Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Some quantitative unique continuation results for eigenfunctions of the
  magnetic Schr\"odinger operator   We prove quantitative unique continuation results for solutions of $-\Delta u
+ W\cdot \nabla u + Vu = \lambda u$, where $\lambda \in \mathbb{C}$ 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 $M(R) \gtrsim \exp(-C R^{\beta_0}(\log
R)^{A( R)})$, where $\beta_0 = \max\{2 - 2P, \frac{4-2N}{3}, 1\}$. Under
certain conditions on $N$, $P$ and $\lambda$, 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 $-\Delta u +
W\cdot \nabla u + Vu = \lambda u$, $|V(x)| \lesssim \langle x\rangle^{-N}$,
$|W(x)| \lesssim \langle x\rangle^{-P}$ and $|u(x)| \lesssim
\exp(-c|x|^{\beta_0}(\log |x|)^C)$.

13
The categories of this paper are: quant-ph
29
The categories of this paper are: funct-an
23
The categories of this paper are: solv-int
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
36
The categories of this paper are: cmp-lg
15
The categories of this paper are: nucl-ex
3
The categories of this paper are: physics
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
33
The categories of this paper are: plasm-ph
11
The categories of this paper are: hep-lat
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
7
The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex