A Liouville-type theorem for Schrödinger equations with nonnegative potential

TL;DR

This work establishes a Liouville-type theorem for solutions of $\Delta u = V u$ with $V\ge0$ bounded, showing that if the $L^2$ mass on unit-width annuli tends to zero along some sequence $r_j\to\infty$, then $u$ must be identically zero, thereby validating a Landis-type phenomenon for continuous nonnegative potentials in any dimension. The proof combines a localized energy estimate, a Caccioppoli inequality, and a careful cutoff/covering argument to conclude $\int_B V u^2 = 0$ on a ball, leading to global vanishing by unique continuation; a corollary yields a universal lower bound on annular mass for nontrivial solutions. The results extend to exterior domains and to a nonlinear Schrödinger setting when the potential $V(u)$ has an isolated zero set, with a quantitative corollary and a nonlinear analogue under suitable decay criteria. Overall, the paper shows that sufficiently fast algebraic decay on annuli suffices for triviality, without requiring global decay, and broadens the Landis-type framework to nonnegative potentials and certain nonlinear models.

Abstract

Let $u$ be a solution of $Δu=Vu$ 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 Schrödinger equations under suitable conditions on the zero set of the potential.

Theorems & Definitions (8)

• Theorem 2.1
• Remark
• Corollary 2.2
• proof
• Theorem 2.3
• Proposition 3.1
• proof
• Lemma 3.2