Abstract
We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width ε ≪ 1) which have a square cross section. This spectrum coincides with the union of segments which all go to +∞ as ε tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length O(e−δ/ε), δ > 0, while the length of the next spectral segments is O(ε). To establish these results, we need to study in detail the properties of the Dirichlet Laplacian AΩ in the geometry Ω obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max–min arguments as well as a well-chosen Poincaré–Friedrichs inequality, we prove that AΩ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for AΩ, that is no non trivial bounded solution at the threshold frequency for AΩ. This implies that the correct 1D model of the lattice for the next spectral segments is a system of ordinary differential equations set on the limit graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.