Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials
Abstract
In this article, we study unique continuation properties at infinity for solutions to generalized Schrödinger equations with potential functions that belong to the reverse Hölder class. For equations of the form −div (A ∇ u) + V u = 0 in ℝn, where A is bounded and elliptic and V ∈ RHp for some p ∈ [n/2, ∞], we prove that if a solution doesn't grow too quickly at infinity, then it must be trivial. We use dV, the Agmon distance function associated to V, to quantify the threshold growth rate. More precisely, there exists a constant γ0 > 0 so that if |u(x)| ≲ exp(γdV(x, 0)) for some γ< γ0 and every x ∈ ℝn, then u must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis' conjecture. Our proof techniques are inspired by Z. Shen's exponential decay estimates for fundamental solutions of Schrödinger operators and involve the application of a Fefferman-Phong inequality.
Related
- Unique continuation at infinity for potentials with arbitrary radial growth
- Weighted and Sobolev Estimates for Fractional Schrödinger Operators with Reverse Hölder Potentials
- Spectral-Edge Continuation, Dimensional Crossover, and Endpoint Stability for Partially Confined Nonlinear Schrödinger Equations
- Normalized solutions to mass supercritical Schrödinger equations with radial potentials
- Doubling inequalities and propagation of smallness for Schrödinger equations with singular potentials
BibTeX
@misc{davey2026unique,
title = {{Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials}},
author = {Blair Davey},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21048},
eprint = {2609.21048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21048},
}
RIS
TY - UNPB
TI - Unique continuation at infinity for Schrödinger equations with Reverse Hölder Potentials
AU - Blair Davey
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.21048
UR - https://arxiv.org/abs/2609.21048
AB - In this article, we study unique continuation properties at infinity for solutions to generalized Schrödinger equations with potential functions that belong to the reverse Hölder class. For equations of the form −div (A ∇ u) + V u = 0 in ℝ^n, where A is bounded and elliptic and V ∈ RH_p for some p ∈ [n/2, ∞], we prove that if a solution doesn't grow too quickly at infinity, then it must be trivial. We use d_V, the Agmon distance function associated to V, to quantify the threshold growth rate. More precisely, there exists a constant γ_0 > 0 so that if |u(x)| ≲ exp(γd_V(x, 0)) for some γ< γ_0 and every x ∈ ℝ^n, then u must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis' conjecture. Our proof techniques are inspired by Z. Shen's exponential decay estimates for fundamental solutions of Schrödinger operators and involve the application of a Fefferman-Phong inequality.
ER -