  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.
