Sharp decay rate for eigenfunctions of perturbed periodic Schrdinger operators

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, $V: mathbb{Z}^d
ightarrow mathbb{R}$ is a periodic potential, and $v: mathbb{Z}^d
ightarrow mathbb{C}$ 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 $lambda$.

Sharp decay rate for eigenfunctions of perturbed periodic Schrdinger operators Institutional Repository Document