Sharp constructions of eigenfunctions of the magnetic Schr\"odinger operator

Abstract

We prove sharpness of quantitative unique continuation results for solutions of $-\Delta u + W\cdot \nabla u + V u = \la u$, where $\la \in \C$ and $V$ and $W$ are complex-valued decaying potentials that satisfy $|V(x)| \lesssim ^{-N}$ and $|W(x)| \lesssim ^{-P}$. For $M(R) = \inf_{|x_0| = R}||u||_{L^2(B_1(x_0))}$, it was shown in a companion paper that if the solution $u$ is non-zero, bounded, and $u(0) = 1$, then $M(R) \gtrsim \exp(-C R^{\be_0}(\log R)^{A(R)})$, where $\be_0 = max{2 - 2P, (4-2N)/3, 1}$. Under certain conditions on $N$, $P$, $\la$, and the dimension, 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 $-\Delta u + W\cdot \nabla u + V u = \la u$, $|V(x)| \lesssim ^{-N},$ $|W(x)| \lesssim ^{-P}$ and $|u(x)| \lesssim \exp(-c|x|^{\be_0}(\log |x|)^C)$.