Asymptotics of the Eigenvalues and Eigenfunctions of a Thin Square Dirichlet Lattice with a Curved Ligament

Abstract

The spectrum of the Dirichlet problem on the planar square lattice of thin quantum waveguides has a band-gap structure with short spectral bands separated by wide spectral gaps. The curving of at least one of the ligaments of the lattice generates points of the discrete spectrum inside gaps. A complete asymptotic series for the eigenvalues and eigenfunctions are constructed and justified; those for the eigenfunctions exhibit a remarkable behavior imitating the rapid decay of the trapped modes: the terms of the series have compact supports that expand unboundedly as the number of the term increases.