Abstract:
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,
$$
u (x,y) = O \left(e^{-Nx^{4/3}}\right) \quad \forall \ N \qquad \Longrightarrow \qquad u \equiv 0.
$$