Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder

The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness h ≪ 1) in the shape of an infinite two-dimensional ladder. Passage to the limit as h → +0 is discussed. It is shown that the asymptotically correct transmission conditions at nodes of the corresponding one-d...

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Bibliographic Details
Published in:Computational mathematics and mathematical physics 2014-08, Vol.54 (8), p.1261-1279
Main Author: Nazarov, S. A.
Format: Article
Language:English
Subjects:
Asymptotic properties
Computational mathematics
Computational Mathematics and Numerical Analysis
Dirichlet problem
Eigenvalues
Graphs
Ladders
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Physics
Quantum physics
Spectra
Studies
Two dimensional
Waveguides
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Summary:The Dirichlet problem is considered on the junction of thin quantum waveguides (of thickness h ≪ 1) in the shape of an infinite two-dimensional ladder. Passage to the limit as h → +0 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 T-shaped waveguide that the boundary layer phenomenon.
ISSN:0965-5425
1555-6662
DOI:10.1134/S0965542514080090