A Liouville-type theorem for Schr\"odinger equations with nonnegative potential

TL;DR

This paper proves a Liouville-type theorem for solutions of Schrödinger equations with nonnegative, bounded potentials, showing that certain decay conditions imply the solution is identically zero, and extends results to exterior domains and nonlinear cases.

Contribution

It establishes a new Liouville-type result for Schrödinger equations with nonnegative potentials, including decay conditions and generalizations to nonlinear and exterior domain cases.

Findings

Decay conditions imply trivial solutions for Schrödinger equations.

Results confirm the Landis conjecture under algebraic decay.

Extensions to nonlinear equations and exterior domains are provided.

Abstract

Let $u$ be a solution of $\Delta u=Vu$ on $\mathbb{R}^d$, where $V$ be continuous, nonnegative and bounded. We prove that the condition $$\int_{r_j\leq|x|\leq r_j+1}|u(x)|^2dx\to 0,$$ along any sequence $(r_j)$, $r_j\nearrow+\infty$, implies $u\equiv 0$ on $\mathbb{R}^d$. In particular, this implies the Landis conjecture for solutions satisfying a sufficiently fast algebraic decay. These results are generalized to exterior domains as well as for a class of nonlinear Schr\"odinger equations under suitable conditions on the zero set of the potential.