Some quantitative unique continuation results for eigenfunctions of the magnetic Schrödinger operator

We prove quantitative unique continuation results for solutions of $-Δu + W\cdot \nabla u + Vu = λu$, where $λ\in \mathbb{C}$ 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 $M(R) \gtrsim \exp(-C R^{β_0}(\log R)^{A( R)})$, where $β_0 = \max\{2 - 2P, \frac{4-2N}{3}, 1\}$. Under certain conditions on $N$, $P$ and $λ$, 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 $-Δu + W\cdot \nabla u + Vu = λu$, $|V(x)| \lesssim \langle x\rangle^{-N}$, $|W(x)| \lesssim \langle x\rangle^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{β_0}(\log |x|)^C)$.

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