Transmission Conditions in One-Dimensional Model of a Rectangular Lattice of Thin Quantum Waveguides
- Authors: Nazarov S.A.1,2,3
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Affiliations:
- St.Petersburg State University
- Peter the Great St. Petersburg State Polytechnical University
- Institute of Problems of Mechanical Engineering RAS
- Issue: Vol 219, No 6 (2016)
- Pages: 994-1015
- Section: Article
- URL: https://journals.rcsi.science/1072-3374/article/view/238754
- DOI: https://doi.org/10.1007/s10958-016-3160-z
- ID: 238754
Cite item
Abstract
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 BboR 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 BstR 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 BstR, and classification of “trapped”/“almost standing” waves.
About the authors
S. A. Nazarov
St.Petersburg State University; Peter the Great St. Petersburg State Polytechnical University; Institute of Problems of Mechanical Engineering RAS
Author for correspondence.
Email: s.nazarov@spbu.ru
Russian Federation, 7-9, Universitetskaya nab., St. Petersburg, 199034; 29, Polytechnicheskaya ul., St. Petersburg, 195251; 61, V.O., Bolshoj pr., St. Petersburg, 199178