St. Petersburg State University, Department of Mathematics and Mechanics
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DOI: https://doi.org/10.7868/S0044466914080110
Abstract: The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness $h\ll1$) in the shape of an infinite two-dimensional ladder. Passage to the limit as $h\to+\infty$ is discussed. It is shown that the asymptotically correct transmission conditions at nodes of the corresponding one-dimensional quantum graph are Dirichlet conditions rather than the conventional Kirchhoff transmission conditions. The result is obtained by analyzing bounded solutions of a problem in the $\mathrm{T}$-shaped waveguide that the boundary layer phenomenon.
Key words: lattice of quantum waveguides, Dirichlet spectral problem, quantum graph, Kirchhoff transmission conditions, Dirichlet condition, cross-shaped waveguide, bounded solutions at threshold.
Received: 12.02.2014
English version:
Computational Mathematics and Mathematical Physics, 2014, Volume 54, Issue 8, Pages 1261–1279
DOI: https://doi.org/10.1134/S0965542514080090
Bibliographic databases:
Document Type: Article
UDC: 519.634
MSC: 78A50, 81V80
Language: Russian
Citation: S. A. Nazarov, “Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of the Dirichlet ladder”, Zh. Vychisl. Mat. Mat. Fiz., 54:8 (2014), 1299–1318; Comput. Math. Math. Phys., 54:8 (2014), 1261–1279
Citation in format AMSBIB
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    • This publication is cited in the following 28 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    		Журнал вычислительной математики и математической физики Computational Mathematics and Mathematical Physics
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