  We show that for a Schr\"odinger operator with bounded potential on a
manifold with cylindrical ends the space of solutions which grows at most
exponentially at infinity is finite dimensional and, for a dense set of
potentials (or, equivalently for a surface, for a fixed potential and a dense
set of metrics), the constant function zero is the only solution that vanishes
at infinity. Clearly, for general potentials there can be many solutions that
vanish at infinity.
  These results follow from a three circles inequality (or log convexity
inequality) for the Sobolev norm of a solution $u$ to a Schr\"odinger equation
on a product $N\times [0,T]$, where $N$ is a closed manifold with a certain
spectral gap. Examples of such $N$'s are all (round) spheres $\SS^n$ for $n\geq
1$ and all Zoll surfaces.
  Finally, we discuss some examples arising in geometry of such manifolds and
Schr\"odinger operators.
