Spectrum of the Dirichlet Laplacian in a thin cubic lattice

TL;DR

This paper analyzes the low-energy spectrum of the Dirichlet Laplacian in a 3D periodic lattice of thin bars, revealing extremely tight first spectral segments and detailed properties of junction regions.

Contribution

It provides a detailed description of the spectrum in thin cubic lattices and introduces a new analysis of junction regions, including eigenvalue uniqueness and absence of threshold resonance.

Findings

First spectral segment length is exponentially small, of order $O(e^{-rac{ ext{const}}{ ext{width}}})$.

Next spectral segments have length proportional to the bar width, $O( ext{width})$.

The analysis confirms the 1D graph model with Dirichlet conditions accurately describes the spectrum.

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 $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\varepsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-\delta/\varepsilon})$, $\delta>0$, while the length of the next spectral segments is $O(\varepsilon)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^{\Omega}$ in the geometry $\Omega$ 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 $A^{\Omega}$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^{\Omega}$, that is no non trivial bounded solution at the threshold frequency for $A^{\Omega}$. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.