On the Landis conjecture in a cylinder

The equation $-\mathrm{\Delta }u+Vu=0$ in the cylinder $\mathbb{R}\times (0,2\pi {)}^d$ with periodic boundary conditions is considered. The potential $V$ is assumed to be bounded, and both functions $u$ and $V$ are assumed to be real-valued. It is shown that the fastest rate of decay at infinity of non-trivial solution $u$ is $O\left(e^{-c\mathrm{\mid }w\mathrm{\mid }}\right)$ for $d=1$ or $2$, and $O\left(e^{-c\mathrm{\mid }w{\mathrm{\mid }}^{4\mathrm{/}3}}\right)$ for $d\ge 3$. Here $w$ is the axial variable.