Year: 2025
Type: article
Abstract: This paper investigates uniqueness results for perturbed periodic Schrödinger operators on Z d . Specifically, we consider operators of the form H = − Δ + V + v , where Δ is the discrete Laplacian, V : Z d → R is a periodic potential, and v : Z d → C represents a decaying impurity. We establish quantitative conditions under which the equation − Δ u + V u + v u = λ u , for λ ∈ C , admits only the trivial solution u ≡ 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 λ.
Source: Nonlinearity
Authors: Wencai Liu, Rodrigo Matos, John N Treuer
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G4968886043, Hamiltonian Systems and Related Phenomena, G5336022802, (Semi)algebraic Geometry in Schrödinger Operators and Nonlinear Hamiltonian Partial Differential Equations, FRG: Collaborative Research: Non-Perturbative Analysis for Multi-Dimensional Quasiperiodic Systems, Research and Education in Several Complex Variables
OpenAlex ID: W4409280806