Sharp decay rate for eigenfunctions of perturbed periodic Schr\"odinger operators

TL;DR

This paper establishes sharp decay rates for eigenfunctions of perturbed periodic Schrödinger operators on Z^d, providing conditions for the absence of embedded eigenvalues and extending prior results with precise impurity decay criteria.

Contribution

It introduces new quantitative conditions for impurity decay that guarantee eigenfunction decay and uniqueness, advancing understanding of spectral properties of perturbed periodic operators.

Findings

Eigenfunctions decay at sharp rates under certain impurity conditions.

Embedded eigenvalues are absent for impurities decaying faster than exponential.

Provides precise decay criteria extending previous spectral analysis results.

Abstract

This paper investigates uniqueness results for perturbed periodic Schr\"odinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -\Delta + V + v$, where $\Delta$ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-\Delta u + V u + v u = \lambda u$, for $\lambda \in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $\lambda$.