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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) forsolutions satisfying a suitable exponential decay. The Landis conjecture wasdisproved by Meshkov in the case of complex-valued functions, but it remainedopen in the real case. In the 2000s, several papers have addressed the issue ofthe UCI for linear elliptic operators with real coefficients. The results thathave been obtained require some kind of sign condition, either on the solutionor on the zero order coefficient of the equation. The Landis conjecture isstill open nowadays in its general form. In the present paper, we start withconsidering a general (real) elliptic operator in dimension $1$. We derive theUCI property with a rate of decay $\kappa$ which is sharp when the coefficientsof the operator are constant. In particular, we prove the Landis conjecture indimension $1$, 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 differentapproach, we prove the same result for positive supersolutions of generalelliptic equations.

The Landis conjecture with sharp rate of decay