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

We prove quantitative unique continuation results for solutions of $-\Delta u

W\cdot \nabla u + Vu = \lambda u $, w h e r e$ \lambda \in $C$ $a n d$ V $a n d$ W $a r e c o m p l e x - v a l u e d d e c a y i n g p o t e n t i a l s t h a t s a t i s f y$ |V(x)| \lesssim \langle x\rangle^{-N} $a n d$ |W(x)| \lesssim \langle x\rangle^{-P} $. F o r$ M® = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))} $, w e s h o w t h a t i f t h e s o l u t i o n$ u$ is non-zero, bounded, and $u (0) = 1$ , then $M® \gtrsim \exp(-C R^{\beta_0}(\log R)^{A( R)}) $, w h e r e$ \beta_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 $-\Delta u + W\cdot \nabla u + Vu = \lambda u $,$ |V(x)| \lesssim \langle x\rangle^{-N} $,$ |W(x)| \lesssim \langle x\rangle^{-P} $a n d$ |u(x)| \lesssim \exp(-c|x|^{\beta_0}(\log |x|)^C)$.

We prove quantitative unique continuation results for solutions of $-\Delta u

W\cdot \nabla u + Vu = \lambda u $, w h e r e$ \lambda \in $C$ $a n d$ V $a n d$ W $a r e c o m p l e x - v a l u e d d e c a y i n g p o t e n t i a l s t h a t s a t i s f y$ |V(x)| \lesssim \langle x\rangle^{-N} $a n d$ |W(x)| \lesssim \langle x\rangle^{-P} $. F o r$ M® = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))} $, w e s h o w t h a t i f t h e s o l u t i o n$ u$ is non-zero, bounded, and $u (0) = 1$ , then $M® \gtrsim \exp(-C R^{\beta_0}(\log R)^{A( R)}) $, w h e r e$ \beta_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 $-\Delta u + W\cdot \nabla u + Vu = \lambda u $,$ |V(x)| \lesssim \langle x\rangle^{-N} $,$ |W(x)| \lesssim \langle x\rangle^{-P} $a n d$ |u(x)| \lesssim \exp(-c|x|^{\beta_0}(\log |x|)^C)$.

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