On the rate of decay at infinity of solutions to the Schrödinger equation in a cylinder domains

We consider the equation
$$ -\Delta u + V u = 0 $$
in the cylinder ${\mathbb R} \times (0,2\pi)^d$ with periodic boundary conditions on the side surface. A potential $V$ is assumed to be real-valued and bounded. We are interested in the possible rate of decay of a non-trivial solution $u$ at infinity. Clearly, a solution can decrease exponentially. For $d=1$ or $d=2$, a solution can not decrease faster: if
$$ u (x,y) = O \left(e^{-N|x|}\right) \quad \forall \ N, $$
then $u \equiv 0$. Here $x$ is the axial variable. For $d \ge 3$, we construct an example of a non-trivial solution decreasing as $e^{-c |x|^{4/3}}$, and it is known that it is optimal,
$$ u (x,y) = O \left(e^{-N|x|^{4/3}}\right) \quad \forall \ N \qquad \Longrightarrow \qquad u \equiv 0. $$

This is a joint work with S. Krymskii.