Mathematics – Differential Geometry
Scientific paper
2007-01-10
Mathematics
Differential Geometry
Scientific paper
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.
Colding Tobias H.
Lellis Camillo de
Minicozzi II William P.
No associations
LandOfFree
Three circles theorems for Schrodinger operators on cylindrical ends and geometric applications does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Three circles theorems for Schrodinger operators on cylindrical ends and geometric applications, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Three circles theorems for Schrodinger operators on cylindrical ends and geometric applications will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-448191