We prove quantitative unique continuation results for solutions of $\Delta w - k^2 w = V w + W\cdot \nabla w$ in a neighborhood of infinity, where $k > 0$, and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim |x|^{ -N}$ and $|W(x)| \lesssim |x|^{ -P}$ for some $N, P > 1$. For $M(R, 4n/k) = \inf \{ ||w||_{ L^2(B_{ 4n/k}(x_0))} : |x_0| = R \}$, we show that if the solution $w$ is non-zero, bounded, and normalized, then $M(R, 4n/k) \gtrsim \exp(-kR - G \log R)$, where $G > \frac{ n-1}{ 2}$ is a constant. An examination of radial solutions to $\Delta w - k^2 w = V w + W\cdot \nabla w$ shows that this new estimate for $M(R, 4n/k)$ is sharp up to logarithmic terms.
Categories: math.AP