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

  • S. A. NazarovEmail author
Article

Abstract

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.

Keywords

lattice of quantum waveguides Dirichlet spectral problem quantum graph Kirchhoff transmission conditions Dirichlet condition cross-shaped waveguide bounded solutions at threshold 

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Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.St. Petersburg State UniversityStaryi Peterhof, St. PetersburgRussia

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