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Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides

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We consider the transmission conditions at vertices of the graph modeling a periodic rectangular lattice of thin quantum waveguides described by the spectral Dirichlet problem for the Laplace operator. The type of transmission conditions is determined by the structure of the space B R bo of bounded solutions to the boundary layer problem in a cross-shaped waveguide with a circular core of radius R. We describe all variants of the structure of the space B Rst of nondecaying solutions and present methods for constructing hardly probable and very probable examples. Based on the method of matched asymptotic expansion, we construct all possible transmission conditions. We discuss numerical methods for computing critical radii, construction of the space B Rst , and classification of “trapped”/“almost standing” waves.

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Authors and Affiliations

  1. St.Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russia

    S. A. Nazarov

  2. Peter the Great St. Petersburg State Polytechnical University, 29, Polytechnicheskaya ul., St. Petersburg, 195251, Russia

    S. A. Nazarov

  3. Institute of Problems of Mechanical Engineering RAS, 61, V.O., Bolshoj pr., St. Petersburg, 199178, Russia

    S. A. Nazarov

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Correspondence to S. A. Nazarov.

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Translated from Problemy Matematicheskogo Analiza 87, October 2016, pp. 153-172.

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Nazarov, S.A. Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides. J Math Sci 219, 994–1015 (2016). https://doi.org/10.1007/s10958-016-3160-z

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