Some quantitative unique continuation results for eigenfunctions of the magnetic Schr\"odinger operator

TL;DR

This paper establishes quantitative unique continuation estimates for eigenfunctions of magnetic Schrödinger operators with decaying complex potentials, and demonstrates the sharpness of these bounds through explicit constructions.

Contribution

It provides new decay rate estimates for solutions of magnetic Schrödinger equations with decaying potentials, including sharpness results via explicit counterexamples.

Findings

Derived explicit lower bounds for the decay of eigenfunctions.

Constructed examples showing the bounds are optimal.

Extended unique continuation results to complex-valued potentials.

Abstract

We prove quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim \langle x\rangle^{-N}$ and $|W(x)| \lesssim \langle x\rangle^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, we show that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{\beta_0}(\log R)^{A( R)})$, where $\beta_0 = \max\{2 - 2P, \frac{4-2N}{3}, 1\}$. Under certain conditions on $N$, $P$ and $\lambda$, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for $M(R)$ is sharp. That is, we construct functions $u, V$ and $W$ such that $-\Delta u + W\cdot \nabla u + Vu = \lambda u$, $|V(x)| \lesssim \langle x\rangle^{-N}$, $|W(x)| \lesssim \langle x\rangle^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{\beta_0}(\log |x|)^C)$.