On the rate of decrease at infinity for solutions to the Schrödinger equation in a half-cylinder (on a joint research with S.Krymskii)

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