Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder
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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.
Keywordslattice of quantum waveguides Dirichlet spectral problem quantum graph Kirchhoff transmission conditions Dirichlet condition cross-shaped waveguide bounded solutions at threshold
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- 1.M. Sh. Birman and G. E. Skvortsov, “On square summability of highest derivatives of the solution of the Dirichlet problem in a domain with piecewise smooth boundary,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 5, 11–21 (1962).Google Scholar
- 7.Quantum graphs and their applications, Ed. by P. Kuchment, special issue of Waves Random Media 14(1) (2004).Google Scholar
- 10.S. A. Nazarov, “Asymptotic analysis and modeling of the junction of a massive body with thin rods,” J. Math. Sci. 127, 2172–2263 (2003).Google Scholar
- 14.N. A. Umov, Equations of Energy Transfer in Bodies (Tipogr. Ul’rikha i Shul’tse, Odessa, 1874) [in Russian].Google Scholar
- 16.L. I. Mandelstam, Lectures on Optics, Relativity Theory, and Quantum Mechanics (Akad. Nauk SSSR, Moscow, 1947), Vol. 2 [in Russian].Google Scholar
- 22.M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in Hilbert Space (Leningr. Gos. Univ., Leningrad, 1980; Reidel, New York, 1986).Google Scholar
- 29.S. A. Nazarov, Asymptotic Theory of Plates and Rods: Dimension Reduction and Integral Estimates (Nauchnaya Kniga, Novosibirsk, 2002) [in Russian].Google Scholar
- 33.J.-L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications (Mir, Moscow, 1971; Springer-Verlag, Berlin 1972).Google Scholar