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Zh. Vychisl. Mat. Mat. Fiz., 2014, Volume 54, Number 8, Pages 1299–1318 (Mi zvmmf10077)  

This article is cited in 18 scientific papers (total in 18 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

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.

DOI: https://doi.org/10.7868/S0044466914080110

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English version:
Computational Mathematics and Mathematical Physics, 2014, 54:8, 1261–1279

Bibliographic databases:

UDC: 519.634
MSC: 78A50, 81V80
Received: 12.02.2014

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
\by S.~A.~Nazarov
\paper Bounded solutions in a $\mathrm{T}$-shaped waveguide and the spectral properties of~the Dirichlet ladder
\jour Zh. Vychisl. Mat. Mat. Fiz.
\yr 2014
\vol 54
\issue 8
\pages 1299--1318
\jour Comput. Math. Math. Phys.
\yr 2014
\vol 54
\issue 8
\pages 1261--1279

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Spectra of three-dimensional cruciform and lattice quantum waveguides”, Dokl. Math., 92:1 (2015), 514–518  crossref  mathscinet  zmath  isi  elib  scopus
    2. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Discrete spectrum of x-shaped waveguide”, St. Petersburg Math. J., 28:2 (2017), 171–180  mathnet  crossref  mathscinet  isi  elib
    3. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “Multifarious transmission conditions in the graph models of carbon nano-structures”, Mater. Phys. Mech., 29:2 (2016), 107–115  isi
    4. S. A. Nazarov, K. Ruotsalainen, P. Uusitalo, “Localized waves in carbon nano-structures with connected and disconnected open waveguides”, Mater. Phys. Mech., 29:2 (2016), 116–124  isi
    5. S. A. Nazarov, “The spectra of rectangular lattices of quantum waveguides”, Izv. Math., 81:1 (2017), 29–90  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    6. S. A. Nazarov, “Open waveguides in a thin Dirichlet ladder: I. Asymptotic structure of the spectrum”, Comput. Math. Math. Phys., 57:1 (2017), 156–174  mathnet  crossref  crossref  isi  elib
    7. S. A. Nazarov, “Open waveguides in a thin Dirichlet lattice: II. Localized waves and radiation conditions”, Comput. Math. Math. Phys., 57:2 (2017), 236–252  mathnet  crossref  crossref  isi  elib
    8. K. Pankrashkin, “Eigenvalue inequalities and absence of threshold resonances for waveguide junctions”, J. Math. Anal. Appl., 449:1 (2017), 907–925  crossref  mathscinet  zmath  isi  scopus
    9. F. L. Bakharev, S. G. Matveenko, S. A. Nazarov, “Rectangular lattices of cylindrical quantum waveguides. I. Spectral problems in a finite cross”, St. Petersburg Math. J., 29:3 (2018), 423–437  mathnet  crossref  mathscinet  isi  elib
    10. B. Delourme, S. Fliss, P. Joly, E. Vasilevskaya, “Trapped modes in thin and infinite ladder like domains. Part 1: Existence results”, Asymptotic Anal., 103:3 (2017), 103–134  crossref  mathscinet  zmath  isi  scopus
    11. S. A. Nazarov, “Enhancement and smoothing of near-threshold wood anomalies in an acoustic waveguide”, Acoust. Phys., 64:5 (2018), 535–547  crossref  isi  scopus
    12. S. A. Nazarov, “Breakdown of cycles and the possibility of opening spectral gaps in a square lattice of thin acoustic waveguides”, Izv. Math., 82:6 (2018), 1148–1195  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. S. A. Nazarov, “Asymptotics of eigenvalues in spectral gaps of periodic waveguides with small singular perturbations”, J. Math. Sci. (N. Y.), 243:5 (2019), 746–773  mathnet  crossref
    14. F. L. Bakharev, S. A. Nazarov, “Eigenvalue asymptotics of long Kirchhoff plates with clamped edges”, Sb. Math., 210:4 (2019), 473–494  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    15. S. A. Nazarov, “Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament”, Math. Notes, 105:4 (2019), 559–579  mathnet  crossref  crossref  mathscinet  isi  elib
    16. S. A. Nazarov, “Waveguide with double threshold resonance at a simple threshold”, Sb. Math., 211:8 (2020), 1080–1126  mathnet  crossref  crossref  mathscinet  isi  elib
    17. F. L. Bakharev, S. A. Nazarov, “Kriterii otsutstviya i nalichiya ogranichennykh reshenii na poroge nepreryvnogo spektra v ob'edinenii kvantovykh volnovodov”, Algebra i analiz, 32:6 (2020), 1–23  mathnet
    18. S. A. Nazarov, “The preservation of threshold resonances and the splitting off of eigenvalues from the threshold of the continuous spectrum of quantum waveguides”, Sb. Math., 212:7 (2021), 965–1000  mathnet  crossref  crossref  isi
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