On (Global) Unique Continuation Properties of the Fractional Discrete Laplacian

TL;DR

This paper investigates the unique continuation properties of the fractional discrete Laplacian, revealing that discretization weakens these properties but can be compensated with correction terms, enabling stability in inverse problems.

Contribution

It demonstrates that the fractional discrete Laplacian lacks strong unique continuation properties but can regain them through exponential correction terms, leading to stability results.

Findings

Discretization weakens unique continuation properties.

Exponential correction terms restore these properties.

Uniform stability for inverse problems is achieved.

Abstract

We study various qualitative and quantitative (global) unique continuation properties for the fractional discrete Laplacian. We show that while the fractional Laplacian enjoys striking rigidity properties in the form of (global) unique continuation properties, the fractional discrete Laplacian does not enjoy these in general. While discretization thus counteracts the strong rigidity properties of the continuum fractional Laplacian, by discussing quantitative forms of unique continuation, we illustrate that these properties can be recovered if exponentially small (in the lattice size) correction terms are added. This in particular allows us to deduce uniform stability properties for a discrete, linear inverse problem for the fractional Laplacian. We complement these observations with a transference principle and the discussion of these properties on the discrete torus.