## T. H. Colding - C. De Lellis - W. P. I. I. Minicozzi

# Three circles theorems for Schrödinger operators on cylindrical ends and geometric applications

created by delellis on 22 Jan 2007

modified on 03 May 2011

[BibTeX]

*Published Paper*

**Inserted:** 22 jan 2007

**Last Updated:** 3 may 2011

**Journal:** Comm. Pure App. Math.

**Volume:** 61

**Pages:** 1540-1602

**Year:** 2008

**Abstract:**

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.

For the most updated version and eventual errata see the page

http:/www.math.uzh.ch*index.php?id=publikationen&key1=493
*