Spectra of three-dimensional cruciform and lattice quantum waveguides
Complete conclusions for three-dimensional junctions of quantum waveguides are formulated. The main and technically most complicated results are that the discrete spectrum of the Dirichlet problem on the union of the two mutually perpendicular circular cylinders contains a single eigenvalue, while the homogeneous problem with a threshold value of the spectral parameter has nontrivial bounded solutions. As a result, the one-dimensional model of a square (or rectangular) lattice of thin circular quantum waveguides involves Dirichlet conditions rather than Kirchhoff transmission ones at the graph nodes. It is noted that, as in the case of the two-dimensional formulations, a detailed study of the boundary layer reveals various phenomena not typical of the Pauling model; specifically, in the low-frequency range of the spectrum, oscillatory processes localized near lattice nodes, leaving bridges weakly moving. In the mid-frequency range, on the contrary, they shift onto the bridges and the nodes are affected little in view of Dirichlet limit conditions.
scientific article; zbMATH DE number 6507793
scientific article