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