Spectrum of the Dirichlet Laplacian in a thin cubic lattice

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 $\eps\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\eps$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-\delta/\eps})$, $\delta>0$, while the length of the next spectral segments is $O(\eps)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{\Om}$ in the geometry $\Om$ 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 Friedrichs inequality, we prove that $A^{\Om}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{\Om}$, that is no non trivial bounded solution at the threshold frequency for $A^{\Om}$. 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.