Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Landis conjecture with sharp rate of decay   The so called Landis conjecture states that if a solution of the equation
$$\Delta u+V(x)u=0$$ in an exterior domain decays faster than $e^{-\kappa|x|}$,
for some $\kappa>\sqrt{\sup |V|}$, then it must be identically equal to $0$.
This property can be viewed as a unique continuation at infinity (UCI) for
solutions satisfying a suitable exponential decay. The Landis conjecture was
disproved by Meshkov in the case of complex-valued functions, but it remained
open in the real case. In the 2000s, several papers have addressed the issue of
the UCI for linear elliptic operators with real coefficients. The results that
have been obtained require some kind of sign condition, either on the solution
or on the zero order coefficient of the equation. The Landis conjecture is
still open nowadays in its general form. In the present paper, we start with
considering a general (real) elliptic operator in dimension $1$. We derive the
UCI property with a rate of decay $\kappa$ which is sharp when the coefficients
of the operator are constant. In particular, we prove the Landis conjecture in
dimension $1$, and we can actually reach the threshold value $\kappa=\sqrt{\sup
|V|}$. Next, we derive the UCI property -- and then the Landis conjecture --
for radial operators in arbitrary dimension. Finally, with a different
approach, we prove the same result for positive supersolutions of general
elliptic equations.

5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
14
The categories of this paper are: math-ph
16
The categories of this paper are: stat
25
The categories of this paper are: patt-sol
7
The categories of this paper are: hep-th
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
24
The categories of this paper are: dg-ga
22
The categories of this paper are: chao-dyn
19
The categories of this paper are: eess
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
6
The categories of this paper are: astro-ph