On the Three Balls Inequality for Discrete Schr{ö}dinger Operators on Certain Periodic Graphs

TL;DR

This work develops a discrete Carleman framework for magnetic Schrödinger operators on periodic graphs, introducing a graph-adapted weight $|\cdot|_{\Gamma}$ and showing a Carleman inequality with a parameter $τ$ bounded by $\mathcal{O}(h^{-1})$. From this, the authors derive a quantitative Three Balls inequality for 1-point periodic graphs and extend it to a large class of graphs, including the Hexagonal lattice, by graph-reduction techniques that preserve $\ell^2$-norm equivalence. The Hexagonal lattice is treated by embedding two 1-point problems, while Star periodic graphs are handled via similar reductions to the 1-point setting. Overall, the paper advances discrete unique continuation on periodic graphs and broadens the class of graphs for which Three Balls-type propagation of smallness can be established, with potential implications for spectral properties of crystalline structures.

Abstract

We investigate quantitative unique continuation properties for discrete magnetic Schr{ö}dinger operators in certain periodic graphs. This unique continuation property will be quantified through what is known in the literature as a Three Balls Inequality. We are able to extend this inequality to another family of periodic graph which contains the Hexagonal lattice. We also give a sketch of the proof for general star periodic graph.Our proofs are based on Carleman estimates.

Figures (7)

• Figure 1: Three 2-dimensional one point periodic graphs. Dots are the vertices and lines joining them the edges. The thick (blue) lines are the edges that are reproduced periodically
• Figure 2: The octogonal lattice (4 points) and the Kagome lattice (3 points) are periodic graphs.
• Figure 3: Sketch of $\mathcal{N}_{\tau,\mathcal{C}}$ (hatched area) in dimension 2 where $\tau=1$.
• Figure 4: The hexagonal lattice (2 points) and the triangular lattice (1 point) associated
• Figure 5: The standard Hexagonal lattice, $g_1$ would be the restriction of $g$ to the blue (circle) points) and $g_2$ the restriction to the red (triangle) points

Theorems & Definitions (28)

• Definition 2.1
• Lemma 2.2
• proof
• Definition 2.3
• Theorem 3.1: Carleman Estimate
• Lemma 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof