On the Landis Conjecture in a Cylinder

The equation $$- \Delta u + V u = 0$$ 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 nontrivial solution $$u$$ is $$O\left(e^{-c|w|}\right)$$ for $$d=1$$ or $$2$$ , and $$O\left(e^{-c|w|^{4/3}}\right)$$ for $$d\ge 3$$ . Here $$w$$ stands for the axial variable. DOI 10.1134/S1061920824040058