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

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 B_bo^R 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_st^R 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_st^R, and classification of “trapped”/“almost standing” waves.