A further quantification of the unique continuation properties of eigenfunctions of the magnetic Schr\"odinger operator

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.