Inserted: 22 jan 2007
Last Updated: 3 may 2011
Journal: Comm. Pure App. Math.
We show that for a Schrödinger 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.
One of the key ingredients in these results is a three circles inequality (or log convexity inequality) for the Sobolev norm of a solution $u$ to a Schrödinger 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 Schrödinger operators.
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