Breakdown of cycles and the possibility of opening spectral gaps in a square lattice of thin acoustic waveguides

Abstract

We study the spectrum of a planar square lattice of multidimensional acoustic waveguides (the Neumann problem for the Laplace operator), constructing and justifying asymptotic formulae for solutions of the spectral problem on a periodicity cell. A detailed study of corrections to expansions of eigenvalues and eigenfunctions enables us to construct a model of improved accuracy which is free from the drawbacks of the classical model on a one- dimensional graph (the skeleton of the lattice) with Kirchhoff's classical conjugation conditions at the vertices. In particular, we demonstrate the breakdown of cycles (localized eigenfunctions occurring in the classical model but almost always absent from the improved one) in the multidimensional problem. We discuss the opening of gaps and pseudogaps in the spectrum of the problem on an infinite multidimensional lattice.