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

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.