Rectangular lattices of cylindrical quantum waveguides. I. Spectral problems on a finite cross

Abstract

The spectrum of truncated cross-shaped waveguides is studied under the Dirichlet conditions on the lateral surface and various boundary conditions on the ends of the column and the cross bar. The monotonicity and asymptotics of the eigenvalues are discussed in dependence on the size of a cross whose section may be fairly arbitrary. In the case of a round section, the estimates found for the second eigenvalue agree with the asymptotic formulas obtained. Such information is needed for the spectral analysis of thin periodic lattices of quantum waveguides.2010 Mathematics Subject Classification. Primary 81Q37. Key words and phrases. Infinite and truncated cross-shaped quantum waveguides, discrete spectrum, stable and decaying solution at the threshold of the continuous spectrum, asymptotics.Supported by the Russian Science Foundation (grant no. 14-21-00035).